THE W.K.B. APPROXIMATION FOR THE
QUANTUM RADIAL DISTRIBUTION
FUNCTION
By
JONG KOOK JAEN
A DISSERTATION PRESENTED TO 'IIIE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
April, 1966
Copyright by
Jong Kook Jaen
1966
DEDICATION
0)
641
(Dedicated to the Memory of my
Beloved Mother and Father)
0
ACKNOWLEDGMENTS
On completion of this dissertation, I gratefully acknowledge the help and advice which I have received from many quarters.
First of all I want to express my very deep gratitude to
Prof. Arthur A. Broyles, my major professor;for his many suggestions and instructive discussions related to the problems which arose during the course of writing this dissertation and for reading the manuscript and' his careful correction of the wrong parts, and especially in eliminating the worst Koreanisms. My thanks are also due to the following: Dr. Charles F. Hooper, a member of my supervisory committee, for reading the manuscript and correcting the wrong parts, and to Prof. Stanley S. Ballard, Chairman of the Department of Physics, to Dr.*Thomas A. Scott, Dr. Billy S. Thomas, and Dr. Charles B. Smith, i.e. the other three members of my supervisory committee, for their warm-hearted assistance throughout my graduate studies.
I am particularly grateful to Dr. Lyun Joon Kim, the president and to all the faculty members of Hanyang University, i.e. my home university in Seoul, Korea, for their hearty assistance extended to my family who have remained in Seoul, Korea, and also to my wife, Mrs. Jung Im Lee, for her supporting my family in Seoul, Korea, by herself and offering her constant encouragement, and unbounded patience during my graduate study in the foreign country of the United States of America farthest from my native country of Korea.
Finally, I am grateful to Mr. R. A. Smith and Mr. C. V.
Gardiner for their carrying out the complicated and tedious Fortran programs on the IBM 709 Computer related to this dissertation, and to Mr. G. Scheffer for his artful execution of the graphic work, and also to Mrs. Philamena Pearl for her typing excellently the vast pages of my manuscript involving many long mathematical expressions with complicated indices.
The author believes deeply that his graduate study leading to the Ph.D. degree could not have been achieved without all of the warmhearted assistance of the above people.
PREFACE
This is the fifth dissertation written by a member of the statistical mechanics group under the program and direction of Dr. Arthur A. Broyles, Professor of Physics. The purpose, value, history, and future plan of the program can be seen from the booklet entitled "Progress Report on the Physics Department Project entitled the Equation of State of Dense Fluids" written by Professor Arthur A. Broyles (unpublished).
The theoretical part of classical statistical mechanics of the program has been studied thoroughly by the author's predecessors, i.e. Drs. H. L. Sahlin, A. A. Khan, D. D. Carley and F. Lado in their elegant ways.
The next step subsequent to their successful studies was to contemplate quantum-mechanically the problem. The tape of beginning this important and difficult task has been cut off by the author with his several colleagues. The main subject and methodology treated by him will be seen succinctly in Chapter I entitlted "Introduction" The author feels very humble and grateful that he has been very fortunate in successfully solving the problem with his limited ability. The theoretical result obtained by him is compared, for example, with the experimental result of ieon quantum-fluid at four cases of temperatures and particle densities.
He feels now as follows: The field of sciences is broad and long. However, it seems that the depth of the philosophical principle vi
of the nature given once upon a time by the creator is rather mysteriously deep and far from the science constructed artificially by human beings. Poor is the scientific knowledge and idea of human being. The ultimate doubt is that the human being can, in the long run, detect surely enough and exactly the essential principles by which the natural world has been created ever before. At this point, he wants finally to qdote the following H. Weyl's point of view with his resonance: Statistical physics, through the quantum theory, has already reached a deeper stratum of reality than is accessible to field physics; but the problem of matter is still wrapt in deepest gloom. We must state in unmistakable language that physics at its present stage can in nowise be regarded as lending support to the belief that there is a causality of physical nature which is founded on rigorously exact laws. It is yet able to follow the intelligence which has planned the world, and that the consciousness of each one of us is the center at which the One Light and Life of Truth comprehends itself in phenomena.
Jong K. Jaen
Gainesville, Florida
March,, 1966
vii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ......... ........................ . iv
PREFACE ......... ....................... ....... vi
LIST OF FIGURES ............ ........................ x
Chapter
I. INTRODUCTION ....... .....................1
II. THE FUNDAMENTAL THEORY OF QUANTUM
STATISTICAL MECHANICS ..... ................ . 13
2.1. The Introduction of the Statistical Density
Operator . ............. ......13
2.2. The General Properties of Neumann's Density
Operator and the Determination of its
Concrete Form ..... ................. . 32
2.3. The Identical, Indistinguishable Particle
System and the Symmetrization of Neumann's
Density Matrix ........ ................ 40
2.4. A New Formal Expansion Theory of the Quantum
Partition Function of Canonical Ensemble . . . 53
2.5. The Formal Theory of Quantum Pair
Correlation Function ... ............. . 64
III. THE W.K.B. APPROXIMATION FOR-THE QUANTUM RADIAL DISTRIBUTION FUNCTION .... .............. ...72
3.1. The Concrete Determination of the Function
F (k,r).................... 72
3.2. The Power Series Forms of the Diagonal Element
of the Neumann's Density Matrix and the
Quantum Pair Correlation Function ....... . i.100
3.3. The Practical Determination of the Approximated
Quantum Radial Distribution Function ..... . 134
3.4. The Experimental Determination of the Quantum
Radial Distribution Function by X-ray and
Neutron Scatterings .... .............. . 154
viii
3.5. The Numerical Calculation of the Quantum
Radial Distribution Function of the
Neon Fluid ........... . ......... 166
3.6. The Results of Computation and Comparison,
and Conclusion . . .............. 177
APPENDICES
I ... . . . . . . . . . . . . . . . . . . . . . . . .. .# 205
II ............ ............................ ...207
III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2116
LIST OF REFERENCES . . . . . . . . . . ... . . . . . . . . * . . 222
BIOGRAPHICAL SKETCH ....................... 228
LIST OF FIGURES
Page
Figure
1. The Lennard-Jones Quantum Effective Potentials
for Neon ........ ......................... 178
2. The Modified Buckingham-Corner Quantum Effective
Potentials for Neon ...... ................. ...180
3. The Exchange Effective Potentials for Neon ...... 181 4. gpy-function at T = 44.20K ............. . . . . 184
5. gCHNC-function at T = 39.40K .................. . 187
6. gpy-function at T = 33.1ï¿½K . * . ... . . . 189
7. gCHNc-function at T = 33.1ï¿½K ............. 190
8. gpy-function at T = 24.70K . . . ......... . 191
9. gCHNCfunction at T = 24.7 K ... ............. ...192
10. Jpy-curves at 44.20K . o . ............... 197
0
11. JcCC-curves at 39.4 K ....... ......... . 198
12. Jpy-curves at 33.1ï¿½K ............ .. . 199
13. JcHNC-curves at 33.1ï¿½K ................... o200
14. Jp-curves at 24.7 K . ...... . . . . . . 20
15. JCHN-curves at 24.70K . . . . . . . . . . . . . . . . 202
CHAPTER I
INTRODUCTION
-But the problem of matter is still wrapt in deepest
gloom. ..., our ears have caught a few of the fundamental chords from that harmony of the spheres of
which Pythagoras and Kepler once dreamed-(Herman Weyl)
This dissertation is concerned with the problem of equilibrium quantum-statistical mechanics, based on the SchrBdinger-Heisenberg quantum mechanics, and the purpose is six fold:
(1) The presentation of the fundamental ideas and conceptions of
the equilibrium quantum-statistical mechanics by taking advantage of the Dirac ket-bra vector description.
Standing on the Mach's point of view, viz. the Mach's principle of the economic use of the thought,* this purpose is trivial since this description leads us naturally to the correct result by means of the deductive method only under a simplified consideration, i.e. the arrival of the correct answer with the least consumption of our thought as well as the minimum introduction of necessary notations concerned with the given problem.
(2) The presentation of the natural arrival at the most basic
conception of "the Neumann's statistical density operator"
in quantum statistical mechanics starting from "the Born's
density operator" in the Schr~dinger-Heisenberg quantum
*E. Mach; Die Mechanik in ihre Entwicklung, Leipzig, 1883.
1
mechanics by introducing a new conception of "trajectory of
the state vector of the system' in Hilbert space.
The presentation of this is due to the author's dissatisfaction with the obscure but correct conception of "the statistical weight" used first by Neumann himself and still used by many recent authors in formulating the Neumann's statistical density operator.
(3) The presentation of the new expansion theory of the quantum
partition function and pair correlation function on the basis
of the group-theoretical discussion of the symmetric group.
This theory will be developed basically by taking significant advantage of the unified property of all the elements in the same class of the symmetric group, i.e. the cyclic structures of all the elements in the same class are the same.
(4) The presentation of the new expansion theory of the exchange
effective potential formally universal for all the quantummechanical systems of many particles.
We will, in this expansion theory, introduce the HamiltonCayley's theorem as our basic starting point, and extend it to a point appropriate for our expansion theory. Our theory may be called the expansion theory in terms of the matrix-traces of the first, second, third orders, and so on.
(5) The presentation of the second order approximation of the
quantum radial distribution function by the W.K.B. approach
within the framework of the Broyles program.
The theoretical result of this approximation is applicable universally for any kind of quantum fluid with one component in its equilibrium state. Two results applicable for the quantum fluids at high and fairly low temperatures will be presented by taking into account the quasi-quantum effect as well as the exchange effect of the quantum particles of the fluid under consideration. It would, in this presentation, be interesting to the reader that some hypersurface integrals related to this problem vanish for the volume of the quantum fluid large enough and in equilibrium. The new hypervolume integrals appearing in the course of the treatment of this problem are probably useful also for the treatment of other statistical-mechanical problems, and are presented in Appendices with their evaluated steps.
(6) The presentation of the comparison of the theoretical result
with the experimental result of the quantum radial distribution function of the neon fluid to testify to the order of
the accuracy of our approximation theory.
We will, in this presentation be concerned with testing the order of the correctness of our approximation theory by comparing it with the radial distribution function determined experimentally by the measurement of the scattering intensity of x-rays by the neon liquid. This comparison will give us some aid for our further better idea leading to a better solution to this problem.
Finally, we will set up some notes about the notations to be used in the development of our theory immediately subsequent to this chapter. We are, in this dissertation, concerned with the quantum mechanical system of N(large enough) identical, indistinguishable, interacting particles. We will use the notation- and summation-conventions of tensor analysis in this discussion.
We assume the Euclidean character of the 3-dimensional realistic space* with the line element measured by the unit"cm," where each particle of the system occupies a point. We regard the realistic space as the covariant vector space of the position vector of a particle. Furthermore, we introduce the 3-dimensional abstract vector space reciprocal to the realistic space and also with the Euclidean character. Then, the line element of this reciprocal vector space is measured with the reciprocal unit "cm'1, i and therefore, this abstract reciprocal space can be regarded as the wave (or propagation) vector space of the particle.
*In the strict sense, this assumption is not correct from the standpoint of the theory of general relativity, even though we will introduce the interaction potential of N particles, in addition to this Euclidean character, corresponding to the fundamental metric tensor (not equal to Kronecker's 5) in the theory of general relativity.
5
Since the Euclidean character has been assumed for the realistic space, the contravariant vector space of the position vector of the particle is identical to its covariant vector space, i.e. the realistic space. Similarly, this is also the case for the reciprocal space. Thus, we see that the four kinds of spaces are mathematically identical to each other with no distinction between the covariant and contravariant components of a vector and two kinds of different units (cm and cm- ) used for measurements of their line elements. Hence every mathematical operation can be defined numerically even between the vectors of the realistic and reciprocal spaces. For example, the scalar product k1-rI between two vectors of the reciprocal and realistic spaces can be well defifted even in the mathematically strict sense.
Since we have no distinction between the covariant-component
indices(subscript) and the contravariant-component indices(superscript), we may represent the components (x,x,x) and k k and wave vectors rX and k> of the Xth particle among the ordered N particles by
..El
We will use hereafter the Greek letters for the ordering subscripts of N particles, and the English letters for the ordering superscripts of the components of the vector quantities associated with each particle.
The important quantity q with which we will be confronted in the development of our main theory is, among all, the quantity of the following type of 3-dimensional scalar-product summation over all N particles of the system under consideration: q = IBX,?= A B% [2]
constructed by two vectors taken from one among the reciprocal and realistic spaces, or both. Eq. [2] can be rewritten in the form given by
q 1 3] where 8.. and & are the subscripts and superscript Kronecker's 8, and we will use also the mixed-index Kronecker's 8, i.e., 8,J or 8j%, and we note, in Eq. [3], that the index %, or L runs from I to N, while the index i, or j, from 1 to 3.
Now, let us introduce the 3N-dimensional Euclidean configurationspace (mathematically identical to its reciprocal space) of the system under consideration, and e. (i=1,2,3;x=l,2,...,N) be the fundamental
1
unit vectors (3N in numer) of a 3N-dimensional orthogonal coordinate system established in the configuration space. We have, then,
e.-e = 8. 8X8 , [4] and Eq. [3] is, by using Eq. [4], written in the following alternative form:
q = (A~e')(B e P, or q = AB, [5] by defining the two vectors A and B of the configuration (or reciprocal) space given by
-def .i% * .4def. J-4[6
A A ei (Ai'A2,..., AN); B B e. (B2'...B) [6] Eq. [5] shows us that the 3-dimensional scalar-product summation of Eq. [2] can be interpreted simply as the scalar product of the two vectors A and B defined by Eq. [6] in the 3N-dimensional configuration (or reciprocal) space of the system under consideration. Therefore, we
can, in our theory, recognize that N vector-quantities A (X=I,2,..., N) associated with the same kind of quantities of N particles in the realistic (or reciprocal) space form together a vector A defined by Eq. [6] in the 3N-dimensional configuration (or reciprocal) space of the system under consideration. For example, we can regard the 3-dimensional scalar-product summation of the position vector r% and wave vector k of each particle over all N particles of the system, i.e.. 'r. simply as the following dimensionless scalar product:
k>.-f * ,' k r 7 of two 3N-dimensional vectors k and r defined by
"def
k- (klk 2". -.-,k N)' [8]
-def. _- "0 ..)
and also
-2
r - r.r = (e (xe x
= X=8. X xx rby using Eq. [4], or
= rr, similarly = " ï¿½ [9] Therefore, the total kinetic energy T of the system is given simply by h2 *2
TZ = 1 m [10] Next, the operators playing basic roles, with which we will be confronted in our later theory, are the following types:
ik%4kf, k-xf , i~g. iZ ; :k f, 11
where the double dots ":" represent the scalar product of the two dyadics associated with it, and f, or g is a scalar function of the position vector r = ( I,,. and f deX ei6/6x, [12] with the fundamental unit vectors e, (i=1,2,3) of a rectangular coordinate system (x1,xX, x2 ) set up in the realistic space.
We seen, then, that
k. 3 = ki, i= ki~ 5k%/&x X X XX Xij
= (Ke)"
by using Eq. [4], or
by using the definitions of Eq. [8] and
- 116/0'x 0 [141 Thus, the first of Eq. [11] can be expressed simply in the following form:
kX. Xf = kK'f [15] by using the 3N-dimensional gradient operator 6 defined by Eq. [14]. Similarly, the second of Eq. [11] is written simply as = - [16]
by using the 3N-dimensional Laplacian operator defined by
__X ~)(6/xI) =%6) -~ [17]
The other two of Eq. [9] are written similarly as X X g[18]
The differential operator T of the total kinetic energy of the system under consideration is, in this fashion, expressed simply by T 6 = - T " "19m
2m ?I X m
just as in the case of one particle.
We will, though not so frequently, be confronted with the need of the gradient operator in the reciprocal space. However, we designate this operator by the notation given by e6 k. . [20] and correspondingly, the 3N-dimensional Laplacian operator by
-2def., ~ ~
a '( / k' kX) 2-3 [21] The introductions of those notations stated above are presented in the hope that the complicated many-body problem may be reduced, even notationally, to one particle problem by avoiding the tedious, misleading, various kinds of indices, multiple summation and product notations in the conventional notation-fashion, withinwhich no mistakes arise. Under this idea, we will have below further introduction of still more notations.
The differential operators corresponding to the position and wave vector operators (r,k) of the system is given by =r
We will use the following hypervolume-element notations in the 3N-dimensional configuration and reciprocal spaces respectively:
def. N d3 def.d3N- N 3-def. N3Nr i dr= d= r, dT'r- 1 d3 r d r x 4 =3
d 2o H1ï¿½ d3k -d3 k, [23] X-1
and the following integral-domain notations:
,r -N'de--jR'vN'2 [241
for the integration with respect to the variables (rl,2,...,rN) and (r 'r1,...r) over the volume V of the realistic space respectively.
*Corresponding to V', we will frequently use also the notation written by
,def.[25]
The notations dS and ds will stand for the (3(N-2)-l)-dimensional hypersurface element in the 3(N-2)-dimensional space and the surface element in the realistic space respectively, and the single integral notation will stand for any kind of multiple integral notation, i.e
Jff ..I [26]
and for any kind of hypersurface integral, i.e.
OJ - [27]
In this way, we can deal with the many body problem notationally just as the one particle problem by using the usual 3-dimensional analysis, as long as we are concerned with our theory to be presented in this dissertation.
We have, in quantum statistical mechanics, an important parameter P closely corresponding to the time parameter t in the SchrldingerHeisenberg quantum mechanics. This parameter is defined by P= /(kT) [28] with the absolute temperature T of the system under consideration and the Boltzmann's constant k. We will use the following notations about the derivatives with respect to these two parameters t and P:
t _ def./ . [29] The configuration, or reciprocal space may be regarded as the dimension space, of which each point labels each coordinate axis of the r-representation, or k-representation in the Hilbert space. We represent the total number of dimensions of the Hilbert space of which each dimension (or coordinate axis) corresponds to a point of a curve, surface, and volume, notationally by
Therefore, the total number of dimensions of the Hilbert space of which the dimension space is given by the configuration, or reciprocal space, is given notationally by
03N
We represent the fundamental orthonormal bra-vectors of .3N in number in the r-representation, or the k-representation of the Hilbert space by
idef
(2's ' l ,.,N 2(r2( ... N( Nl,
[30]
We make a distinction between (7'sI and (7i, similarly, (''sJ and i. The meanings of the two bra vectors <71 and (I will be manifested in the middle part of the next chapter.
Since the time variable t is regarded as a parameter in the
Schrbdinger-Heisenberg quantum mechanics as stated above, it is rather reasonable that we express the state function i(r,t) of the system under consideration by
(r, t)-7 ;t), [31] instead of expressing it by
r ( , t) -- r, t I T .
This latter notation is rather reasonable in the Dirac-Feymann relativistic (special, not general) quantum mechanics. Therefore, in our theory to be developed immediately from the next chapter, the ketvector denoted by
L;t> [32]
represents an arbitrary, normalized, state ket-vector, corresponding to the state function (r,t) which is a component of the state ket-vector of Eq. [32] along a coordinate axis of the r-representation labeled by 3N
r(_rl,$r2, ...,N) in the * -dimensional Hilbert space (aa linear manifold).
CHAPTER II
THE FUNDAMENTAL THEORY OF QUANTUM STATISTICAL MECHANICS
-Willst du immer weiter schweifen? Sie! das Gute
liegt so nah. Lerne nur das Gluck ergreifen,
Denn das Gluck ist immer da.-(Goethe)
2.1. The Introduction of the Statistical Density Operator
Let us consider a quantum-mechanical system composed of N particles (atoms, molecules, etc.) with a Hamiltonian operator H. The state ket-vector j;t) of the system develops, in S-picture, according to the time-dependent Schr6dinger's equation given by
i D tJ;t) = HJ;t) (1) or in its equivalent form:
V;t) = e iH(t-to) J;to) (2) if the system is conservative.
The probability density w(t;r,r) of finding the system in a unit volume about a point r at a time-point t in its 3N-dimensional configuration space is
wtr rj r`
w(t=- * I ;t)(;t 1 (4)`
as an operator (or an observable) in functional space (or Hilbert's space) corresponding to the classical phase space.
The probability density w(t;r,r) defined by Eq. (3) may, then, be written in the form:
w(t; 7 W(t) IV> , (5) which is the quantum-mechanical expectation value of the observable W(t) at the state of the position eigenket 17). Eq. (5) may be regarded also as a diagonal element of the representatives of the operator W(t) in the representation of the position eigenkets, i.e., Ir) so that the nondiagonal element of this representatives is given by
w(t;*,') (rjw(t)jr') (6) This representatives forms a continuous matrix which is called the quantum-mechanical probability density matrix of the system.
Now, let us consider the representatives of this operator W(t) in a representation 1q) defined by the eigenkets lq)'s of an observable Q of the system, i.e.
Qjq) = qjq) ï¿½ (7)
It becomes, then,
w(t;qq') = (qjW(t)jq') (qj;t)(;tlq'), of which the diagonal element becomes
w(t;q,q) = ](qj ;t)12 . (8)
This implies physically that the diagonal element of the representatives of the operator W(t) in the representation of the observable Q represents the probability of obtaining the measurable value q of the observable if a measurement of it is done on the system at the state Eqs. (8) and (5) show us, therefore, that the information about the
probability of obtaining the measurable value of every observable of the system at the state j;t) is contained in the operator W(t) defined by Eq. (4), which is intrinsic to the system under consideration. We call, therefore, the operator W(t) the quantum-mechanical probability density operator, or briefly the Born's density operator of the system, since M. Born has given first the probability interpretation of Eq. (3) to the state function (Gj;t).
The combination of Eq. (4) with Eq. (2) gives the following alternative form of Born's density operator:
W(t) =e -Htt)Y ;to)(;tole iH(t'tï¿½)/h
or
W(t) e'fit/h ;O)(;Ole iHt/t, (9)
if we take to = 0, which expresses the time-development of the Born's density operator of the system starting from the initial state t;O) It is easy, from Eq. (9), or Eq. (4), to see that
d(;tlw(t)I;t) 0 , (10) by noting
(;tlw(t)l;t) 1
coming from the normalized character of the state vector j;t). Eq. (10) may be reduced to its equivalent form:
iii-t W(t) + [W(t),H] = 0 (11) by using Eq. (1), or Eq. (2). It can be shown physically that Eq. (11) is the formal form of the law of conservation of the probability of finding the system at the state j;t) at a time-point t, or at a point r
in its 3N-dimensional configuration space. The value of the invariant trace of Born's density operator under any similar transformation is unity, i.e.
Tr(W(t)) = 1, (12) and Born's density operator WH(t) in H-picture is time-independent and given by the initial Born's density operator W(O) in S-picture, i.e.
WH(t) = w(O) = I;0><;O1
The expectation value (Q) of an observable Q at the state I;t) is given by
(Q) = Tr(WQ) = Tr(QW) (13)
Born's density operator W(t) is an Hermitian projection-operator (an idempodent) to project any vector onto the state vector j;t) and has the state vector as its eigenket vector corresponding to the eigenvalue
1 among two possible eigenvalues 1 and 0. Besides those properties stated above, this operator has many interesting properties. A systematic and complete presentation of these properties of Born's density operator may be seen from several papers published recently by several
authors! 2032,,6, 7,8)
However, we are not, in this dissertation, concerned directly with the Born's density operator W(t). We omit, therefore, our further discussions about it. We are going to introduce another density operator useful for quantum statistical mechanics starting with Born's density operator by taking another point of view different from those of recent authors.
We consider, at first, the behavior of the state vector I;t) in the functional space of dimension c3N. Suppose the continuous set S
is composed of every ket vector normalizable to unity in this functional space. It is, then, evident that this continuous set S forms the hypersurface of (03N-l)-dimensional unit sphere in the 03N -dimensional functional space and all accessible state vectors j;t) normalizable to unity form a subset s of the continuous set S, i.e s C S . (14) Thus, all accessible state vectors occupy a part of S since every element of the subset s must be expanded in the following limited form:
t - iEt/hI)(
n
in addition to the normalization requirement. a- is, in general, the
n
complex constant coefficient independent of the time parameter t, and I'd satisfies
H't) = EJIt) 1 (16)
2
Furthermore, the state vector jP';t) starting with an initial state JP') moves in accordance with Eq. (2), i.e.
JP' ;t) = e- ift ' ) ,(17) and the terminus of 10;t) describes a state trajectory in the domain s on the hypersurface of the unit sphere described above. It can be, then, shown easily not only that every state vector corresponding to every time point, i.e. every point on a state trajectory cannot, in
it is a lattice vector in quantum number vector space, and we note here that the subset s is a closed linear manifold.
2 , is a labeling parameter which has very significant physical meaning as we will see later.
general, be orthogonal to each other but also that every state trajectory starting from different initial state never intersect with each other. This proposition may be proved by using the quantum-mechanical law of causality, i.e., the law of the unique determinancy of the state vector for a given initial state!7) This property of non-intersection of every state trajectory is very important in formulating the basic principles of quantum statistical mechanics.
Now, we classify every accessible state trajectory into two categories according to whether the initial state lp') is an energy eigenstate 1i) or not. Let us call the system starting initially from an energy eigenstate the first kind of system, and the system starting initially from nontenergy eigenstate the second kind of system. This classification is reasonable since the state of the system at an arbitrary time point can be determined uniquely only by its initial state according to the quantum-mechanical law of causality. We may, thus, decompose the set (IP')} of every accessible initial state into two subsets, i.e., (lit)), the set of every eigenstate of energy, and tj )1, the set of every non-eigenstate of energy of the system, or ) = ir)] + ) (18) Correspondingly, the subset s is decomposed again into two subsets s1 and s2, or
s s1 + s2P (19)
where sI corresponds to the first kind of system, and s2' the second kind of system.
Next, we will analyze the properties of each of the systems belonging to these two categories respectively.
A. The First Kind of System sl:
Since the first kind of system is defined by
the equation of time development of the state of this system becomes, in accordance with Eq. (17), 1n;t) =e itH/h I
or
fle;t) n e )'in (20) by using Eq. (16) and putting F. Yr (21) Eq. (20) refers to a stationary state of the system with a conservative energy eigenvalue E., so that the terminus of the state vector given by Eq. (20) is moving along its state trajectory on the hyperspherical surface of unit sphere with an invariant energy N.3
Our next question is whether this trajectory is closed or not. If we can find a time point t such that
1l;t) = I->, or e Ln) = i11- (22) We can, then, conclude that this trajectory is a closed curve on the hypersurface of unit sphere. Actually, this conclusion is correct. It is as follows: If we put, in Eq. (22),
t = 2A n/-., (P=0,-l,ï¿½2, ... ),
n
Eq. (22) is, then,satisfied since this value of t makes the exponential
3We note here that I ) and e'ict-A) represent the same physical state, but they are different ket (-or state) vectors from each other.
function of Eq. (22) be unity. This shows us that the state vector comes back to its initial state vector with a period T given by = 2v/wa = h/E , (23) that is, the state vector is moving periodically along its trajectory with a definite angular frequency ai, given by nn
This is a different point of view for a), from those of de Broglie and
4
Bohr. We should note here that every trajectory of the stationary state vector labelled by a different lattice vector n of quantum numbers never, as we have seen already, intersectswith each other in addition to its property of closeness. The number of trajectories of stationary state vectors is the same as the number of the lattice points Vs in quantum number vector space, which is, in general, infinite.
Next, we will investigate Born's density operator W_(t) for this system. The second term (commutator) of Eq. (11) is equal to
= e iHt/h [W.A(O),H]e iHt/h
= e-iHtl (HJ<9)nJ-Jn)(nJH)e iHt/ = 0 by using Eq. (16). We see, therefore, that t W-(t) = 0,
i.e., time-dependent of Born's density operator for the first kind of system. Actually, we have, from Eq. (9), w (t) = 1n)(nI = w.Am. (24)
4For example, see A. Messiah; Quantum Mechanics, Vol. 1, Chapter 1 and 2, (1961).
We have, so far, assumed tacitly that the energy eigenvalue E. of the
n
system is not degenerate. In systems made up of very large numbers of molecules, the energy levels will, in general, be highly degenerate. Let
,t e n,j), (j=l,2,...,f) (25) be orthonormalized degenerate (f-fold) stationary states corresponding to an energy level E.. The Born's density operator W. (t) for the jth n n,j degenerate stationary vector j ,j;t) is, then, given by W+ ,(t) = I ,j;t)ci,j;tl
n, j
There are f Born's density operators W. (t) (j=1,2,... f) corren,j "
sponding to this one energy level E.. All of these density operators
n
of f in number are density operators of pure states. For the statistical information of the isolated system specified by the energy E-., it is plausible to take the time-average of those f Born's density operators Wd,j(t) (j=l,2,...,f) under some statistical weight. As we will prove in Eq. (37)(.take 1P) = jTn,j) in Eq. (37)), this statistical weight is independent of the time variable t and the subscript j. Therefore, the time averaged Born's density operator D-. is given by
n
def. tWMj ft)/7 dt, (26) j=l j=l f
where c. is the closed trajectory of the stationary state vector j;t), and the contour integral goes along the closed trajectory curve c..
Equation (26) reduces to
f f n W Tj
with the period T. of the state vector I ,j;t). As we have shown
(27)
already in Eq. (23), the period T. depends upon only the energy level
but not upon its degenerate states, i.e. the subscript j. We have, therefore,
1I =2 = = 2 Itff(- h/Et) . (28) The combination of Eq. (28) with Eq. (27) gives
f f
it f yt~j(o) = I J,<(jJ " (29) j=l j=l
This is the statistical density operator of mixed states useful for the statistical information of an isolated system with a specified energy level Et. We call this density operator the Neumann's density operator of first kind. The macroscopic quantity corresponding to an observable Q of the isolated system may be defined by the time-average of the expectation value (Q) of the observable Q, and shown easily to be
(Q) = Tr(QD) . (29)' The right-hand side of Eq. (29)' represents the microcanonical ensembleaverage in the ensemble theory. Therefore, Eq. (29)' shows us the correctness of the quantum-mechanical ergodic hypothesis, i.e., the hypothesis that the time-average of an observable is equal to the ensemble average of the observable.
It is instructive to note here that we never have used the postulate of equal a priori probability in deriving the correct form of Eq. (29) of the Neumann's density operator of first kind, and instead, we have used the postulate of Eq. (37) that the statistical weight in the time-average of Born's density operator is inversely proportional
to the speed of the terminus of the state vector at a time point t on its trajectory
Next, we are in the position to analize the properties of the second kind of system.
B. The Second Kind of System s2
Since the second kind of system is, as stated in Eq. (18), defined by
we have, from Eq. (17), the following equation of time development of the state of this system: lp;t) -iHt/ (30) with 1P) not being an energy eigenket at the time t = 0. A further change in Eq. (30) may be made by inserting the identity operator given by
6 (31)
n
between the exponential operator and the vector 1P) in the right-hand side of Eq. (30). It is as follows:
= Ze_~i)(l (32)
5This postulate is more basic than other authors have recognized so far. For example, see T. L. Hill; Statistical Mechanics (1956), p 40 et seq.
6We assume here that the Hamiltonian operator H has no continuous eigenvalues. But our subsequent discussions can be extended easily even to the case of continuous eigenvalue of H.
by using Eqs. (21) and (16).
Our next problem is to see whether the state trajectory given by Eq. (32) on the hyperspherical surface in functional space is open, or closed. In other words, does there exist a time-point t such that lp;t) = 10 ? (33)
The substitution of Eq. (32) into the left-hand side of Eq. (33) and the use of the identity operator given by Eq. (31) for the right-hand side of Eq. (33) lead to
EJ-n) (lnJP)(1-e~i') = 0.
We must, therefore for all n, have e ict= 1 , (34)
since the IP)'s are linearly independent of each other and (njp) 0 in accordance with no intersection (Durchschnitt) between two subsets s and s2 defined by Eq. (19). Eq. (34) requires that dtt
with no dependence of t on 'n, where i is an arbitrary integer which is possibly dependent on 'n. This shows us that
E./,- = h/t (independent of i), or
..... =:A ,......
The energy level Efl of the system under consideration must, therefore, be proportional to integer ï¿½ in order that the state trajectory of Eq. (32) can be closed. This requirement is, in general, impossible
except in the case of non-interacting harmonic oscillator system. Therefore, the state trajectory of the second kind of system does not, in general, close as in the case of the first kind of system. The terminus of the state vector lp;t) describes an open curve in the domain s2 on the hypersurface of the unit sphere in functional space. Our subsequent question is as follows: Is it a Peano curve, or an open Jordan curve?(9) It seems that this question is a difficult problem to be solved.7 However, fortunately, we are not here concerned directly with this question in setting up our fundamental principles of quantum statistical mechanics.
Our next problem is to examine whether the Born's density operator of this kind of system is dependent on time, or not. The commutator in the second term of Eq. (11) is, by using Eqs. (32), (31), and (24),
n e iHt/h *
n
Therefore, the Born's density operator W(t) for this system depends upon time, i.e. W(t) is changing as the state vector lp;t) moves along its trajectory. Actually, this operator is, by using Eq. (32), expressed explicitly as follows:
W(t) r
n
+ Z e-i(u'C')t i, (35) n, n
7We note that Peano curve corresponds to Baltzmann's ergodic hypothesis and Jordan curve corresponds to Neumann-Birkoff's quasiergodic hypothesis.
where we used the definition given by Eq. (21). It should, in Eq. (35), be noted that the first summation term is time-independent while the second double-summation term is time-dependent but sinusoidally oscillatory.
Now, we are going to prove the validity of the postulate of a priori probability for this system. We are, at first, interested in seeing how much time is needed in changing the state vector jp;t) by a given infinitesimal norm of its displacement II dJP;t)JJ, during its course of motion, and how the corresponding infinitesimal time dt depends upon the position of 11;t) on its trajectory.8 We start with the differential form of the following Schr~dinger's time-dependent equation:
iidlp;t) = dtHlp;t) , or -i(P;tld = (P;t]Hdt. The Hermitian scalar product of these two equations gives
(P;tJH2Il;t)dt2 = h2I1d l;t) i.2 (36) But, we have, by using Eq. (30) in the left-hand side of Eq. (36), (P:tJH 21p;t) = (PIjH 2 0
and we express the given infinitesimal norm by dsdef.
We have, then from Eq. (36),
dt = ids/((PJH21p))2 (37)
8
8L dIP;t)IIdf'(the norm of dlp;t)) =[(dJ;t))t(dJP;t))]2 t n (Hermitian conjugate).
which is not dependent upon the point, i.e. the time-point on the trajectory in the course of motion of the state vector. In other words, the infinitesimal time dt during which the state vector stays at every point on its trajectory is the same. This implies physically that the probability of finding the system at every point on its trajectory is the same, i.e. the validity of the postulate of a priori probability for the quantum-mechanical state.
Equation (37) is written also as
ds (38),
n
by having a change given by:
(=JHfP
n
n
Equation (38) may be interpreted physically as the speed of the terminus of the state vector IJ;t) on its trajectory, which does not depend upon time but does depend on its initial state vectorip). That is, the representative point of state of the system has an open hypercurvilinear motion with an uniform speed on the hypersurface of unit sphere in Hilbert space. This situation is very much analogous to the realistic motion of uniform speed of a force-free particle constrained on a surface in realistic space, which moves, in general, along an open geodesic line on the constraining surface.(I0) The same can be seen also from an Einstein's postulate of the theory of general relativity that a realistic mass point moves, without any 4-acceleration, along an open
geodesic line in 4-dimensional Riemanian time-space. Now, we are in a position to introduce Neumann's statistical density operator D by using our preliminary knowledge discussed so far.
Let us rewrite here Eq. (35) in the following form: W -i((L -a),O )t
(t) = e (39)
As stated already, this Born's density operator of the second kind of system is varying with the position of the terminus of the state vector jI;t) on its trajectory, and the validity of the postulate of a priori probability, i.e. the possibility of the state vector staying equally at every time-point on its trajectory, has been proved. It is preferable to take the time-average of Born's density operator W(t) given by Eq. (39) over the whole time-interval from the ififinite past t r -0 to the infinite future t = = for the statistical information of the system under consideration. It is, then, sufficient to take the time-average of W(t) in usual sense in accordance with the proof or validity of the postulate of a priori probability. We have, therefore,
i n, t
l-m 1 e dt,
-TO
or
D n n n (40)
by noting
9
lim I e ai(C t = it,9') (40)'
This operator D is the time-averaged Born's density operator over whole trajectory of the state vector started initially from an accessible state specified by a parameter P. We will call this operator D the Neumann's statistical probability density operator, or briefly the Neumann's density operator of the system under consideration, since J. von Neumann has introduced firstly this operator according to his intuitive foresight without the basic discussions which are presented here in this paper.(11,12) The advantage of the Neumann's density operator D is especially in its universal character of use at every time-point irrespective of the time-point coordinate. Furthermore, this operator D is invariant in the sense that the initial state vector JP) can be replaced by any state vector !p;to) (to is arbitrary) corresponding to a point on the trajectory. For we have, by using Eqs. (30) and (16),
1 ei'tï¿½/ J p;to)(p;toli /n n
n
il to, n i e n n< ;ox<;tol-nx-Rie
9Note that this integral represents also the curvilinear integral along the trajectory of state vector.
or
n n n= D n1tifJ;oj(t n. (41)
This invariant character of D assures us that we can write the initial state with 1P) instead of writing with IJ;O), so that P can be regarded as the labeling parameter of the trajectory instead of the labeling parameter of initial state vector, and also the ket vector I) can be interpreted as the statistical state vector intrinsic to the trajectory specified by the labeling number P.
Now, we are going to modify our point of view for the trajectory of the state vector regarded as an orbit (or locus) of a moving point. Let us pick out a trajectory specified by a labeling number P, and look at all points on it statically. . Every point of it determines uniquely a state vector in Hilbert space respectively. We may, then, make a system correspond to each of the states so obtained as above. Thus, we obtain collection, or an ensemble of systems corresponding to the trajectory specified by P. Mathematically, this ensemble of systems is equivalent to the set of all points on the trajectory. Therefore, this ensemble of systems must be a continuous set specified by a labeling number P. Every system contained in this ensemble has to have the same Hamiltonian H, since the trajectory is intrinsic to not only the parameter P but also the Hamiltonian operator of the system under consideration. We must, therefore, use the Hamiltonian operator H and the parameter P in specifying the trajectory, or the ensemble of systems. Thus, we can consider that the abstract conception of state trajectory on the hypersurface of the unit hypersphere in Hilbert space
is nothing but an ensemble of realistic systems with a common Hamiltonian operator H and a common parameter P. Let us express this ensemble of realistic systems by the following symbol: s2(pN,V;H)
where s2 comes from Eq. (19), and N is the common total number of particles contained in the common volume V, of which the property of "commonness" has been assumed tacitly in our discussion done so far.
According to this point of view, the Neumann's density operator D defined by Eq. (40) can be interpreted as the ensemble-average of Born's density operator W(t) over all elements of the continuous set s2( ,N,V;H) with continuous elements. It is instructive to note here that the time-average of the Born's density operator of the pure state is changed into the Neumann's density operator of the mixed state. In this sense, the Neumann's density operator can be used for the quantum statistical-mechanical information of the realistic equilibrium-system under consideration. The real number l OI's of the pure states
1 ) 's. Quantum-mechanically, this weight of n
n
of the system when the system is at the state 1p) right before the measurement done.
Finally, it should be mentioned that the Neumann's density
operator of Eq. (40) can be used for the statistical description of an isothermal equilibrium system irrespective of the magnitude of the
particle number N and the volume V of the system. We will, hereafter, treat exclusively the quantum mechanical system in thermal equilibrium. Especially, in the theory of the quantum correlation function, we will assume also Hi((H , where H. is the interaction with a isothermal bath.
2.2 The General Properties of Neumann's Density Operator
and the Determination of its Concrete Form
Before we determine the concrete form of Neumann's density operator, we will study the general properties of it preliminary to its determination. We will choose brief descriptions of proofs below. Theorem 2.1:
D~ n (1=0,1,2,3,.... (42)
n
Proof: From Eq. (40), we have
D2 = T -)I ( 12_ ,1,1>2<,
n , n'
= Z IE>I< I >I4( J.
n
by using the orthonormality of the complete set of eigenkets jn)'s. This procedure is continued to obtain the general form given by Eq. (42). Theorem 2.2:
1 = (q]DIq) = Tr(D )
q
= r3N~
1OThe ket vector I') is the same vector as Eq. (7).
Proof:
3N 2 (-
n
-- ( j > = .
The others come immediately from the general character of invariance of trace of any operator in any representation. Theorem 2.3: D is Hermitian and commutable with the Hamiltonian operator H, i.e., Dt = DP, [DPH] = 0, (44) i.e. D P is an invariant operator (a constant of motion).
Proof:
n
-~1IP1 > 12 E( If>)C1- f>
and the Hermitian property can be seen immediately from Eq. (40). Theorem 2.4: The eigenvalue P,(p) and eigenkets of D are given by D j ) = P-(p)jf> , (45) P.(P) dgf< I >I
Proof: From Eq. (40), we have
n
Hence, D I > = 1 1P:)1< '112<. 1-) = I
which shows us that P.(4) - J(W 1 >12 is the eigenvalue of D correnP
sponding to the eigenket i). This is trivial since two operators H and D P are compatible with each other in accordance with Eq. (44). Theorem 2.5: The time-average 7Q of the expectation value (Q) of a time-independent observable Q at a state I;t) is, in terms of D., given by
= Tr(QD ) (46) Proof: See Appendix I. This theorem guarantees our foregoing
statement that - all statistical information of a system under consideration is contained in the Neumann's density operator D of the system. Equation (46) is comparable with Eq. (13), i.e.: the quantummechanical average.
Now, we are going to determine uniquely the concrete form of Neumann's density operator by referring to the knowledge about D obtained above.
Theorems 2.2 and 2.5 shows us that the operator D may be
regarded as an observable (with no classical analogy) concerned with the quantum-statistical probability density which corresponds to the classical distribution function. Furthermore, Theorem 2.4 shows us that the operator D is the observable giving the probability of obtaining the measurable value N of the energy observable H at the
statistical stateIP). Theorem 2.3 shows us that the operator D takes the following form:
DP = F(P,H) (47) since D must be an operator function of invariant operators, e.g. the Hamiltonian operator. This statement will be rather guaranteed by our subsequent discussions on the basis of the so-called "the law of large numbers.11
This law for canonical ensemble is, in terms of the terminology of this dissertation, given by
-E./kT / -E/kT (13)
nne (48)
n
Let us look carefully at the connection between both sides in Eq. (48). The independence of two ket vectors in) and 1p) upon each other corresponds to that of the microscopic energy level E- and the macroscopic absolute temperature T multiplied by the Boltzmann's constant k. Furthermore, we have only two kinds ofparameters, i.e. ( ,P) in the left-hand side, (',T) or (n,lI/kT) in the right-hand side. We may, therefore, take the parameter P as = i/kT . (49) The parameter P introduced originally as a labeling index of the intial state (or statistical state) of the system under consideration is now interpreted as a macroscopic physical-quantity i/kT, i.e. the reciprocal
11See P 210 of the English translation edition of J. von Neumann; Mathematical Foundations of Quantum Mechanics translated by R. T. Beyer, Princeton University Press (1955).
of approximate thermal kinetic energy of a particle. This interpretation is also consistent with the hypothesis established in the end of the previous section since I/kT has a continuous interval (0,00).
We rewrite Eq. (48) in the form:
I( )2= e def. e'En (50)
n
and combine this with Eq. (40), to see that
DP = Q( ne- _I (_j = Q-I1 e- Hj_)(_j n n
= Q_ 1e-PH( !1n)(ni)
n
or
DP = e -H /Q , (51) which is the form consistent with the required form given by Eq. (47). Equation (51) is the concrete form of Neumann's density operator which contains every statistical information concerned with canonical ensemble.
The macroscopic function Q of thermodynamical variables is, by using Eq. (43), determined as Q P Tr(DP) m Tr(e-H) (52) In the Hamiltonian representation, this invariant trace takes the form given by the second of Eq. (50). It has turned out easily that this function Q represents actually the partition function of the system in othermodynamical equilibrium.
Let us insert the identity operator: I~ =jlq)(q
q
constructed from the discrete complete set (1q)) of the eigenket vectors of an arbitrary operator Q between the operator e"PH and the position eigenket vector j?) in the following expression:
= Tr(e-PH) =Jd3!(- Ie-H11). (53) We obtain, then,
Q= fd3ll T, (q I-r) e r CIq) (54+)
by changing the operator H into its equivalent differential operator. The function S (7, ) defined by
Pd'N. E(ql')e-H q) = N!(-rje-HIr (55)
q
is the so-called Slater sum in the discrete representation jq) which has been introduced firstly by J. C. Slater in 1931.(14) The partition function given by Eq. (54) is, then in terms of the Slater sum, written as
1 f, d r krr. (56)
Let us, next, consider the representatives of Neumann's density operator, especially in the ?- and n-representations, respectively. We have, then,
= Qle-H( ,i P ')Ior
D (M,) = Q Ple- r 12 (57) and we have, similarly,
D (k,k') = Qle'H( iQ-'e)ï¿½(k,k' ).. (58)
Next we are going to construct the so-called bridge differential equation between thermodynamics and quantum statistical mechanics by using Neumann's density operator D as a bridge connecting them and the well-known bridge algebraic equation: QP = e-'FP, or F P = -nQ , (59)
where F P is the Helmholtz free energy of the system in question. The differentiation of both sides of Eq. (51) with respect to P gives ( nQP)DP + 16 PD = -HD ,
which is combined again with Eq. (59), to give ( b+H)D P = (F P +-6F P)D . (60) This differential equation of operator is the so-called bridge differential equation connecting between the microscopic world and macroscopic
12s(f,)def'-i]i'), which is not the product of Dirac's 5-function in the usual sense. See Section 2.3, Eq. (90).
world. Equation (59) is derivable from Eq. (60) by using the temperature-boundary condition:
liO (PF +inQ ) = 0. (61)
This equation (61) has not been quoted so far by any author of statistical mechanics in addition to the following Bloch equation:(15)
+H)U= 0, U P (2T)3N/2(-lefHI) , (62) which is also derivable from Eq. (60).
The combination of Eq. (59) with Eq. (51) gives our following final form of Neumann's density operator to be used in our later theory:
DP = e (F-H) . (63) This is a more useful form than Eq. (51) since it contains automatically, within it, also a bridge equation, and plays the most central role in modern quantum-statistical mechanics of equilibrium system.
We have, thus, arrived very naturally at the conception of Neumann's density operator, i.e. the most powerful basis of modern quantum-statistical mechanics with one hypothesis, i.e. the law of large numbers, starting from two basic postulates of SchrdingerHeisenberg's quantum mechanics, i.e. the SchrBdinger's time-dependent equation of motion and Born's physical interpretation of wave ket vector L;t). In this way, we could avoid the obscure conception of "the statistical weight" which was first used by Neumann himself and still by other recent authors in formulating Neumann's density operator.(4'6' 11,16,17) A further analysis about this operator can be seen in the quoted paper (7).-
2.3. The Identical, Indistinguishable Particle System and the
Symmetrization of Neumann's Density Matrix
We shall, in this section at first, for our preliminary knowledge of the symmetrization of Neumann's density matrix, present the general quantum-mechanical character of a system of identical, indistinguishable particles of N in number in a different way from those presented by recent authors by using the properties of symmetric group as our mathematical tool.
The system composed of identical, indistinguishable particles is essentially a quantum-mechanical system, of which the generic position, momentum, and spin operators RPZ corresponding to the so-called "generic distributions'I3'(AII) of them in 3N-dimensional configuration spaces of position, momentum, and spin respectively are commutable every permutation operator H. i.e.
[R,'] = 0, [P,] = 0, [,] = 0, (A) (64) so that the Hamiltonian operator H of the system is commutable also with ji, i.e.
[H,n] = 0. (65) The system of identical, indistinguishable particles is to be defined by Eq. (64) instead of Eq. (65), since the system must remain unchanged for the exchange of any two particles which is equivalent physically to the exchange of every observable (including even the observables not commutable with H) concerned with any two particles. As is well-known
13See p 136 of the quoted book 34.
in group theory, every permutation operator T with the same degree forms together a group which is called the symmetric group, or the permutation group. We denote this group by a symbol O(N). The group O(N) corresponding to the system of N identical, indistinguishable particles has N! order and N degree, and is not abelian. The number C(N) of distinct classes of this group O(N) is equal to the number of distinct positive-integer solutions to the following equation:
N
Z jx = N. (66) j=l
This equation implies that a positive integer N is decomposed into x, parts of 1, x2 parts of 2, x3 parts of 3, ... , and xN parts of N. A. Young and G. Frobenius called the numbers (Xl,x2,x3,...,xN) "the partition of positive integer N." Some author represents this structure of partition of positive integer N by a symbole: (Xl x23x3 XN)
i , 1 2 x , 3 , 2...,)N x ) .
It is very convenient to use the so-called Young Tableaux in the study of the group O(N) with the degree N not so large.
The order mi(the number of distinct elements) of the ith class C. corresponding to the partition: S11,22 1 N
is given by
N (~i) 18
M. =N F(j 3 x(i)1) (67)
1 j=l i
and the total number of elements contained in C(N) classes must be equal to the order of the symmetric group O(N), i.e.
0(N)
Nm. =N! (68)
The particularly important knowledge in connection with quantum-statistical mechanics is, as will be seen in a later section, the property of the group O(N) that every element belonging to the same class has the same cyclic structure of permutation and vice versa.
Any permutation of degree N is a product of commuting cycles, and this decomposition is unique. Therefore, every element of the group O(N) can be completely defined by (a) its cyclic structure, i.e. the number of its cycles and their respective lengths, and (b) the numbers appearing each cycle and their order to within a circular permutation. If we reverse the order of the numbers in each cycle, we obtain the inverse element, i.e. the inverse permutation. Every cycle of a given length L is equal to the product of (i-l) transpositions, so that any permutation can be decomposed into a product of transpositions. This decomposition is not unique, but the number of transpositions involved will have a definite parity, either even or odd which we shall denote) (-1) . A permutation 11 is called even, or odd according to (-=) +1, or -1.
Let 1i.. be a transposition operator exchanging #i and #j particles of the system. It is, then, evident that
IT 1 . H 1 2 = I (identity) (69)
ij ji ij ij
so that the eigenvalues of 1I.. are +1 and -1.
UJ
Every element of the group O(N) must, at least, be unitary, i.e.
fl M IIi", (70) since for an arbitrary eigenket 11l? 2, ...,'rN_ of the generic positionoperator R of the system, we must have, from
that
,F. 7 . . . , r N 1 t I r ~ 2 . . r
(ii,2,.., N 1' 2c N.
1,I rl,,r2, ,N N
N N
I 5(l = X- r = 5(lr('l2''N . 1.,r2, ..,rN) X=l x >=I
.* lilih-I.
Among N! permutations contained in the group O(N), N!/2 permutations are even, and the remaining N! permutations are odd. These even permutations form together an invariant subgroup(called the alternating group) of order N!/2 of the group O(N), while odd permutations of N!/2 in number do not since the product of any two odd permutations is even and the identity(an even permutation) is not contained in them.
Since the Hamiltonian operator of the system is invariant under the group operations of O(N) according to Eq. (65), i.e.
[H,O(N)] = 0, (71)
the symmetric group O(N) belongs to a Schr~dinger group so that the state vector must be chosen to be also the eigenket of every element of the group O(N). This implies that the representatives of the state
vector in f-representation must be completely symmetric since the group D(N) contains every kind of possible exchange pattern of N particles in it. Therefore, we introduce the following operator S constructed from the linear combination of all elements of the symmetric group O(N):
def. def. 1 57(ï¿½)I
NS+ ' - N
(72)
where S is the compound operator of two operators S+ and S_ defined by
def. 1 ' def. S+ H N' , S_
N T N!
(73)
with the summation over all N! elements of the group Z(N). As we will see immediately, the operator S plays a role of symmetrizing completely every vector to which it is applied. This situation may be seen also as follows:
Let jI)be an arbitrary ket-vector, and I)' be the ket-vector produced by applying the operator S on .1), i.e.
Now, let us vector
DI = S 1). (74) apply an arbitrary element II' of the group O(N) on the ketWe see, then,
HI
N!ï¿½)~ (lIT IJ (ï¿½l)~ = S1)
(75)
I' I)'= (+)III )' , or O(N)I)' = (+, or -)1)' 14
14The sign "+" corresponds to the operator S+, and the sign "-" corresponds to the operator S-.
by noting the group property of O(N), i.e. the combination of two elements H' and H must be an element of the group O(N) with changing the original parity (ï¿½1) into its own parity (ï¿½1) '' multiplies by
(ï¿½1) .Equation (75) shows us that the ket-vector I)' obtained by the operation of the operator S on as in Eq. (74) remains unchanged within its sign by the operation of the group O(N). This implies that the ket-vector ID' is completely symmetric, or antisymmetric for every possible exchange pattern of N particles in the system under consideration. In this sense, we call the operator S+ the complete symmetrizer, and the operator S_ the complete antisymmetrizer.
We see easily that the following properties of the compound operator S of the complete symmetrizer S+ and antisymmetrizer S_: st =s, s =s, (76)
and
= = i (77) Equation (71) leads, furthermore, to [S,H] = 0, (78)
that is, the operator S is a constant of motion, and Eq. (76) shows us that the operator S can be regarded as an Hermitian projection operator(idempodent):
S = Sn (n=2,3,4,...). (79) We have also
s+ = s_ s+ = 0. (8o)
Since the generic position- and spin-operators of the system are commutable with an arbitrary permutation operator I as discussed already in Eq. (64), they are commutable also with the symmetric group O(N), i.e.
[R,O(N)] = 0. (81)
so that
[Rs] = 0 13 (82) Equation (82) implies that there must be simultaneous eigenket vectors between R and S. Let j ) be one of these simultaneous eigenket vectors, i.e.
1?) = ?I-) , siL) = s[r) . (83) The eigenvalue s of S can, by using Eq. (76), be found easily as follows:
2 = S21-) = SJ-) =
. 2 - s = o,= 0, 1. (84) This is trivial according to the general character of idempodent since the complete symmetrizer S is an idempodent. We note here that for symmetrizer S+ the eigenkets corresponding to s = 1,0 are symmetric and antisymmetric, and for antisymmetrizer S-, vice versa respectively.
We are now interested only in the eigenket 1r) corresponding to the eigenvalue s = 1, i.e.
sj7) = 17) . (83) One of the eigenkets of the generic position-operator R is
R~ 1' 2' r N r r1,r2,-, N)
but this eigenket IrI,r2,...,r N) is not the simultaneous eigenket of the of the compatible symmetrizer S with R. Now, let us apply the group
15We symbolize totally the generic position- and skin-operators by one notation f this time on, so that the eigenvalue of R contains also the spin coordinates.
operation O(N) on this eigenvalue equation by noting Eq. (81). We see then that
so that the N! ket vectors given by Sl,2... rN (86)
are also the eigenkets of R belonging to the same eigenvalue i. Furthermore they are linearly independent of each other since we have for the Hermitian scalar product of two arbitrary members of Eq. (86),
by using the group property of O(N) and Eq. (77) of the unitary property of I, where 11" = H-it,. We may say thus that a continuous eigenvalue I of the generic position-operator I is N!-fold degenerate with N! eigenkets given by Eq. (86). This means physically that we have N! possible ways of distributing N particles of the system for one generic distribution chosen. Therefore, we make, according to the ususal procedure used in quantum mechanics, a linear combination of these N! eigenkets of Eq. (86) to give the simultaneous eigenket of the completely symmetrizing operator S. It is as follows:
which is written actually in the following form: I ) = SIrl PN2,...,rN) (87) by using the definition given by Eq. (72). The proof showing that Eq. (87) is the eigenket of S corresponding to its eigenvalue 1 among two possible values is, by using Eq. (76), done simply as follows:
SI) r S Iel'2,...,rN) S I' i,'2, *ï¿½., )
or
"j) = IWe have not discussed, so far, the applications of the complete symmetrizer S+ and antisymmetrizer S_ for our physical theory. However, it has turned out that the operator S+ can be used for the symmetrization applicable to a bason-system, while the operator S_ can be used for the antisymmetrization applicable to a fermion-system.(19) We will distinguish hereafter every physical object concerned with a boson-system and a fermion-system to be needed in our further theory by using the subscripts, or superscripts, "+" and "-" respectively, i.e. the subscripts, or superscripts, "+" stands for a boson-system, and
"-", for a fermion-system.
Our above theory leads us to the generic position-ket vector,
1[ 1,2,.,rN) , (88)
and similarly, for the generic momentum-,or propagation-ket vector,
[~:] k [:: jk~2,..kN). (89)
The Hermitian scalar product between two eigenket vectors of
the compound form S of two kinds of symmetrizers S+.and S_ is, by using Eq. (76), given by
(.,.,)+ def 8(i, ') 6 (rI=i (90) I[ =
Since the ket vector rl,?2, rN) may be regarded as a simultaneous eigenket vector of N commutable indivitual-operators R.(j=l,2, ...,N), this ket vector can be decomposed into direct products of N indivitual eigenket vectors of R. given by
J
t i)j, (j=,2,...,N)
that is,
r' r2''". r N) r 1 )1 2)2 .......i-N)N "
(91)
At this point, it should be mentioned that we should distinguish substantially the two possible kinds of permutation operators 11 and Y, in which the former II is the permutation operator permuting the order of N indivitual particles and the latter q is the permutation operator permuting the order of N indivitual eigenket vectors, so that it should be
I1" Il7 2,. ."'N) r
rl, r2, ..., r = and, likewise, we have
Ilk2, ..., kN = The combination of Eq. (91)
7,12
(92)
kl,k2,...'N , ()I#I),9 with Eq. (87) makes us see the following
16We note here that we are concerned necessarily with writing the subscript j on these N indivitual eigenket vectors for the distinguishability of N indivitual eigenket vectors coming from the distinguishability of N particle's positions still remained, i.e. K.Irxj r jr x. for the jth position operator within the generic posioi6n operator R.
Vrl 2,-.2 ) '
alternant forms of two eigenket vectors J'> and jiz) of the generic position- and propagation-operators: != I l I2) ...... N S r2)2 ...... r 2
I~ 1)N 1 2) N N'N ï¿½ or in an abbreviated form: 1- 1 i) (93) NJ.) ï¿½
similarly,
1 > + = ., Iki) (94)
- ï¿½
which are formally similar to the so-called Slater determinant (or permanant) with a different realistic meaning.(720)
All of above basic discussions lead us to a conclusion that the state-ket vector I;t) of a system of identical, indistinguishable particles must, in the configuration representation, given by
= I(ï¿½l)" ( Jrl'K~r ''''rj;t, (95) IT
and, in the momentum or propagation representatign,
(kI ;t)+ = (kl~k2$,...,kNISI ;t)
wZ(ï¿½l)" ( ekl k2,...,rrtn. (96) which can be extended to that of any representation.
We are, now with the preliminary knowledge discussed above, in
a position to have the symmetrization of the representatives of Neumann's density operator in a representation. As shown already, this density operator has the form given by Eq. (63), i.e. a negatively exponential operator-function of single operator-variable H which is the Hamiltonian operator of the system under consideration, and Eqs. (78) and (71) show us that
[D(N),D =0, [S,D ] = O. (97) Equation (97) requires that the representatives DP(',') defined by D (-, ')ef' ('I D 1-, (98)
must be symmetric, or antisymmetric under the symmetric group operation D(N) in the system of identical, indistinguishable particles. This requirement forces us to use the symmetrized representation I-> in Eq. (98). We have, thus by combining Eq. (98) with the symmetrized representation I) given by Eq. (87),
D P(7,7') = e FP(71.72"..., N Is e- HIrl~r2, ...,)r N>
which is, by using the second of Eq. (97) and Eq. (76), reduced to a simpler form given by
D (, ) eF l,, .. ~-PHsI-I -,I'
P 2). .rI2,.-.,rN> (99) Similarly, we have, in k-representation,
P ) = e ,2,...,kNIe Skl,k2,...,0)) .
Likewise, we can construct also the representatives of Neumann's density operator in any representation other than I-, or i-representation according to our convenience for the practical purpose with which we are concerned. However, the form given by Eq. (99) is most preferable since the potential function contained in the Hamiltonian operator H is, in the most cases of realistic problems, given by a complicated form in terms of the coordinate ( 2...N) of the representative point of the system in 3N-dimensional configuration space.
Now, we introduce the following notation to avoid duplicate writings in our subsequent discussions:
s) rdef " .~2... r N); 1 ' s) k ke-f I~,2, k. N
Let us, for example, insert the identity operator I taking the form given by
f N- dN Z's)< (st (100)'
between two operators exp(-PH) and S in Eq. (99), to have the following expression of diagonal element convenient for a practical calculation:
. exp(pFpï¿½) ZdNe(101)
D P r,r)= )3N, (ï¿½i)" dNe e'PH(-ih',)e ..- . (101)
(2t) N.1 0
by using
i.7/( )3N/2
(-r'sl'k's) z- e /2ir, (102) where the superscript "+" stands for Boson-system, while "'" for Fermion-system as stated already, and we note here that r= (103)
in accordance with the unitary character of every element of the symmetric group O(N).
As shown in Eq. (101), the explicit determination of Neumann's
density matrix is dependent upon finding the explicit function obtained by operating out the operator exp(-PH) on the function
exp(iZ.*), i.e.
e'PH(-ih, r) eilt.7 14
e (104 )
in addition to the completion of the given integral in i-space, which compels us to define the so-called Bloch function UP(',) given by Eq. (62). We shall return to this problem again in Chapter III.
2.4. A New Formal Expansion Theory of the Quantum Partition Function of Canonical Ensemble
We are, in this section, interested in developing a new formal expansion theory of the quantum partition function Q using the characters of symmetric group O(N) within the framework of our fundamental theory of quantum statistical mechanics. As well known, this partition function Q plays a role of a bridge between the microscopic and macroscopic worlds, and also between canonical and grand canonical ensembles. This function Q is found from the normalization character of the trace of Neumann's density operator D stated in Eq. (53) already. Physically, this normalization character implies the following meaning: Let us suppose that a quantum-mechanical system composed of N identical particles is closed in a volume V which is in contact with an isothermal bath. It is, then, evident that every particle can be found quite certainly in the volume V. The trace of the density matrix DP( , ')
must, therefore, be normalized for the integration over the 3N-dimensional domain V in the 3N-dimensional configuration space of the system in question, i.e.
V3N
fNrlD I 1) = 1, (105)
V
which leads us to have the same result as Eq. (53).
For our later reference, we shall start with deriving the socalled quasi-quantum partition function Q(0) of the system of N identical particles. The generic position-eigenket vector fr) is, for the system of identical, distinguishable particles, given simply by the direct product of the indivitual position-eigenket vectors, i.e.
N
[:) = 11 1jr-- j's) (106)
j=l
The substitution of Eq. (106) into Eq. (53) give us
Q O) = d3N ( II ('rij) e-PH( 111 ij)) V i=l j=l
(107)
=YN d3Nr(-' sIe- H: r
V
Equation (107) is the quasi-quantum partition function corresponding to the original Maxwell-Boltzmann statistics. On the other hand, the quasiquantum partition function Q(0) corresponding to the corrected MaxwellBoltzmann statistics is given by Q(O) = Q(c)/N, (108) according to Gibb's intuitive foresight which is even incorrect under the point of view of modern quantum mechanics.
If the identical particles of the system are independent, i.e. there is no interaction between particles, the total Hamiltonian operator H of the system is split into the sum of the individual Hamiltonian operators h.(j=l,2,...,N),i.e.
N
H = Z hj ,
and Eq. (108) is, then, given by
Q(c) = N ~ def 19
where the determination of the individual partition function qP belongs to the problem of single particle. If the single particle is structureless, the operator h takes simply the following form: h2
-m
where m is the mass of single particle. For the particle with an inner structure, we need to solve the Schr~dinger's time-independent equation of single particle given by
hfn) =njn)
and we calculate, then, the function q in the way like
j -I d3-(-1n)(nle-hI = e"Pen( n o n
by using the identity operator constructed from the complete set (In) of Hamiltonian eigenket vectors of single particle.
Now, we introduce the quantum partition function Q P This is done by combining the symmetrized eigenket vector f of generic
110)
)}
position-operator R given by Eq. (87) with Eq. (53). We have, thus,
Q - e-F P f d3 (s I e'-H S1's) (ii vN
or
e~ = N-Z (+I) N rrsje-PHI's) , (112)
V
by using Eq. (72). The summation in Eq. (112) runs over all the N! elements contained in the symmetric group O(N). We rearrange this summation in such a way that every term of Eq. (112) belonging to the same class of the group O(N) is collected together. Before doing this, we examine the simplest solution to Eq. (66). It is
xI = N, x. =0, (j=2,3,...N), which leads to the following cyclic structure: (,N,20,30,.3 .0 .,NO0)
and the number m1 of distinct elements contained in the class with this cyclic structure is, in accordance with Eq. (67), given by mI = ,
i.e. only one element. This is nothing but the identity I which forms by itself a subgroup and a class of the group O(N). Since the parity
(ï¿½i)Pi of the ith class Ci with the cyclic structure given by
X ix(i), 40~i x(i) i ) x~i)
(3 2 3 4 ,..N ],(113)
is, in general, determined by the positive integer pi given by
( Xi) + (~i) + (i) + . +(i)+. (21) (114)
Pi 4 x6)+x + ' 2j "'
the parity (+I)pl of this class is Pl = 0+0+ ... (even)
We call, hereafter, this simplest class the first class of the group O(N).
We put, thus, the term corresponding to this first class at the first place in the rearrangement of all terms of Eq. (112). We have then,
Zmi
Q~ fd r r S(ï¿½l) (115)
N!N .. . . . . . i.
V i=2 X=l
where the summation inside of-------represents the total sum of integrals J x defined by
1
corresponding to all elements H of the ith class Ci with the parity determined by the positive integer pi of Eq. (114), and C(N) is the total number of classes of the group O(N).
Now, we are going to prove that the integral J. defined by
1
Eq. (116) does not depend upon the superscript X but only the subscript i. In another word, the integrals J i(X=l,2, ...,m1) corresponding to the permutations lli(=l,2,...,mi) belonging to the same class C. are all the same. The proof for this is as follows:
Let H i([=l,2,...,mi) be in the same class C.. Then, there
1
must be such elements i(Xl'2'''"mi) that
11 '-" r/ lï¿½i '(x l'2,''''mi),
in accordance with the definition of class, and the element is unitary in accordance with Eq. (76). Furthermore, it, or its Hermitian conjugate is commutable with the operator exp(-PH) in accordance with Eq. (65). These characters of lead to
X Ple-HI-as) ï¿½ 7sze Iï¿½~>i
. 1 1
and
d3Nr = d3N(n X r)
U
Therefore, the integral given by Eq. (116) is written also as
J dN((D '-r)<( O 'e -PH On,\X
=i '
VN
y putting V' r~, which is equal to the integral J!, i.e.
1 1J!,(X=2,3,..,m.) (117) This shows us that the integral J. defined by Eq. (116) is independent
1
of the superscript X. We substitute Eq. (117) into Eq. (115), and then use Eq. (67) to have
Qï¿½ = I'. N -P s
+ (N) ï¿½ N)xPi -i
i=2 Lj=
jdef . fd3N..<,s .e'HI-+"S), J- d r.,IvN
with
(118) (119)
where H! is any one permutation-operator among all elements belonging to the ith class Ci of the symmetric group O(N).17
We call this expansion of Eq. (118) the symmetric-group-class expansion formula for the quantum partition function of canonical ensemble. The quasi-quantum partition function Q(c) given by Eq. (108) and corresponding to the corrected Maxwell-Boltzman statistics appears as the first class term in our expansion formula of Eq. (118), which does not make the distinction between a boson and fermion on their specific behaviors. This distinction is first seen in the second class term. It is supposed that this second class term plays a main role in the quantum-effect due to the spin-characteristics of indivitual particle since the other class terms of the expansion with more complicated cyclic structure than that of this second class term may be considered to be quite small in their numerical contributions to the function Q at a temperature different from zero.
As Eq. (118) shows us that the quantum partition function Q may be regarded as the superposition of every partition function Q(i)(i=1,2, 3,...,C(N)) belonging to every indivitual class Ci(i=I,2,3,..,C(N)) of the symmetric group 4(N), i.e.
C(N)
Q Qi . (120) i~l
171t is a very much impressed fact that the cyclic structure of H! is equivalent to the structure of irreducible cluster integral in the
L
Mayer's classical theory. We may call the integral Ji the Quantum irreducible cluster integral in analogy with the classical nomenclature. We may apply the Mayer's cluster theory to our expansion theory.
This decomposition of Q into its components is unique since a symmetric group with a given degree and therefore a given order determines uniquely the structures of its classes, and the number C(N) of total components of Q P is equal to the number of distinct classes, i.e. the number of inequivalent irreducible representations of the symmetric group. In other words, the components of Q given by
(1) (2)Q(C(N)))
have one-to-one correspondences to the inequivalent irreducible representations of the symmetric group corresponding to the system under consideration. In this sense, our expansion theory of the quantum partition function is considered to be reasonable from the grouptheoretical point of view.
There are two kinds of jobs to determine our quantum partition functions Q exactly, i.e.
(a) the complete determination of (C(N)-l) numerical
coefficients of the integrals Ji(i=2,3,...,C(N)),
in addition to the positive integer pi(i=2,3,...,C(N))
concerned with the parity of the ith class, i.e. to
find the solutions of C(N) in number to Eq. (66),
(b) the complete determination of C(N) integrals
J.(i=1,2,... C(N)), where J is the integral of
first class term of expansion in Eq. (118).
Realistically, since we are dealing with the macroscopic system containing a vast number of particles, the number C(N) may be also large enough. According to the Hardy-Ramanujans asymptotic formula,(22,23)
this number C(N) is, if N-o, given by
=(N e r2/ /V N. (121) We have, for example, for N = 6.02486x1O23(gmol)" , C (N ) -- 10 1 1
We are, therefore, concerned with the determinations of numerical coefficients and integrals of the order of 101013 in number respectively in the calculation of Q of even one gmol substance. However, as stated somewhat already, the first few class terms of Eq. (118) with simpler cyclic structures would be decisively significant. This makes feasible the solution of the realistic problem with which we are concerned.
We will show below these first few class terms which we have obtained by using the identity of Eq. (100)' constructed from the ' -eigenket vectors jk)'s!?) In the use of this identity, the C(N) components of Q are given by
(1) Q (c) ( 1/2)N 3N 3N, (122)
Q~l d~ J r JUNI:_"-' i~k .r (22 VN . .
= (i) -i
Q ~j =l (ï¿½l 1 Jx~i)' J.
with
def. 3 3N(-,s 1111,s~e r k VN ds
(i=2,3,4,...,C(N))
We should take the following binary cyclic structure:
flN-2,21,30,...,OO...,NO
corresponding to the second class term Q , of which the coefficient is given by
N x! 2)
11 j J x(2). = 2[(N-2)!],
j=l I
and the parity integer P2 is, in accordance with Eq. (114), given by
P2 = + 0 + 0 + ... (odd). We have, thus, after a rearrangement,
+ (1/2,)3N fd3Nï¿½
2-(N-2)'
ï¿½ v
3N 1 2e,- *(P i -l.
Ite e- . .
! .. ....... ('123)
Likewise, we have
Q(3) (i/2)3N d3Nrd3Nk Exp[i(kl-k2).71+i(k2 -k 3) 2+i(k3-ki)- 3]
3"(N-3)! Li(
ilt-r_ -P iq.r'
xe e i (124)
----------------------------------Corresponding to the following tenary cyclic structure:
, 4 0...,j ,...)N03 )p3 = 0, and, in general, we have
0~) = (ï¿½tl)jl(l/2jt)3 f N VN
d3NH ik.r
:e e e
00 -- - - -
j-1
x Exp[i , ( -'+i)*}'r + i(-l)r
X= 1
Q (2)
(125)
63
corresponding to the following cyclic structure:
(1,N-203 ,..-, 1,...,N J, pj =j-1.
In this way, we can determine any kind of desired class term Q(i).(7)
As we have seen above, the function F (k,r) defined by
(k,r ) def. -ite -PH i!' (126) with a temperature-boundary condition given by = 1, (127)
plays, in some sense, a basic role in the practical determination of Q (i) (i=1,2,3,...,C(N)) in our expansion theory. This function F(_k+r) is connected with the function of Eq. (104), or the Bloch function U defined by Eq. (62) in the difference of only the pure imaginary exponential function. This function has many-interesting but unaccessible characters(7) which give us sometimes even a hopelessness in the concrete solution of a problem of quantum statistical mechanics. We will be confronted with the same problem in our subsequent theory of correlation function in next section.
As our first trial approximation of Q , we may take
Q Q ()= Q(C) (128) This approximation does not give the distinguishability between the boson and fermion systems. Our next possible approximation is to take ï¿½ Q(1) + Q(2)
Q+ P Q P P , (129) or in its explicit form:
+ (1/2,)3N 3N 3N- , N(N-l)ei( l'k2)'( l-r)]
N djr Yd kF--+ 2 e
VN 0 (130)
where FW( ,r) was defined already by Eq. (126). The different behavior between the boson and fermion systems is seen firstly in this second class-term approximation. The following diagram may give us an aid to distinguish the quantum partition functions Qand for a boson and fermion systems:
Q 1st class (2nd class
Q "approximation "approximation Q(Boson)
Boson --"
Fermion".
Splitting "'. Q(Fermion)
This situation of splitting due to the exchange effect of particles is, in some sense, similar to the perturbed splitting of energy level due to the orbital-orbital, orbital-spin, spin-spin intetactions, the relativistic effect of electrons and other kinds of interactions in an atom. Finally, it should be mentioned that our expansion theory is comparable with Kahn-Uhlehbeck's quantum cluster expansion theory. However, there is a great difference between them ideologically.
2.5. The Formal Theory of Quantum Pair Correlation Function
A diagonal element D. ,r), i.e.
of the Neumann's density matrix given by Eq. (98), is interpreted physically as the probability density of finding the representative point of a quantum-mechanical system of identical, indistinguishable
particles at any one point among all points corresponding to the same generic configuration of the system specified by (J23...r-N) in 3N-dimensional configuration space. Therefore, the probability density of finding simultaneously any two particles of the system at two points specified by two position-vectors 71 and 72 in a volume V in realistic space, is given by
dV'rrID P rM
where
]def N 3 , ,def. vN-2 j=3
The quantum pair correlation function gP(n;7lr2) is, then, defined by ,n def.lim V2 fdT'(lDI (131) (V,N)-.o i,'
with
,lim = (finite) (V,N)-ooN/V) n A further change in Eq. (131) can, by using the equation right above Eq. (99), be made in the following form:
e(n; r2 (132) (V,N)- ,
where we note, in accordance with Eqs. (111) and (76), that
Eutoa d3r Ivd3r dd'(-rc'soSien o-usa) p(133)
e V2 Ir IU
Equations (133) and (132) gives us a boundary condition of quantum pair
correlation function given by
r d r2m (nvrr22 (134) V-_o
which is obtained also from Eq. (131) and (105).
Next, let us insert the identity operator given by Eq. (31) between two operators exp(-PH) and S in Eq. (132), and use Eq. (16). We have, then, a somewhat concrete form given by
g (n;?l,,2 = A e"E Jd'fy(r)* Z{.(r), (135) n T
where ]{.(r)* is the complex conjugate of the symmetrized energy-eigenn
function ([) defined by
n( r de--f 2 , 2, .. ï¿½ -N IS I >, ( < I >
with
= E.( (r) , (136) and A P is an normalization operator defined by def. im 2 P
A P (VN)-' V e
The physical meaning of Eq. (135) is self-evident in connection with the law of large number of Eq. (48) and Eq. (24). Since the integral in Eq. (135) is a function of FI and ?-2, we may write it also in the following form:
byN,V;r1l, 2) (137) by defining
'
which is temperature-independent. Equation (135), or Eq. (137) is the exact form of the quantum pair correlation function. This exact form will play a role of guidance in the determination of the approximate form of the quantum pair correlation function in a realistic problem.
We are now interested in deriving two specific forms of this pair correlation function at two specific absolute temperatures, i.e. 00K(P=) and Oï¿½K(P=O) from Eq. (137). Let us change Eq. (137) into the following form:
g= A e [g+ e ng], (139)
n
where represents the summation over all possible except t = 0 corresponding to the ground state of the system. We note, in Eq. (139), that
(E.-E) > 0, g = (a bounded function independent of P).
This character of boundness of g, comes from the wave character of energy eigenfunction T_.. We have, thus in the limit of P- in Eq. (139), g = im Ae gPE (140)
and then integrate Eq. (140) with respect to 1I and 2 over whole realistic space by noting Eq. (134) and the character of g, normalizable to unity, to have
im F
,V - e e = 1. (141) The combination of Eq. (141) with Eq. (140) gives us =V r im V2 jdr'Yo(-)*o(r), (142) V 2(n ) =(r)
by using the defintion given by Eq. (138). This is the relationship between the quantum pair correlation function g. at absolute zero temperature and the ground state function (r) of the system. It should be pointed out from Eq. (142) that there is no contribution from any excited state to the quantum pair correlation function at absolute zero temperature, which implies physically that the system under con.0
sideration is exactly at its ground state at 0 K. The exact determination of the function g. depends upon finding the exact energy eigenfunction of ground state by solving directly, or indirectly Eq. (136).
At extremely high temperature P-0, we have, by using Eq. (137),
i= 2 Ldi2 y I T y ,, (143) g o V-- V2 =_ i ,_n n n n
which shows us that the system is likely equally at any possible excited state including its ground state at -0K. As shown in Eq. (143), we need to find all the energy eigenfunctions in order to determine the exact form of the quantum pair correlation function g at - K in contrast to the case of g.
As shown in Eqs. (143), (142), and (135), it is quite obvious that the quantum pair correlation function has, in general, an oscillatory character due to the wave character of the energy eigenfunction of the system. Since we have, from Eq. (134), fd1 id3'2(g -1) = O; (144) this oscillatory character must occur around the value 1, and its
feature depends upon the type of interaction between particles in the system. In the realistic problem, this interaction is varying very slowly with the increase of I 2-i 'l for 1?2 -? 1 large enough. This causes the slowly varying feature of 2, or g-, defined by Eq. (138) for Ir2- 1 large enough, so that we may have, from Eq. (138),
lim 1im2 = im iY 2rd,
= Jd3NkI{.-2 = 1 (145) by noting the normalization condition of -.(7). Now, we combine Eq. (145) with Eq. (137) by replacing the operator A by its original one, to see that
ï¿½ im
im (n;' = 1 (146)
by using also Eq. (49) and the second of Eq. (50).
Equation (132) defining the quantum pair correlation function may be written also as
g(n;l, = AdT 'r'sISe' s (147) by using the normalization operator A,, Eqs. (96) and (76), and also
g Z() d ' slle-PH S)s (148) P ~ ~ I --2 = ! II
by substituting Eq. (72) into Eq. (147). We may, in analogy with Eq. (115), expand Eq. (148) in the following form.
(n1 'r A 1 (-'sle- Hr's
C(N) m.
+ r(ï¿½)Pi:Z (,nl,2): (149) i_2
i=2 ' X=l
with
p n;l,)ef A r 'd (r' s e rI J l2 N! j , r (150) This is the symmetric group class expansion formula for the quantum pair correlation function. This expansion has a different feature from that of the quantum partition function Q of canonical ensemble. All of the element functions P (X=l,2,...,mi) in the same class (the ith class C.) are not the same. They have their own structure, and some of them are equivalent to each other. We call this the fine structure of the same class element. This feature is due to the volume element given
N
by dT' = H d r. instead of d3Nr. However, there is always a number of
j=3 J
the different fine structures less than the total number m. of the elements in a class C.. In another word, we can find always the same integrals defined by Eq. (150) in a class. This fact is formally very much similar to the diagram expansion theory for the classical radial distribution function!25) It may be possible that we develop a new formal expansion theory of the quantum pair correlation function in analogy with the idea of the classical expansion theory. In Eq. (149), the first integral is the first class term corresponding to the identity element, which gives us no distinguishability between the boson and fermion systems, even though it makes the quantum effects but no exchange
effects of the system under consideration. We call, therefore, the function g(l) defined by
(1, .)def. (11H
1 irs2 N!e
the quasi-quantum pair correlation function of the system in question in analogy with the nomenclature in the case of the quasi-quantum partition function defined by Eq. (108). Eq. (151) corresponds to-the corrected Maxwell-Boltzmann's statistics. The second part of double summation represents the exchange effects of identical, indistinguishable particles in addition to the quantum effects. According to Landau and Lifshitz, these exchange effects are small at high temperature but very significant at very low temperature. 26) We can develop also the approximation theory of our quantum pair correlation function in analogy with that of the quantum partition function of canonical ensemble developed in the previous section.
Let us change Eq. (151) into the form convenient for a concrete calculation by using the identity operator constructed from the eigenkets of wave vector operator. It gives, then, us
(1) = N: 2 )3N rdT d eHik
n; 2 . ,.... . ..... .... 152)
We see here again the same function F (k,r) defined by Eq. (126) inside the dot rectangles !as that encountered in the previous section.
We will, in Chapter III, study the concrete determination of the quantum pair correlation function defined by Eq. (152) and the function F (QI,') by using the W.K.B. approximation method.
CHAPTER III
THE W.K.B. APPROXIMATION FOR THE QUANTUM RADIAL DISTRIBUTION FUNCTION
-For those well ordered motions, and regular paces,
though they give no sound unto the ear, yet to the
understanding they strike a note most full of
harmony-(Sir Thomas Browne)
3.1. The Concrete Determination of the Function F l,7)
We rewrite, here again, the definition of the function F (,) given by Eq. (126):
F(k,r) =e e e . (153) This function can be constructed also from the diagonal element D r r of the so-called quasi-Neumann's density matrix D (rr')(disregarding the particle exchange effects) defined by
D r , 'ldef-feres) . (154) by inserting the identity operator I given by Eq. (100)' between the operator exp(-PH) and the ket vector 17"s) in Eq. (154), where 17's) is defined by the expression immediately subsequent to Eq. (100). Historically, the use of this identity operator I was introduced firstly by Kirkwood in developing his expansion theory of the so-called quasiSlater sum 27) Uhlenbeck and Beth have repeated the Kirkwood's expansion calculation and extended it in their paper,28) and later they have used it for calculating the quantum virial coefficient at low temperature.29) 72
Furthermore, de Boer has taken also their expansion expression in developing his theory of the equation of state by introducing the socalled de Boer factor %-h/ai ).30'31) On the other hand, the approach of finding the Slater sum by solving the Bloch equation has been done by Husimi, Mayer, and Band!32)33) Recently, ter Haar, Landau and Lipshitz have, in an elegant way, developed this expansion theory in their books!26,34) All of these expansion theories have been constructed so that they are rather powerful for the theory of the equation of state of quasi-quantum fluid over some low temperature.
However, we are, now in this dissertation, concerned with a
somewhat different expansion theory of the function F (k,r) defined by Eq. (153) from those done by our predecessors. Our expansion theory to be developed hereafter must be rather useful for the theory of quantum pair correlation function g (n;l,' 2) than for the theory of the equation of state. Furthermore, it should be made within that it can be led finally to the formal classical case with an effective potential varying also with the temperature. This requirement comes from the Broyles' program18 that the Percus-Yevick, or the Convolution HyperNetted chain nonlinear integral equation of the classical radial distribution function can be applied to the numerical determination of the function gP(n;l,r2) by using the computing machine. We will discuss about these two nonlinear integral equations in next section.
18This program is suggested first by Dr. A. A. Broyles, Professor of Physics, University of Florida, Gainesville, Florida, U.S.A. To the author's best knowledge of this paper, this program is the most powerful method in the theory of fluid, at least at present, compared with the several methods suggested by other authors.
Now, let us, in terms of the function F (Ur) defined by
Eq. (153), write the diagonal element of the exact Neumann's density matrix given by Eq. (101) in the following form:
...= exp(PF +) 11;3N- ik. (-11F(r (
DP(r,r) (2 )3NN:Z (ï¿½l) ke (155)
and at first consider the case of an ideal fluid. Our function F( k,_r) becomes, in this case, simply F (_k,) = e"Phk /2m, (156) and therefore Eq. (155) takes the following explicit form:
Dï¿½0- exp(-F*) rI - 2 A2 19 (157) D P3N Z (ï¿½l.O
where m is the mass of a particle of the ideal fluid under consideration, and X, its thermal de Broglie's wave length. This result shows us that there is the apparent attraction between Bose-Einstein particles, while the apparent repulsion between Fermi-Dirac particles.
It seems that the function F(kr) may not, in general, be
expressed explicitly by an elementary function in the case of realistic imperfect fluid except that of the simple imperfect fluid. We will quote below the three results of DP(7,7) obtained in the three simple systems respectively.
19We have used here the formula given by
dx exp(-a2x2+ibx) = (2a exp(-b2/4a2))
00
(i) A system of one particle enclosed in a 1-dimensional
box with a finite size:
D (x,x) = exp(lF-) i e 4gx2/%2
( - X
(ii) A system of a linear harmonic oscillator with an angular frequency a):
exp(PF 1)[ 2Wia ... 12 __ x2tah2
D (x,x) = l-exp exp( 2h h)
(iii) A system of two identical, indistinguishable particles
enclosed in a 1-dimensional box with-a finite size:
+ . exp(4F) /4X/i2)( x 2 (x,x) 2 2 (le )(l-e )
ï¿½ (e"A(xlx2) 2/2- e-I(xl+X2)2/X2 (17,35)20
Incidentally, it is valuable to note that the Neumann's density matrix DP(rr') can be found also by solving a differential equation with a boundary condition, instead of calculating the integral given by Eq. (1i5). We will show the correctness of this statement as follows: Let us consider the function X (?,?') defined by
dxf. , "eH11'> (158)
X ') ' Fe DP (7,7') = ( e
or D ( , ') = e X (i, '). (158)' This function XP(7,7') with 7'='r is equivalent to the Slater sum with
20In these 3 systems, exp(PFP) can be easily found by the normalization condition of D and the calculation of the Gauss' probability integral.
the difference of the factorial coefficient N! (see Eq. (55)). The differentiation of Eq. (158) with respect to the parameter P gives us the following result:
X (irt) = - (riHePHPI)
or
X P - HX 21 (159) by taking out the Hamiltonian operator H before the bra vector (rj and changing it into its equivalent differential operator H(-ih6,r) in the r-representation. Eq. (159) is the Bloch equation. The Bloch equation is, thus, satisfied also by the function XP(r,r') as well as the Bloch function UP(k*,) defined by Eq. (62). But their boundary condition for temperature are different from each other. The temperatureboundary condition for the function XP(W,'') can be found immediately from Eq. (158), i.e.
lim <(/Y,>)<(,sIS S), (160)
where the function ('[:') is defined by Eq. (90). The variable r' goes in the solution of Eq. (159) as a parameter through this boundary condition.
Now, we are going to have a change in the function F (rF)
defined by Eq. (153). Before doing this, we introduce the total kinetic energy operator T and potential energy operator 0 of the system
21It is noticeable that this equation is formally equivalent to the Schr~dinger's time-dependent equation.
77
under consideration, and assume that the operator (I does not depend upon the particle-momentum operator, i.e.
ef _h-2 def
T .r 2m c2, =" ). (161) The Hamiltonian operator H of the system is, then, written as
-*2
H =- + O(r) = T + 0 (162) by defining a microscopic-characteristic constant c of the particle of the system given by
c 'h2/2m , (163) and Eq. (153) takes the following form:
F (k,r) = e .e e (164) We insert, next, the identity operator exp(PT)exp(-PT) between the operator exp(-Pï¿½-PT) and the operand exp(ik.r), to have
F (k,r)e =eikre'--T eT(e-rPTe ik.r)
or
F ( k,-r) = e-Hc f (O i') (165) by noting that the part of the parenthesis ( ) becomes e= e e
-'2
with a c-number exp(-c43k ), and defining the classical Hamiltonian Hc, an operator'lp and a function fP(ce;k,r) given by
def. - .2
d f
f--kr3 e e
As shown in Eq. (165), the explicit determination of the function F'(k,r) is dependent upon finding the explicit form of the function fp(c;g,r) defined by the last of Eq. (166). It is believed that there are two ways to find the function f,(c;k,r) in this kind of formalism. The one way is to change the operatorl)P into such a form that it can operate out on the operand exp(i. ). The other way is to find the function fp(c;k,r) directly by solving a differential equation satisfied by it. According to the author's study of this paper, as we will show immediately, the former method is suitable for very high temperature region, and the latter is believed to be powerful for the region up to the fairly low temperature. However, it seems that our method to be presented hereafter in this dissertation would fail in the region of extremely low temperature.
A. The First Method Effective for Very High Temperature:
We expand the operators given by the second of Eq. (166) in the ascending power series of P as follows:
00
= () X!4v
or
= n. Pnn) (167) (X+4+v=n)
by defining
Z ndA (')dfl_),,n (-T)PTV
P~ ~~ EX1iv (,!4l!n (168) (%+ +v=n)
where the summation goes over all 3-dimensional lattice points (X, ,v) satisfying the condition given by
%++v =n; n = 0,1,2,3,...;
(169)
Xm. 0, P ;- 0, V "7,O
The number f(n) of the lattice points (%,4,v) satisfying the condition of Eq. (169) for a given positive integer n is seen easily to be given by
= = (n+2)(n+l) 22 f(n) n+3-lC3-1 n+2C2 2 (170) by observing the same countings as that of Bose-Einstein statistics. We will calculate the first four terms of Eq. (167), i.e. the terms corresponding to n = 0,1,2,3, below, respectively. The first term: The number f(O) of the lattice point (X,4,v) satisfying
the condition X+g+v = 0 is, from Eq. (170), f(O) = 1,
and it is (X,pv) = (0,0,0). We have therefore from Eq. (168), = I (identity) (171)
2PThe notation, e.g. nCm represents the total number of combination ways choosing m things among n things (n7?-m).
The second term: The number f(l) under the condition X+4+v = I is,
from Eq. (170),
f(l) = 3,
and the 3 lattice points are
(X,4,v) = (1,0,0);
(0,1,0); (0,0,1).
Hence, we have, from Eq. (168),
4'2(i) ___ + T T =
! 0!0! 0!1!0! + 0 -0.
(172)
The third term: The number f(2) under the condition X+4+v = 2 is
f(2) = 6, and the 6 lattice points are given by
(2,0,0); (1,1,b); (0,2,0); (1,o,1); (0,0,2); (0,1,1).
We have, thus from Eq. (168),
- 2 + + T2 T2 + T
2! 2! 2 1! 1. 1! '
r
;B 2-[ T,O] C'-] [ () + 2(q -]
(cD+T)T
11!
(173)
The fourth term: The number f(3) is given by f(3) = 5C2 = 10, and the
calculated result of the operator j(3) is
P 3!
+ 2( 0
L 3 = -- [(- C),, ï¿½]]
24 -2-.
= _U 2 ((S ) + 4(.3
3!
as follows:
+ 4~1
(174)
double dot product of Methods of Theoret-
23The double dot ":" represents the scalar two dyadics. e.g. see P. M. Morse and H. Feshbach; ical Physics, Vol. 1, p54 et seq., (1954).
(% . .V) =
The number f(4) of the lattice points for the fifth term is f(4) = 6C2
15, and for the sixth term, f(5) = 7C2 = 21, and so on.
We have, thus on substituting Eqs. (174), 173), (172), and (171) into Eq. (167),
33.2 + + a 2
2. 3:
a ((6 0) + 4(6237 0 + 4(6aO)*6)] + (175) The substitution of this expansion of the operator af2 into the last of Eq. (166) gives us the following result:
2f ( ) + 2il.-_(Z] + 1 [2(O)
4 ,-*
- d( 4 c) + 4ik' 0 - 4k._' '"6)] + .... (176) This result allows us to find the function F (k, ) in the power series form of P by combining Eqs. (176) and (165). This is the case only at the point (",) in phase space where the power series of given by Eq. (176) is convergent under the assumption that the given potential
(r) is an analytic function at the point r in configuration space. It is evident, from the power series form of Eq. (176), that this expansion result is useful only for the region of very high temperature. B. The Second Method Effective for the Region up to
the Fairly Low Temperature:
Before we construct the differential equation satisfied by the function f (x;7,'), we investigate, at first, the several properties of this function for the sake of the familiarity with it, and also for the
reference of our later discussions. We will describe briefly the proofs needed for us in our discussions. Property 3.1: The dimensionless function fP (1,) is neither a real
function nor a pure imaginary function. It is really a complex function having the formal form given by fo = f(r) + if(i), (i= _T), with (176)'
f (r) cos k.r cos k.r + sin k. sin '.',
f(') cos _k-rd2 sin Kr - sin ZiJ2 cos i.
Proof: The formal forms of fr) and f(i) are resulted from
the use of Euler's theorem given by
e = cos k.r ï¿½ i sin k.r and it is obvious, from Eq. (176), that f(r)f(i)to at every
point (-,i*) in the phase space.
Property 3.2: The function f contains both of the independent variables
r and T within it, i.e. we must write fp = f P(r,) r
Proof: The dependence on k is self-evident in accordance with
the form of the operator lfl. Actually, it is proved as
follows: Let us apply the gradient operator o in k-space on
the function f We have, then after a simple calculation,
or = i" . or af o,
since we have, always, [d. P, ] A 0, (p#O). This shows us that
f contains, at least, the variable k in it. Similarly, we can
show that
6f 0.
Property 3.3: The boundary value fP (ik,) at = 0 (T-o) is unity at
every point (k,r) in the phase space, i.e.
=f)] 1 . (177) Proof: This is obvious, from the boundary condition of the
function F (;,7) given by Eq. (127) and Eq. (165). Or, we can
see this property from the second and the third of Eq. (166)
by noting (P)P=0 = I.
Property 3.4: The function fP(k,r) is completely symmetric for the
inversion, reflection (within its complex conjugate), and
exchange operations in phase space, i.e.
f (-k,-r) = f (kr) I
f (-k, r) = f(k,r), (178) 11 f P(kr) = f (k , r)
where 1 4 is the exchange operator exchanging two particles at the two points (r.,k) and (r ,k ) in the phase space, and the
symbol "*" represents complex conjugate.
Proof: This property is quite clear, from the definition of the
function fP(k,r) given by Eq. (166), on the basis of the fact
that k-r = (invariant), T = (invariant), and (D(r) = (invariant
due to the isotropic property of the realistic space) under these
operations.
Property 3.5: The function fP(k,r) is related to the Bloch function
UP(k, r)by
-(PHc-i-.r) 0
U (k,r) = e fPk)f(kr) (179)
Proof: We have from Eqs. (165) and (153),
-Ik - Hi~ - H ï¿½-k
e e e e f(k,r)
This leads to Eq. (179) with the use of the definition of the
Bloch function given by
.+ _4 24
U tk r)def -PH ik-r
(k, r) =*e- e (180)
which satisfies the Bloch equation:
(6+H)UP = 0. (181)
We finish our brief description about the property of the function fP with this. Next, we construct a differential equation satisfied by the function f,, which makes the most important basis of our subsequent theory.
Let us differentiate the last of Eq. (166) with respect to the parameter P referring to the second of Eq. (166). We have, then,
Pf = e -i A,'* ( )eik'
= e - ir- Teik-r (e-ik- eTe-e ik r)(e-ik ' eikr) orT
Sf =k f - e (182) 24Exactly speaking, this function may be called the quasi-Bloch function(disregarding the particle exchange effects) in contrast with the definition of Eq. (62).
by operating out the operator T on its operand in the first term of the right-hand side. The operator inside of k . in Eq. (182) is the similar transformation of the kinetic energy operator T of the system by an operator exp(PD-ik-7). We change the differential equation of second order given by Eq. (182) into a more accessible one. We operate out the operator T on its operand exp(-++ik.r) in the right hand side of Eq. (182). We have, then after some calculations,
P f 2 _ - 00( 2 0) 25
+Oa2 (SO") - 2ii. (P(Sï¿½() - iJfP . (183) We note here that the operator inside of [ 3 in the right-hand side of Eq. (183) is a linear operator. Therefore, Eq. (183) is a linear partial differential equation of second order, and the solution to this equation must satisfy the boundary condition given by Eq. (177). The last complex term of Eq. (183) given by 2iCak. (PfP - 'f } (184) in an unfortunate term in finding the function f . We can not see the general reason to neglect this unlucky term. Nevertheless, it is desirable to exclude this unlucky term for the sake of our more accessible treatment of Eq. (183). However, we will show below the fact that we are led to a very dangerous result if we neglect this unhappy term.
The neglection of this unlucky term corresponds to regarding
25This is, in fact, an alternative form of the Bloch equation, since Eq. (183) can be derivable from the Bloch equation of Eq. (181) by substituting Eq. (179) into it, and vice versa.
the function f possibly as a real function, and makes Eq. (183) be written as
Sf = f - C + [ )-( )]f. This equation can be reduced to the following form: f =- epoT e-O fp, (185) which is obtainable also by putting k = 0 in Eq. (182). Since Eq. (185) does not contain the variable k, the function f takes the form given by
This result obtained by neglecting Eq. (184) implies physically that the contribution of the particle-momenta (=hk) to the function f comes mainly from the zero momenta (k=0), i.e. the particles at rest. However, this is not physically the case. Now, Eq. (185) can be, after a simple arrangement, changed into the form of the Bloch equation given by
(T+Z)v Vv, or 3v = Hv (186) by putting
v e f.C~r (187) Noting that Eq. (186) is a linear equation, we may assume the solution to it in the following form: = e n(r) (188)
with a constant E-.. We have, then on substituting Eq. (188) into
n
Eq. (186),
This is the familiar time-independent Schr6dinger's equation of the system under consideration. Therefore, the function r{.{( ) contained in a solution of Eq. (188) to the Bloch equation of Eq. (186) is really equal to the eigenfunction corresponding to the energy eigenvalue Eof the system. Since Eq. (186) is a linear partial differential equation, the general solution of it is given by the linear combination of the types of the functions given by Eq. (187), i.e.
E n
v= C-n e- E () , (189)
n
where C-.'s are the coefficients of the linear combination. We find
n
from Eqs. (189) and (187), finally
n o(i (r) " (190)
n
Now, let us impose the boundary condition of the function f given by Eq. (177) on this solution of Eq. (190), to have
1 = Z *n(r).
n
We have, from this equation, formally
6 JVNd r !*) (191) VN n
by using the orthonormalization condition of the energy eigenfunction f(r) given by
J 3N ' Tt,( -') Tt(_) -n n(, (191),
The substitution of Eq. (191) into Eq. (190) gives us
f e () e'E[ fd3N T' (r')]T(r).' (192) VN
n
This is the formal form of the function f corresponding to the neglection of the unlucky term of Eq. (184). We can, then, find the Neumann's density matrix D (rr) by combining Eqs. (192), (165), and (155), and then using
d~~~~~~~~~ Ne C i ï¿½(-1) (1Ne (-I1/2 (AIII) = , (193)
where the quantity ? is defined in Eq. (157). It is as follows:
+ 7,r e e Y d r ,
n ~ ' N.X (194)
Now, let us find the exact formal form of the Neumann's density matrix D P(',) in terms of the complete set of the energy eigenfunctions f's. This can be done easily by inserting the identity I of Eq. (31) between the bra vector (71 and the operator D in Eq. (98) with "' = r', and referring to the definition of Tv() immediately above Eq. (136). It is , then, as follows:
= e- n e e n _r ) T(rt() (195) n n
We see, thus on comparing Eq. (195) with Eq. (194), that the neglection of the term of Eq. (184) is equivalent to approximating the complex conjugate Y(r) of the energy eigenfunction to the function inside of the dot rectangle ;-: in Eq. (194), i.e.
fnd r A 3Nr'(7)) 3+N) -If
- -- N e ( (195)' N nN1%k3
This implies that the complex conjugate 7n(i) for every possible lattice vector A of quantum number is approximated by the Neumann's density matrix of quantum ideal fluid given by Eq. (157) within their different constant-coefficients. Furthermore, this approximation of the righthand side of Eq. (195)1 may become a function given by the product of the real function and complex constant, which can make the approximated density matrix of Eq. (194) be complex in non-consistence with the realistic physical menaing of the Neumann's density matrix. There is also a possibility leading to a serious difficulty that the integral given by Eq. (191) is divergent even if the normalization of Eq. (191)' is satisfied. Thus, it is very dangerous to neglect the term given by Eq. (184) in Eq. (183), even though it is an unlucky term desirable to be neglected.
Now, we want to return to the discussion of solving our basic equation given by Eq. (183). We are going to find the solution f ( ;k,) in its series expansion form of the parameter (X, or contained in it. Before we develop our theory of the solution f,, we examine, at first, the numerical characters of Ce and P for the sake of our intuitive foresight into the convergence of the series expansion
form of the solution f to be assumed in advance. The parameter defined by Eq. (163) has the dimension given by
[a] = [ML4T-2] = [energy][length]2 = [erg cm ]
and its numerical vlaue depends upon the mass of the particle under consideration with the inverse proportionality of the mass. For example, for the elctron, proton and Ne-atom, they are roughly as follows:
5.5xi0"28 for e-; 3.0xlO for p; .xlO32 for Ne, 2,
where the unit of a has taken as "erg cm'. It is noticeable that we have, for Ne-atom, numerically a k2 (l.9xlO32)
The parameter P defined by Eq. (49) has the dimension given by
[P] = TMIL-2T2] = [energy]"I = [erg- , and its numerical value is given roughly by P 7.2xlOIS/T (erg-') The numerical value of O is therefore roughly as follows:
W 4: 4.OxlO' 12/T for e-; 2.2xlO-15/T for p; 1.3xlO'16/T for Ne with the dimension and unit given by [p] = [L2] = [area] = [cm2 Since the numerical value of a is very small as shown above, the
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