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The equation of state in quantized kinetic theory and quantum statistical mechanics
Physica XV, no 10
October 1949
THE EQUATION OF STATE IN QUANTIZED KINETIC THEORY AND QUANTUM STATISTICAL MECHANICS b y H. S. GREEN Edinburgh University
Summary A rigorous derivation is given of the equation of state of ~ fluid on the basis of quantum statistical mechanics, and it is confirmed that the thermodynamic pressure so obtained differs from the kinetic pressure derived from quantum hydrodynamics or otherwise. The difference is evaluated and it is shown that it becomes appreciable at very low temperatures. 1. Introduction. Although the application of the quantum theory to gases and crystalline solids has been known for some time, a rigorous quantum theory of liquids has been lacking until recently, partly Owing to the fact that there existed no classical counterpart to which the correspondence principle could conveniently be applied. Some progress towards the elimination of this difficulty has been made b y K i r k w o o d l ) , and b y B o r n and G r e e n * ) , so that it has become possible to attempt the formulation of a quantized kinetic theory of liquids. That has been done b y B o r n and G r e e n 8), and independently b y G u r o v 4). It is not difficult to derive from the quantum theory the macroscopic equations of hydrodynamics in their usual form; this provides incidentally an expression for the pressure in terms of the intermolecular forces, which does not differ formally from the accepted classical formula, though numerically there are corrections arising from the quantum theory which have been recognized for some time and calculated b y U h l e n b e c k and B e t h 6 ) , and b y D e B o e r 6 ) for condensed gases. It has been pointed out in a recent paper b y D e B o e r ,) that the resulting equation of state is easily obtained b y an application of the virial theorem. The pressure defined and derived in this way will be called the kinetic pressure. There exists, - -
882
- -
THE EQUATION OF STATE IN QUANTIZED I~INETIC THEORY
883
however, and alternative definition of the pressure in terms of the rate of change of the free energy of the fluid with variation of volume, and the question arises whether this thermodynamic pressure is the same as the kinetic pressure, obtained previously. There is no a priori reason for supposing that this should be so: it is well known that the 'temperature' defined in kinetic theory in terms of the mean kinetic energy of the molecules deviates from the thermod y n a m i c temperature at low temperatures. At high temperatures the distinction between the two pressures can indeed be shown to disappear, and' the thermodynamic method of U h 1 e n b e c k and B e t h gives the kinetic pressure; but at low temperatures this method fails and the two pressures are not necessarily the same. The physical consequences of such a difference in the kinetic and thermodynamic pressures at low temperatures are, however, farreaching; it is implied that the work done in adjusting a difference in pressure need not appear as macroscopic kinetic energy, but may be converted reversibly into thermal energy b y a process other than normal dissipative effects. This fact has been used b y the author s) to explain the abnormal properties of liquid He II, b u t in view of its fundamental importance it is clearly essential to establish the difference between the two pressures beyond reasonable doubt. The derivation of the thermodynamic equation of state is therefore reexamined here. The difference between the thermodynamic and kinetic pressures will be shown to become appreciable at low temperatures, and will be evaluated b y a rigorous mathematical development. 2. The Equation o~ State Derived /rom Quantum Statistics. The method here employed for the evaluation of the thermodynamic pressure is that commonly used, proceeding from the definition
where F is the free energy and V the volume of the fluid, which will be supposed to contain N similar molecules; t h e s u f f i x T indicates that the temperature is to be kept constant in performing the-differentiation. According to statistical thermodynamics 9), F is expressible in terms of V and T b y means of the relation F = - - k r log Z, Z = f . . . . f V exp ( - - fill) (r, r') dr~ . . . dr N (2)
884
H.S. GREEN
where k is B o 1 t z m a n n's constant, ~ ---- (kT)-1, and (exp - - ~ H ) (r, r'), which is short for (exp --/~H) (r I . . . . rN, r'1 . , . . r~v), is the representative of the operator e-/~H in a representation in which the co-ordinates rl, r 2 r N of t h e N molecules are diagonal. The operator H is the Hamiltonian energy of the system, given in terms of the intermolecular potential energy 9(r) and the m o m e n t u m operators p~ b y . . . . .
1 H=
K+
•
=--
2m
Z p ] + ½Z279(r,k ),
rik=r k-r
i, p ~ - - . , (3) z Or~ where m is the molecular mass. To carry out the differentiation indicated in (1), one writes r i = Lri, where L a = V, returning t o the original variables when the differentiation has been performed, thus:
Z = LsNf... f (exp --~H) (L~, Lr) d~... drN; v OZ= L OZ NZ "f ; 0 OV 30-L= +~J...__fr,.-~ri(exp--~H)(r,r)dr...drN = {.;... f~
'y
= ~
...
~.{r,
(exp-/~H)
(r, r)} dr~...dr~v
(4)
{ # ~ . r ~ e x p ( - - / ~ H ) - - e x p (--fill) p~.r~} (r, r) d r 1 . . . d r m
To obtain the final result, the formulae h0 ( p . r A ) (r, r') = - i- - - Or h (Ap. r) (r,r') = ___2_ r '
i
. {rA
0
"~r'
(r, r')};
A (r, r'),
(5)
valid for a n y operator A, have been used. F r o m (1) and (2) it now follows that
kT oZ P --
ikT [ [S{p,.r, 3hVZJ'" 0 ~ .
-Z O-V--
exp (--/~H) - -
exp (--/~H) p~.r~} (r, r) d r t . . . d r N.
(6)
This is the t h e r m o d y n a m i c equation of state, though not in a very convenient form as the integral has still to be evaluated. This requires considerable care. One has
THE EQUATION OF STATE IN QUANTIZ-ED KINETIC THEORY
__i 27(pi.r~H__HPl.r~ ) (r, r ) = - - - - 1 -3h
~
885
27p~ + { 2727r~k. 09(r~k ) _
3m ~
~k
~ik
OH
= - - . { K + {~3= V0--#.
(7)
Hence, i] it could be assumed, ]or any ]unctions ](H) and 9(H) o] H, such as ](H) = e-pu, that .
.
.
/ ( ~ ) v ov --v3-¢- ](~) (9(H)} (r, r) dr1 . . . . drN (a)
vanishes, i.e., that the order of the two operators ](H) and V OH/OV can be interchanged under the 'trace', one could show that the thermodynamic pressure is IFzJ
{(~ N - - ~t3) exp - - ~ N } (r, r) d r 1 . . , drN.
(9)
This integral, as will be shown subsequently, can be reduced to a form representing the kinetic pressure, so that on the assumption stated it would follow that there was no difference between the kinetich and thermodynamic pressures. Such an assumption would clearly require the most careful justification, as by a similar argument (with i/h 27 pl. r i in place of V (O'H/OV)) it would follow from (6) that p itself identically vanishes. In fact, the assumption is incorrect, as a straightforward calculation, in the next section will show. The trace tr (AB) of the product of two operators A and B is equal to tr (BA) 0nly under special conditions, which are discussed with the help of illustrative examples in the appendix to this paper.
3. The Evaluation o/the Thermodynamic Pressure. For the evaluation of the commutator C ~ 27 {p~. r~ exp (--fill) - - exp (--/~H) p~. ri}
(1 0)
appearing on the right-hand side of (6), a lemma in quantum algebra is useful. This is to the effect t h a t if # and v are any two operators, and ](v) is any power series in v, ~, 1(,,) - - l(,,)/, = 1'(,,) ~
+ 1"(,,) ~
+ 1'"(,,) ~
+ ...
(11)
886
H . S . GREEN
where the primes indicate differantiation with respect to v, a n d (k = I, 2, etc.)
/.1 = ~*v--v~*, /*~+t = / * k v - - v / * k
(12)
This obviously holds for/(v) = v, and, assuming that it is correct also for/(v) = v~, one has
/*v"+l - - v " + t # = (/*v"-- v"/*) v + v"(/w
v/*)
(13)
m = 1 kin~
1) , , + , _ ~
__~'(n+
--~= t\
m
/*m,
SO that it is true for/(v) = v "+l as well. The lemma therefore holds for all powers of v and any power series in v. Combining (11) with the conjugate equation /'1
/* l(,) - - 1(,)/* = ~ 1'(,) - - ~
/*2
/*s
I"(~) + ~ 1'"(~)
.....
(14)
one obtains /* l(v) --l(v)/* = ½~/*2k+l ll2k+ll + l12k+ll/*2k+t k=o __
2k + 1)!
½~ ~u2k/C2k)__/12k,/'2,~ k=,
(15)
(2k!)
where the commutators/*2 1 " - - f ' / * 2 , / * 4 / ' v - - [ 'v/*4, etc. can be eliminated by means of the same formula with/*2,/'4, etc. replacing /*. In this way one has finally
/*t(,,)--t(,,)/* = ½ ~. & (/*~.+, l (~'+') + 1(~*+')/*~.+,) (2k+!)! k=O
; t 0 = 1, t x - - - - ½ ,
L2 - -
1, 4 3 -
17 4 )
(16)
....
The general formula for ;~k is k k121 \12 ~.k=Aok+,,=,Z (--1)"A.k{(l+n)2k+L(1--n)2k+t}.A.k=,=. E t n + l )(i)"
(17)
This result can now be applied to the formula (I0), by substituting /* = Z p~. ri, v = H , and/(v) = e-/~u ; one finds i
THE EQUATION OF STATE IN QUANTIZED KINETIC THEORY
887
C = ~ 2.(-/~) 2"+' ½ {/.2.+, exp ( - / ~ H ) + e x p (-/~H)/.2.+,}/(2m+ 1, (18)
ra=l
where
/zl =
Pi. ri H--
/*2 = / . 1
H--
Hpi. ri =
i~ (2K --
•),
H/* 1
/*3 = [12 ~
etc.
H/*2,
(19)
Substituting (18) in (6), one derives p = (3i
vz)-,
~,k~2k f
° = (2k + l)[ oo
"'"
j~ {/*2k+I exp ( - - ~ H )
+ exp (--~H)/.2~+1} (r, r) d r I
....
dr N
(20)
A similar series has been obtained b y another, rather more direct, method, by M a s s i g n o n xo). It shows immediately that the assumption mentioned in t.he previous section is incorrect. The successive terms/*l, #3, etc. are readily evaluated, as follows #1 = ¢h { 2 / / - - 27 E ~ (r,,)}, ~o (r,,)} ---- ~o(r,,) + ½r,,. 0qo(r,,) ." Ori,
/~
=
2---m ,!7 27 Pi," - -
i k ~
+
Orik
- -
Orik
•Pi,
i~ ~ 27 27 / ~lo(r,,) F 2p,, /.3 = ~ ' * ~P,,.P,,. 0r,k~r,---~2727 27
_
m
i
k
t
Ori,
, Pi* =
P* - -
Pi ;
(21)
~Pir'k). P,,-t ~w(r,,) . P,,-p,,] "~r,,Or,-------, -Or,,Or,-----~, Ori~
These m a y now be substituted in (20), and the integrals evaluated in terms of q u a n t u m mechanical averages. The average value of a n y function [(r I . . . . ru, Pl . . . . PS) of coordinates and m o m e n t a is defined to be ? ---- 2 ,
...
½ l / e x p ( - - ~ N ) + exp ( - - ~ H ) / } (r, r) d r l . . , drN. (22)
If ] is a function depending on the coordinates r I . . . . r e of q molecules only, such an average m a y be simplified b y introducing the molecular distribution functions nq (r I . . . . rq) defined b y
%-- ( N - - q ! ) Z
"'"
(exp--~N). ( r , r ) drq+ l . . .
drN.
.(23)
H.S. GREEN
888
Thus one has
a~(r12! -- V. / n 2(r12) r12" arl2 dr 12
Zi 27krik" ~ 272727 0~°(rik) . O~o(r~) _
~ ~ 0r, k
0r~
(24)
V f f n 3 (r12 , r13 ) 0~°(rl-------2) 0~v(rl3) drl2 drla. .,fJ
0r12
0rla
Similarly a generalized temperature T¢(r 1 . . . . rq) may be defined by the equation N! 2
P~[ (r, r) dr, "'" drN; -}- exp ( - - ~ H ) 2m/
(25)
for q = 1, Tq reduces to the kinetic temperature, which, however, in quantum statistics deviates from the thermodynamic temperature at low temperatures. With the help of these definitions, one obtains "Or
~--2ihaV{/n2(r)kT2(r)O2~p(r~)m
(26)
Or.Or / ;na(r's) O§°(r~)Or~P(s) ' o drs ds}
It follows by the substitution of (26) into (20) that P = Pl + P3 + Ps + . . . . .
Pl = -# kTi
f
- - ~ •2 (r) r. 09(r) dr" Or
P3 = - - ~ -~
.
n2 (r)
kT2(r ) 0~v(r) d r Or. Or
(r, s) O~°(~r) . O~:)drds} Or The higher terms P2k+l in the series contain the factor (h/kT) 2k and clearly become significant at sufficiently low temperatures. A rough estimate is sufficient to show that P3 attains a value comparable with p~ in liquid helium at a temperature of about 2°K. It is, therefore, clear that in liquid helium the difference between the kinetic and thermodynamic pressures is sufficiently large to account for anomalies of the kind suggested in the introduction to this paper. ~ffn3
THE EOUATION OF STATE IN ~UANTIZED KINETIC THEORY
(~89
4. Appendix. It has been pointed out in the text t h a t it is not in general permissible to assume t h a t the trace of a c o m m u t a t o r vanishes, and is in fact incorrect in connection with expressions such as (8) a n d (10). If A and B are matrices, one has trace (AB - - BA) = Z Z (A,,,.B.~ - - B~.A.m)
(I)
nlq¢
which vanishes provided Z ' Z C.,,, = 2:27 C . .
(II)
where C.,,, = A,..B,,,.. This condition is always satisfied when m and n assume a finite n u m b e r of values, b u t not always when the double series is infinite, even t h o u g h b o t h sides of (II) exist and are finite. F o r example, if Cmn
Sin. - - - -
=
Smn-
, m >
Sm , n--I -
1, n >
S r a - - 1~n -~- S m - - I s~t--I ,
1; Sin. =
0, m <
0, n <
0,
(III)
one has
Z Z C ~ . = lim Sin1 = 1 ; Z Z C ~ . = lim Sl. = - - 1 ?tt
m - - - ~ OO
?ttn
(IV)
n----~ oo
Such double series are characterized b y the fact t h a t t h e y do not satisfy the analytical condition for convergence. A similar s t a t e m e n t can be made concerning the integral operators A (x, x'), B(x, x') ; the trace tr (AB - - B A ) vanishes provided
f f C(x, x') dxdx' = f f C(x, x') dx'dx,
(V)
where C(x, x') = A (x, x') B(x', x). This condition will be satisfied if C is a continuous function of b o t h variables and the integrals exist and converge uniformly. It is not satisfied, however, when either A or B involve the singular m o m e n t u m operator p = h/i O/Ox which is used in q u a n t u m statistics. For example, let A = 9(x) and B = = p (Oq~(x)/Ox) + (09(x)/Ox) p; t h e n tr(AB)= tr
i
~t
dx, tr ( B A ) = - - * ? *" J ~ x )
(AB) - - BA) = 2ih -( ( O*~2dx J \ Ox /
dx; (VI)
which does not vanish for a n y function 90(x). This example is not
890
T H E EQUATION OF STATE IN QUANTIZED KINETIC T H E O R Y
merely of analytical interest, but is important physically, as it embodies in a simplified form the calculation of •3 N # -- #i given in the text. Received J u n e 9th, 1949.
REFERENCES I) J . G . K i r k w o o d , J. chem. Phys. 14,180, 1946; 15, 7 2 , 1 9 4 7 ; 1 7 , 3 3 8 , 1 9 4 9 . 2) M. B o r n and H . S . G r e e n , Proc. roy. Soc. A 188, I0, 1946; J89,103,1947; 190, 455, 1947. 3) M. B o r n and H . S . G r e e n , Proe. roy. Soc. A 141, 168, 1947. 4) K . P . G u r o v, J. exp. and theor. Phys. U.S.S.R., 18, I I0, 1949. 5) G . E . U h l e n b e c k andE. Beth, Physica 3, 729, 1935;4,915,1937. 6) J. d e B o e r, A m s t e r d a m Dissertation, 1940. 7) J. d e B o e r , The Caloric and Termal E q u a t i o n of State in Q u a n t u m Statistical Mechanics, Physiea 15, 843, 1949. 8) H . S . G r e e n , Proe. roy. Soe. A 1 9 4 , 244, 1948. 9) c . f . R . P . F o w l e r & E. A. G u g g e n h e i m , Statlstical T h e r m o d y n a m i s s . 322: Camb. Univ. Press 1939. 0) D. M a s s i g n o n , Comptes Rendus, P a d s , 228, 62, 1949.
October 1949
THE EQUATION OF STATE IN QUANTIZED KINETIC THEORY AND QUANTUM STATISTICAL MECHANICS b y H. S. GREEN Edinburgh University
Summary A rigorous derivation is given of the equation of state of ~ fluid on the basis of quantum statistical mechanics, and it is confirmed that the thermodynamic pressure so obtained differs from the kinetic pressure derived from quantum hydrodynamics or otherwise. The difference is evaluated and it is shown that it becomes appreciable at very low temperatures. 1. Introduction. Although the application of the quantum theory to gases and crystalline solids has been known for some time, a rigorous quantum theory of liquids has been lacking until recently, partly Owing to the fact that there existed no classical counterpart to which the correspondence principle could conveniently be applied. Some progress towards the elimination of this difficulty has been made b y K i r k w o o d l ) , and b y B o r n and G r e e n * ) , so that it has become possible to attempt the formulation of a quantized kinetic theory of liquids. That has been done b y B o r n and G r e e n 8), and independently b y G u r o v 4). It is not difficult to derive from the quantum theory the macroscopic equations of hydrodynamics in their usual form; this provides incidentally an expression for the pressure in terms of the intermolecular forces, which does not differ formally from the accepted classical formula, though numerically there are corrections arising from the quantum theory which have been recognized for some time and calculated b y U h l e n b e c k and B e t h 6 ) , and b y D e B o e r 6 ) for condensed gases. It has been pointed out in a recent paper b y D e B o e r ,) that the resulting equation of state is easily obtained b y an application of the virial theorem. The pressure defined and derived in this way will be called the kinetic pressure. There exists, - -
882
- -
THE EQUATION OF STATE IN QUANTIZED I~INETIC THEORY
883
however, and alternative definition of the pressure in terms of the rate of change of the free energy of the fluid with variation of volume, and the question arises whether this thermodynamic pressure is the same as the kinetic pressure, obtained previously. There is no a priori reason for supposing that this should be so: it is well known that the 'temperature' defined in kinetic theory in terms of the mean kinetic energy of the molecules deviates from the thermod y n a m i c temperature at low temperatures. At high temperatures the distinction between the two pressures can indeed be shown to disappear, and' the thermodynamic method of U h 1 e n b e c k and B e t h gives the kinetic pressure; but at low temperatures this method fails and the two pressures are not necessarily the same. The physical consequences of such a difference in the kinetic and thermodynamic pressures at low temperatures are, however, farreaching; it is implied that the work done in adjusting a difference in pressure need not appear as macroscopic kinetic energy, but may be converted reversibly into thermal energy b y a process other than normal dissipative effects. This fact has been used b y the author s) to explain the abnormal properties of liquid He II, b u t in view of its fundamental importance it is clearly essential to establish the difference between the two pressures beyond reasonable doubt. The derivation of the thermodynamic equation of state is therefore reexamined here. The difference between the thermodynamic and kinetic pressures will be shown to become appreciable at low temperatures, and will be evaluated b y a rigorous mathematical development. 2. The Equation o~ State Derived /rom Quantum Statistics. The method here employed for the evaluation of the thermodynamic pressure is that commonly used, proceeding from the definition
where F is the free energy and V the volume of the fluid, which will be supposed to contain N similar molecules; t h e s u f f i x T indicates that the temperature is to be kept constant in performing the-differentiation. According to statistical thermodynamics 9), F is expressible in terms of V and T b y means of the relation F = - - k r log Z, Z = f . . . . f V exp ( - - fill) (r, r') dr~ . . . dr N (2)
884
H.S. GREEN
where k is B o 1 t z m a n n's constant, ~ ---- (kT)-1, and (exp - - ~ H ) (r, r'), which is short for (exp --/~H) (r I . . . . rN, r'1 . , . . r~v), is the representative of the operator e-/~H in a representation in which the co-ordinates rl, r 2 r N of t h e N molecules are diagonal. The operator H is the Hamiltonian energy of the system, given in terms of the intermolecular potential energy 9(r) and the m o m e n t u m operators p~ b y . . . . .
1 H=
K+
•
=--
2m
Z p ] + ½Z279(r,k ),
rik=r k-r
i, p ~ - - . , (3) z Or~ where m is the molecular mass. To carry out the differentiation indicated in (1), one writes r i = Lri, where L a = V, returning t o the original variables when the differentiation has been performed, thus:
Z = LsNf... f (exp --~H) (L~, Lr) d~... drN; v OZ= L OZ NZ "f ; 0 OV 30-L= +~J...__fr,.-~ri(exp--~H)(r,r)dr...drN = {.;... f~
'y
= ~
...
~.{r,
(exp-/~H)
(r, r)} dr~...dr~v
(4)
{ # ~ . r ~ e x p ( - - / ~ H ) - - e x p (--fill) p~.r~} (r, r) d r 1 . . . d r m
To obtain the final result, the formulae h0 ( p . r A ) (r, r') = - i- - - Or h (Ap. r) (r,r') = ___2_ r '
i
. {rA
0
"~r'
(r, r')};
A (r, r'),
(5)
valid for a n y operator A, have been used. F r o m (1) and (2) it now follows that
kT oZ P --
ikT [ [S{p,.r, 3hVZJ'" 0 ~ .
-Z O-V--
exp (--/~H) - -
exp (--/~H) p~.r~} (r, r) d r t . . . d r N.
(6)
This is the t h e r m o d y n a m i c equation of state, though not in a very convenient form as the integral has still to be evaluated. This requires considerable care. One has
THE EQUATION OF STATE IN QUANTIZ-ED KINETIC THEORY
__i 27(pi.r~H__HPl.r~ ) (r, r ) = - - - - 1 -3h
~
885
27p~ + { 2727r~k. 09(r~k ) _
3m ~
~k
~ik
OH
= - - . { K + {~3= V0--#.
(7)
Hence, i] it could be assumed, ]or any ]unctions ](H) and 9(H) o] H, such as ](H) = e-pu, that .
.
.
/ ( ~ ) v ov --v3-¢- ](~) (9(H)} (r, r) dr1 . . . . drN (a)
vanishes, i.e., that the order of the two operators ](H) and V OH/OV can be interchanged under the 'trace', one could show that the thermodynamic pressure is IFzJ
{(~ N - - ~t3) exp - - ~ N } (r, r) d r 1 . . , drN.
(9)
This integral, as will be shown subsequently, can be reduced to a form representing the kinetic pressure, so that on the assumption stated it would follow that there was no difference between the kinetich and thermodynamic pressures. Such an assumption would clearly require the most careful justification, as by a similar argument (with i/h 27 pl. r i in place of V (O'H/OV)) it would follow from (6) that p itself identically vanishes. In fact, the assumption is incorrect, as a straightforward calculation, in the next section will show. The trace tr (AB) of the product of two operators A and B is equal to tr (BA) 0nly under special conditions, which are discussed with the help of illustrative examples in the appendix to this paper.
3. The Evaluation o/the Thermodynamic Pressure. For the evaluation of the commutator C ~ 27 {p~. r~ exp (--fill) - - exp (--/~H) p~. ri}
(1 0)
appearing on the right-hand side of (6), a lemma in quantum algebra is useful. This is to the effect t h a t if # and v are any two operators, and ](v) is any power series in v, ~, 1(,,) - - l(,,)/, = 1'(,,) ~
+ 1"(,,) ~
+ 1'"(,,) ~
+ ...
(11)
886
H . S . GREEN
where the primes indicate differantiation with respect to v, a n d (k = I, 2, etc.)
/.1 = ~*v--v~*, /*~+t = / * k v - - v / * k
(12)
This obviously holds for/(v) = v, and, assuming that it is correct also for/(v) = v~, one has
/*v"+l - - v " + t # = (/*v"-- v"/*) v + v"(/w
v/*)
(13)
m = 1 kin~
1) , , + , _ ~
__~'(n+
--~= t\
m
/*m,
SO that it is true for/(v) = v "+l as well. The lemma therefore holds for all powers of v and any power series in v. Combining (11) with the conjugate equation /'1
/* l(,) - - 1(,)/* = ~ 1'(,) - - ~
/*2
/*s
I"(~) + ~ 1'"(~)
.....
(14)
one obtains /* l(v) --l(v)/* = ½~/*2k+l ll2k+ll + l12k+ll/*2k+t k=o __
2k + 1)!
½~ ~u2k/C2k)__/12k,/'2,~ k=,
(15)
(2k!)
where the commutators/*2 1 " - - f ' / * 2 , / * 4 / ' v - - [ 'v/*4, etc. can be eliminated by means of the same formula with/*2,/'4, etc. replacing /*. In this way one has finally
/*t(,,)--t(,,)/* = ½ ~. & (/*~.+, l (~'+') + 1(~*+')/*~.+,) (2k+!)! k=O
; t 0 = 1, t x - - - - ½ ,
L2 - -
1, 4 3 -
17 4 )
(16)
....
The general formula for ;~k is k k121 \12 ~.k=Aok+,,=,Z (--1)"A.k{(l+n)2k+L(1--n)2k+t}.A.k=,=. E t n + l )(i)"
(17)
This result can now be applied to the formula (I0), by substituting /* = Z p~. ri, v = H , and/(v) = e-/~u ; one finds i
THE EQUATION OF STATE IN QUANTIZED KINETIC THEORY
887
C = ~ 2.(-/~) 2"+' ½ {/.2.+, exp ( - / ~ H ) + e x p (-/~H)/.2.+,}/(2m+ 1, (18)
ra=l
where
/zl =
Pi. ri H--
/*2 = / . 1
H--
Hpi. ri =
i~ (2K --
•),
H/* 1
/*3 = [12 ~
etc.
H/*2,
(19)
Substituting (18) in (6), one derives p = (3i
vz)-,
~,k~2k f
° = (2k + l)[ oo
"'"
j~ {/*2k+I exp ( - - ~ H )
+ exp (--~H)/.2~+1} (r, r) d r I
....
dr N
(20)
A similar series has been obtained b y another, rather more direct, method, by M a s s i g n o n xo). It shows immediately that the assumption mentioned in t.he previous section is incorrect. The successive terms/*l, #3, etc. are readily evaluated, as follows #1 = ¢h { 2 / / - - 27 E ~ (r,,)}, ~o (r,,)} ---- ~o(r,,) + ½r,,. 0qo(r,,) ." Ori,
/~
=
2---m ,!7 27 Pi," - -
i k ~
+
Orik
- -
Orik
•Pi,
i~ ~ 27 27 / ~lo(r,,) F 2p,, /.3 = ~ ' * ~P,,.P,,. 0r,k~r,---~2727 27
_
m
i
k
t
Ori,
, Pi* =
P* - -
Pi ;
(21)
~Pir'k). P,,-t ~w(r,,) . P,,-p,,] "~r,,Or,-------, -Or,,Or,-----~, Ori~
These m a y now be substituted in (20), and the integrals evaluated in terms of q u a n t u m mechanical averages. The average value of a n y function [(r I . . . . ru, Pl . . . . PS) of coordinates and m o m e n t a is defined to be ? ---- 2 ,
...
½ l / e x p ( - - ~ N ) + exp ( - - ~ H ) / } (r, r) d r l . . , drN. (22)
If ] is a function depending on the coordinates r I . . . . r e of q molecules only, such an average m a y be simplified b y introducing the molecular distribution functions nq (r I . . . . rq) defined b y
%-- ( N - - q ! ) Z
"'"
(exp--~N). ( r , r ) drq+ l . . .
drN.
.(23)
H.S. GREEN
888
Thus one has
a~(r12! -- V. / n 2(r12) r12" arl2 dr 12
Zi 27krik" ~ 272727 0~°(rik) . O~o(r~) _
~ ~ 0r, k
0r~
(24)
V f f n 3 (r12 , r13 ) 0~°(rl-------2) 0~v(rl3) drl2 drla. .,fJ
0r12
0rla
Similarly a generalized temperature T¢(r 1 . . . . rq) may be defined by the equation N! 2
P~[ (r, r) dr, "'" drN; -}- exp ( - - ~ H ) 2m/
(25)
for q = 1, Tq reduces to the kinetic temperature, which, however, in quantum statistics deviates from the thermodynamic temperature at low temperatures. With the help of these definitions, one obtains "Or
~--2ihaV{/n2(r)kT2(r)O2~p(r~)m
(26)
Or.Or / ;na(r's) O§°(r~)Or~P(s) ' o drs ds}
It follows by the substitution of (26) into (20) that P = Pl + P3 + Ps + . . . . .
Pl = -# kTi
f
- - ~ •2 (r) r. 09(r) dr" Or
P3 = - - ~ -~
.
n2 (r)
kT2(r ) 0~v(r) d r Or. Or
(r, s) O~°(~r) . O~:)drds} Or The higher terms P2k+l in the series contain the factor (h/kT) 2k and clearly become significant at sufficiently low temperatures. A rough estimate is sufficient to show that P3 attains a value comparable with p~ in liquid helium at a temperature of about 2°K. It is, therefore, clear that in liquid helium the difference between the kinetic and thermodynamic pressures is sufficiently large to account for anomalies of the kind suggested in the introduction to this paper. ~ffn3
THE EOUATION OF STATE IN ~UANTIZED KINETIC THEORY
(~89
4. Appendix. It has been pointed out in the text t h a t it is not in general permissible to assume t h a t the trace of a c o m m u t a t o r vanishes, and is in fact incorrect in connection with expressions such as (8) a n d (10). If A and B are matrices, one has trace (AB - - BA) = Z Z (A,,,.B.~ - - B~.A.m)
(I)
nlq¢
which vanishes provided Z ' Z C.,,, = 2:27 C . .
(II)
where C.,,, = A,..B,,,.. This condition is always satisfied when m and n assume a finite n u m b e r of values, b u t not always when the double series is infinite, even t h o u g h b o t h sides of (II) exist and are finite. F o r example, if Cmn
Sin. - - - -
=
Smn-
, m >
Sm , n--I -
1, n >
S r a - - 1~n -~- S m - - I s~t--I ,
1; Sin. =
0, m <
0, n <
0,
(III)
one has
Z Z C ~ . = lim Sin1 = 1 ; Z Z C ~ . = lim Sl. = - - 1 ?tt
m - - - ~ OO
?ttn
(IV)
n----~ oo
Such double series are characterized b y the fact t h a t t h e y do not satisfy the analytical condition for convergence. A similar s t a t e m e n t can be made concerning the integral operators A (x, x'), B(x, x') ; the trace tr (AB - - B A ) vanishes provided
f f C(x, x') dxdx' = f f C(x, x') dx'dx,
(V)
where C(x, x') = A (x, x') B(x', x). This condition will be satisfied if C is a continuous function of b o t h variables and the integrals exist and converge uniformly. It is not satisfied, however, when either A or B involve the singular m o m e n t u m operator p = h/i O/Ox which is used in q u a n t u m statistics. For example, let A = 9(x) and B = = p (Oq~(x)/Ox) + (09(x)/Ox) p; t h e n tr(AB)= tr
i
~t
dx, tr ( B A ) = - - * ? *" J ~ x )
(AB) - - BA) = 2ih -( ( O*~2dx J \ Ox /
dx; (VI)
which does not vanish for a n y function 90(x). This example is not
890
T H E EQUATION OF STATE IN QUANTIZED KINETIC T H E O R Y
merely of analytical interest, but is important physically, as it embodies in a simplified form the calculation of •3 N # -- #i given in the text. Received J u n e 9th, 1949.
REFERENCES I) J . G . K i r k w o o d , J. chem. Phys. 14,180, 1946; 15, 7 2 , 1 9 4 7 ; 1 7 , 3 3 8 , 1 9 4 9 . 2) M. B o r n and H . S . G r e e n , Proc. roy. Soc. A 188, I0, 1946; J89,103,1947; 190, 455, 1947. 3) M. B o r n and H . S . G r e e n , Proe. roy. Soc. A 141, 168, 1947. 4) K . P . G u r o v, J. exp. and theor. Phys. U.S.S.R., 18, I I0, 1949. 5) G . E . U h l e n b e c k andE. Beth, Physica 3, 729, 1935;4,915,1937. 6) J. d e B o e r, A m s t e r d a m Dissertation, 1940. 7) J. d e B o e r , The Caloric and Termal E q u a t i o n of State in Q u a n t u m Statistical Mechanics, Physiea 15, 843, 1949. 8) H . S . G r e e n , Proe. roy. Soe. A 1 9 4 , 244, 1948. 9) c . f . R . P . F o w l e r & E. A. G u g g e n h e i m , Statlstical T h e r m o d y n a m i s s . 322: Camb. Univ. Press 1939. 0) D. M a s s i g n o n , Comptes Rendus, P a d s , 228, 62, 1949.