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The caloric and thermal equation of state in quantum statistical mechanics
P h y s i c a XV, no 10
October 1949
THE CALORIC AND THERMAL EQUATION OF STATE IN QUANTUM STATISTICAL MECHAIklICS by J. DE BOER Department of Theoretical Physics, University of Amsterdam, Nederland
Summary I t is shown t h a t i n classical as well as i n q u a n t u m mechanics the " d y n a m i c " expressions for the caloric a n d t h e r m a l equations of state obtained with the method of q u a n t u m statistical mechanics are identical with the expressions obtained b y the p a r t i t i o n function method of statistical thermodynamics.
§ 1. Introduction. In a recent publication B o r n and G r e e n 1) have raised some doubt in connection with the quantum mechanical expression for the thermal equation of state. In particular it has been suggested that the "dynamical" expression for the pressure obtained b y applying the method of statistical mechanics and general dynamic principles is not identical with the "thermodynamical" expression for the pressure obtained by the partition function method of statistical thermodynamics. As in the hydrodynamical equations of motion, which t h e y derived, the dynamical expression of the pressure occurs, B o r n and G r e e n 1) hoped to be able to base on this distinction between the "dynamical" and the "thermodynamical" pressure in quantum theory, an explanation of the strange phenomena in a typical " q u a n t u m liquid" like liquid He II. It is therefore of importance to investigate both expressions for the equation of state in some more detail, which shows that in the author's opinion no such difference between the two pressures e~ists. § 2. The thermodynamical expressions [or the thermal and caloric eqccations o/state. The calculations of the properties of a system of N identical molecules in equilibrium have to start with some expression for the statistical operator P, or the corresponding density matrix 843 - -
844
j. p
DE BOER
r
P ( r I . . . . rN; r I . . . . r N ) , which depends on the ensemble chosen to represent the system. In this discussion the canonical ensemble will be used, having P = Z -1 exp (-- fill), corresponding to the density matrix P ( r l . . . r N ; r ,* . . . r N, ) = Z - - 1 2:p ~,p ( r~, . . . r Nt ) e x p (--flH)~p ( r l . . . r N ) (1) where r I. . . . . r N are the position vectors of the molecules 1 to N, 9p is any arbitrary orthonormal complete set of wave functions, fl = 1/kT and Z -1 is a normalising factor determined by the fact, that the trace
f....[P(r
I . . . rN; r I . . . r ~ ) d r I . . . dr N = 1
(2)
where dr is written for dxdydz. The fundamental hypothesis of statistical mechanics states, that the average value of some quantity G(p 1 . . . . p~, r~ . . . . r~) of the system considered is given by the average value of G in the ensemble, which according to quantummechanics is given by G =f...
f ( G P ) (r 1 . . . r N ; r 1 . . . rN) dr1 . . . drN
(3)
which expression is independent of the order of P and the operator G corresponding to the quantity G. The matrix (GP) (r; r') is given by the expression analogous to (1) : t
(GP)(rt . . . rN; r I . . . r~) = = Z - ' Xp ~p(r'1 . . . r;v) G exp (-- flH)~0p(r1 . . . rN)
(4)
The caloric equation o/state can now immediately be obtained b y calculating the average value of the Hamiltonian of the system of N molecules. Assuming that the potential energy of the intermolecular forces of this system is built up additively out of the sum of contributions ~0 (rL,) of all pairs ,, × of molecules, the Hamiltonian takes the form H = K + • , where K = X p ~ / 2 m and # = ~ According to (3) the expression for the average energy becomes
U = Z-If...f(If+
#) exp (--fib 0 ( r x . . . r N ; r l . . . r N ) d r i . . . d r N
(5)
where / f represent the operator obtained by replacing in K the momenta p by the corresponding operators p = - - i~ 0/Sr. As the kinetic energy operator consists additively out of single molecule contributions and the potential energy consists additively out of contributions due to pairs of molecules, the.expression simplifies to: [ U = f (1/2m) (p~ nl) (r I ; r l ) d r 1 + ~ f f 9 (rl2) 112(rlr2; rtr2)drldr2 ](6)
THE CALORIC AND THERMAL EQUATION OF STATE
845
where in general nh(r I ...
rh; rt . . . rh) =
- (N N! -- h) ! f . . f V (. r l . . . r. N ; r l ... . r N )
drh+l,
den (7)
rN) dr 2 .
drn (8)
(p~rll) (r, ; r,) =
--
• N, i~
;.j 1)i.
•.
"(p~P) .
.
(r I . .
rn; r I . . .
.
The expression n 2 is the quantum mechanical expression for the probability density for a pair of molecules on positions r t and r2, whereas (1/2m) (p~nl) (rl; r 0 represents the probability density of kinetic energy at r~. In a subsequent publication ~) the method will be given to develop these densities into power series of powers of the molecular number density n. The thermal equation o/ state can be derived b y using the virial theorem. The validity of this theorem in quantum mechanics has been proved b y S 1 a t e r. The averages occurring in the theorem have only to be replaced b y the corresponding quantum mechanical averages. Application of the virial theorem, which states that the virial of the internal forces and external forces is equal to minus twice the average kinetic energy, to a system of N molecules enclosed in a vessel of volume V and exerting a pressure P on de walls, gives ~ - - 3 p V = - - 2K. Here - - 3pV is the virial of the forces which the wall exerts in the average of the system, ~ i s the average value of the virial of the intermolecular forces, ~1 X X - - r,g (d~o/dr~g), and K is the average value of the kinetic energy. So the value of p V is equal to the average of ~2 K + ~1~ . . According to (3) this average value is given b y
p V = Z - i f . . •f(~2K + ~I ,~)exp ~ ( - - / ~ H ) ( r l . . . rn; r l . . . rn)dr I . . dr N (9) Both the expressions (5) and (9) are closely similar, the operator K being replaced b y ~ K in (9) and the potential energy ~ b y 3 - , the virial of the internal forces, in (9). Again (9) can be written in the simplified form
[pV=f(113m) (p~rh)(r I ;rx)drt--~ffrm(dg/dr12)na(rtr 2; r,r2)dridr ~ (10) In the next section it will be shown that both the expressions (5) and (9) for the caloric and thermal equation of state can also be
846
j.
DE BOER
o b t a i n e d b y using the p a r t i t i o n function m e t h o d of statistical thermodynamics. § 3. Thermal and caloric equation o~ state according to statistical thermodynamics. T h e statistical i n t e r p r e t a t i o n of t h e r m o d y n a m i c s is based on the t h e r m o d y n a m i c analogy, which states t h a t P 1 a n c k's function k~ = - - F / T = S - - U / T is equal to ~ = k In Z, where the normalising constant Z Z=f...fexp(--flH)(r
t...rN;r
1...rN)dr l...dr~
(11)
The t h e r m o d y n a m i c relation (12)
d 7-t = U d T / T ~ + p d V / T
t h e n leads to the following expressions U = T2 ST
= kT
./'exp
pV = TV aV
-
.
. r N. ; r .I . . . r .~ ) d. r I
dr N (13)
-
O/fe
= k T Z V ~-# . . .
xp(--/~H)(rl...rN;rl...rN)drl...dr
N(14)
It is of a d v a n t a g e to introduce wave functions of current waves, in which case exp (--/~H) (r I . . . r N; r 1 . . . rN) = =f...fdk
I . . . dk~ exp ( - - i X k,r,) exp ( - - f i b 0 exp ( i X k , r , ) (15)
I n t r o d u c i n g now a scale factor L characterising the linear dimensions of the vessel, as has been done also b y B o r n and G r e e n in the classical case, and changing the variable r and w a v e n u m b e r s k to r = L r and k = L - l k , the p r o d u c t s k . r in the wave functions change into k . r and d k d r into d k d r . T h e H a m i l t o n function in the new variables reads ~2 _ _
H =
0
1
2mL2 ,S,, ~r~ " 8r, + i .S, .S~ 9(~,xL) = / t [ + •
(16)
So the dimensions L of the vessel h a v e disappeared out of the
847
THE CALORIC AND THERMAL E Q U A T I O N OF STATE
whole integral (14) except the Hamilton operator H(f, L). Now substituting (15) in (l 4) and using the fact, that
OH V OV = ~ _ _
_
=
1 L
OH OL
--
2-mL2ft Or, "~r,
a
1
+3 iffLr,~,
.
=
2K+l
=
~.~,(17)
one obtains after transforming again to the old variables
p V = l / Z f . ..f(sK+~,.,)exp(--flH)(r 2 1= I. . . r ~ ; r I. ..rN)dr I .. .drN(18) Use is made of the fact that although exp ( - - fill) and the operator VOH/OV do not in general commute, the order of these two operators can be interchanged because the expression is the trace of the matrix (cf. Appendix). The final expression (18) is identical with the "dynamical" expression (9) for the equation of state, which has been derived in § 2 from statistical mechanics. A similar reasoning, based on the sum-over-states method, has been given b y Y v o n. So no discrepancy exists between the expressions for the equation of state obtained from the dynamical principles and that obtained with the partition function method. This makes also very doubtfull the explanation of the properties of liquid He II, which B o r n and G r e e n based just on this supposed difference between the "dynamical" and "thermodynamical" pressure. APPENDIX
If A is any operator depending on ~ and the operator dA(;()/d~ is indicated b y A', the following general expansion holds" d f exp A(A) = ~A' + [A, A'] + ( da
1
1
+ ~. [AEA, A']] + -~. [A[A[A, A']]] + . . .
} exp A(;t).
(I)
where [A, B] is written for the commutator A B - - BA. This theorem can be applied to the case considered in this paper b y substituting for A(a) = - - 13H(L) and for differentiation to 2, diffe-
848
T H E CALORIC A N D T H E R M A L E Q U A T I O N OF STATE
rentiation to L, giving: d - - exp - - ~H(L) =
dL
---{--~H'--k ,2[H, H ' ] - - ~. [H[H, H']] + . . . } e x p (--~H(L)). (II) If the trace is taken of the whole expression, the trace of each of the c o m m u t a t o r s is zero: trace {[H, H'] exp ( - - fill)} = f . . . f { H H ' exp ( - - ~H) - -
- - H ' H exp ( - - ~H)} (r I . . . rN; r I . . . rN) dr I . . .
dr N
=
0
(III)
all the integrals being finite. So one finally obtains: exp(--flH)
(r l . . . r N ; r
I . . . r N ) dr 1 . . . d r
N=
= ] ( - - f l H ' e x p (--fill)) ( r l . . . r y ; r I . . . r N ) d r I . . . dr N . (IV) which directly leads to (18). Received May 30th, 1949. REFERENCES 1) 2) 3) 4)
M. B o r n and H . S . G r e e n, .Proc. roy. Soc. A 191, 168, 1947. M. B o r n and H . S . G r e e n , Proc. roy. Soc. A 194,244, 1948. J. Y v o n, C. R. Aead. Sci. Paris "*27, 763, 1948. J. d e B o e r , Physica, 15, 680, 1949.
October 1949
THE CALORIC AND THERMAL EQUATION OF STATE IN QUANTUM STATISTICAL MECHAIklICS by J. DE BOER Department of Theoretical Physics, University of Amsterdam, Nederland
Summary I t is shown t h a t i n classical as well as i n q u a n t u m mechanics the " d y n a m i c " expressions for the caloric a n d t h e r m a l equations of state obtained with the method of q u a n t u m statistical mechanics are identical with the expressions obtained b y the p a r t i t i o n function method of statistical thermodynamics.
§ 1. Introduction. In a recent publication B o r n and G r e e n 1) have raised some doubt in connection with the quantum mechanical expression for the thermal equation of state. In particular it has been suggested that the "dynamical" expression for the pressure obtained b y applying the method of statistical mechanics and general dynamic principles is not identical with the "thermodynamical" expression for the pressure obtained by the partition function method of statistical thermodynamics. As in the hydrodynamical equations of motion, which t h e y derived, the dynamical expression of the pressure occurs, B o r n and G r e e n 1) hoped to be able to base on this distinction between the "dynamical" and the "thermodynamical" pressure in quantum theory, an explanation of the strange phenomena in a typical " q u a n t u m liquid" like liquid He II. It is therefore of importance to investigate both expressions for the equation of state in some more detail, which shows that in the author's opinion no such difference between the two pressures e~ists. § 2. The thermodynamical expressions [or the thermal and caloric eqccations o/state. The calculations of the properties of a system of N identical molecules in equilibrium have to start with some expression for the statistical operator P, or the corresponding density matrix 843 - -
844
j. p
DE BOER
r
P ( r I . . . . rN; r I . . . . r N ) , which depends on the ensemble chosen to represent the system. In this discussion the canonical ensemble will be used, having P = Z -1 exp (-- fill), corresponding to the density matrix P ( r l . . . r N ; r ,* . . . r N, ) = Z - - 1 2:p ~,p ( r~, . . . r Nt ) e x p (--flH)~p ( r l . . . r N ) (1) where r I. . . . . r N are the position vectors of the molecules 1 to N, 9p is any arbitrary orthonormal complete set of wave functions, fl = 1/kT and Z -1 is a normalising factor determined by the fact, that the trace
f....[P(r
I . . . rN; r I . . . r ~ ) d r I . . . dr N = 1
(2)
where dr is written for dxdydz. The fundamental hypothesis of statistical mechanics states, that the average value of some quantity G(p 1 . . . . p~, r~ . . . . r~) of the system considered is given by the average value of G in the ensemble, which according to quantummechanics is given by G =f...
f ( G P ) (r 1 . . . r N ; r 1 . . . rN) dr1 . . . drN
(3)
which expression is independent of the order of P and the operator G corresponding to the quantity G. The matrix (GP) (r; r') is given by the expression analogous to (1) : t
(GP)(rt . . . rN; r I . . . r~) = = Z - ' Xp ~p(r'1 . . . r;v) G exp (-- flH)~0p(r1 . . . rN)
(4)
The caloric equation o/state can now immediately be obtained b y calculating the average value of the Hamiltonian of the system of N molecules. Assuming that the potential energy of the intermolecular forces of this system is built up additively out of the sum of contributions ~0 (rL,) of all pairs ,, × of molecules, the Hamiltonian takes the form H = K + • , where K = X p ~ / 2 m and # = ~ According to (3) the expression for the average energy becomes
U = Z-If...f(If+
#) exp (--fib 0 ( r x . . . r N ; r l . . . r N ) d r i . . . d r N
(5)
where / f represent the operator obtained by replacing in K the momenta p by the corresponding operators p = - - i~ 0/Sr. As the kinetic energy operator consists additively out of single molecule contributions and the potential energy consists additively out of contributions due to pairs of molecules, the.expression simplifies to: [ U = f (1/2m) (p~ nl) (r I ; r l ) d r 1 + ~ f f 9 (rl2) 112(rlr2; rtr2)drldr2 ](6)
THE CALORIC AND THERMAL EQUATION OF STATE
845
where in general nh(r I ...
rh; rt . . . rh) =
- (N N! -- h) ! f . . f V (. r l . . . r. N ; r l ... . r N )
drh+l,
den (7)
rN) dr 2 .
drn (8)
(p~rll) (r, ; r,) =
--
• N, i~
;.j 1)i.
•.
"(p~P) .
.
(r I . .
rn; r I . . .
.
The expression n 2 is the quantum mechanical expression for the probability density for a pair of molecules on positions r t and r2, whereas (1/2m) (p~nl) (rl; r 0 represents the probability density of kinetic energy at r~. In a subsequent publication ~) the method will be given to develop these densities into power series of powers of the molecular number density n. The thermal equation o/ state can be derived b y using the virial theorem. The validity of this theorem in quantum mechanics has been proved b y S 1 a t e r. The averages occurring in the theorem have only to be replaced b y the corresponding quantum mechanical averages. Application of the virial theorem, which states that the virial of the internal forces and external forces is equal to minus twice the average kinetic energy, to a system of N molecules enclosed in a vessel of volume V and exerting a pressure P on de walls, gives ~ - - 3 p V = - - 2K. Here - - 3pV is the virial of the forces which the wall exerts in the average of the system, ~ i s the average value of the virial of the intermolecular forces, ~1 X X - - r,g (d~o/dr~g), and K is the average value of the kinetic energy. So the value of p V is equal to the average of ~2 K + ~1~ . . According to (3) this average value is given b y
p V = Z - i f . . •f(~2K + ~I ,~)exp ~ ( - - / ~ H ) ( r l . . . rn; r l . . . rn)dr I . . dr N (9) Both the expressions (5) and (9) are closely similar, the operator K being replaced b y ~ K in (9) and the potential energy ~ b y 3 - , the virial of the internal forces, in (9). Again (9) can be written in the simplified form
[pV=f(113m) (p~rh)(r I ;rx)drt--~ffrm(dg/dr12)na(rtr 2; r,r2)dridr ~ (10) In the next section it will be shown that both the expressions (5) and (9) for the caloric and thermal equation of state can also be
846
j.
DE BOER
o b t a i n e d b y using the p a r t i t i o n function m e t h o d of statistical thermodynamics. § 3. Thermal and caloric equation o~ state according to statistical thermodynamics. T h e statistical i n t e r p r e t a t i o n of t h e r m o d y n a m i c s is based on the t h e r m o d y n a m i c analogy, which states t h a t P 1 a n c k's function k~ = - - F / T = S - - U / T is equal to ~ = k In Z, where the normalising constant Z Z=f...fexp(--flH)(r
t...rN;r
1...rN)dr l...dr~
(11)
The t h e r m o d y n a m i c relation (12)
d 7-t = U d T / T ~ + p d V / T
t h e n leads to the following expressions U = T2 ST
= kT
./'exp
pV = TV aV
-
.
. r N. ; r .I . . . r .~ ) d. r I
dr N (13)
-
O/fe
= k T Z V ~-# . . .
xp(--/~H)(rl...rN;rl...rN)drl...dr
N(14)
It is of a d v a n t a g e to introduce wave functions of current waves, in which case exp (--/~H) (r I . . . r N; r 1 . . . rN) = =f...fdk
I . . . dk~ exp ( - - i X k,r,) exp ( - - f i b 0 exp ( i X k , r , ) (15)
I n t r o d u c i n g now a scale factor L characterising the linear dimensions of the vessel, as has been done also b y B o r n and G r e e n in the classical case, and changing the variable r and w a v e n u m b e r s k to r = L r and k = L - l k , the p r o d u c t s k . r in the wave functions change into k . r and d k d r into d k d r . T h e H a m i l t o n function in the new variables reads ~2 _ _
H =
0
1
2mL2 ,S,, ~r~ " 8r, + i .S, .S~ 9(~,xL) = / t [ + •
(16)
So the dimensions L of the vessel h a v e disappeared out of the
847
THE CALORIC AND THERMAL E Q U A T I O N OF STATE
whole integral (14) except the Hamilton operator H(f, L). Now substituting (15) in (l 4) and using the fact, that
OH V OV = ~ _ _
_
=
1 L
OH OL
--
2-mL2ft Or, "~r,
a
1
+3 iffLr,~,
.
=
2K+l
=
~.~,(17)
one obtains after transforming again to the old variables
p V = l / Z f . ..f(sK+~,.,)exp(--flH)(r 2 1= I. . . r ~ ; r I. ..rN)dr I .. .drN(18) Use is made of the fact that although exp ( - - fill) and the operator VOH/OV do not in general commute, the order of these two operators can be interchanged because the expression is the trace of the matrix (cf. Appendix). The final expression (18) is identical with the "dynamical" expression (9) for the equation of state, which has been derived in § 2 from statistical mechanics. A similar reasoning, based on the sum-over-states method, has been given b y Y v o n. So no discrepancy exists between the expressions for the equation of state obtained from the dynamical principles and that obtained with the partition function method. This makes also very doubtfull the explanation of the properties of liquid He II, which B o r n and G r e e n based just on this supposed difference between the "dynamical" and "thermodynamical" pressure. APPENDIX
If A is any operator depending on ~ and the operator dA(;()/d~ is indicated b y A', the following general expansion holds" d f exp A(A) = ~A' + [A, A'] + ( da
1
1
+ ~. [AEA, A']] + -~. [A[A[A, A']]] + . . .
} exp A(;t).
(I)
where [A, B] is written for the commutator A B - - BA. This theorem can be applied to the case considered in this paper b y substituting for A(a) = - - 13H(L) and for differentiation to 2, diffe-
848
T H E CALORIC A N D T H E R M A L E Q U A T I O N OF STATE
rentiation to L, giving: d - - exp - - ~H(L) =
dL
---{--~H'--k ,2[H, H ' ] - - ~. [H[H, H']] + . . . } e x p (--~H(L)). (II) If the trace is taken of the whole expression, the trace of each of the c o m m u t a t o r s is zero: trace {[H, H'] exp ( - - fill)} = f . . . f { H H ' exp ( - - ~H) - -
- - H ' H exp ( - - ~H)} (r I . . . rN; r I . . . rN) dr I . . .
dr N
=
0
(III)
all the integrals being finite. So one finally obtains: exp(--flH)
(r l . . . r N ; r
I . . . r N ) dr 1 . . . d r
N=
= ] ( - - f l H ' e x p (--fill)) ( r l . . . r y ; r I . . . r N ) d r I . . . dr N . (IV) which directly leads to (18). Received May 30th, 1949. REFERENCES 1) 2) 3) 4)
M. B o r n and H . S . G r e e n, .Proc. roy. Soc. A 191, 168, 1947. M. B o r n and H . S . G r e e n , Proc. roy. Soc. A 194,244, 1948. J. Y v o n, C. R. Aead. Sci. Paris "*27, 763, 1948. J. d e B o e r , Physica, 15, 680, 1949.