Download Partition Functions in Classical and Quantum Mechanics

Partition Functions in Classical and Quantum Mechanics
The canonical partition function in classical mechanics was defined as,
Z
dE
ZC,class =
Ω(E) e−βE
∆
where Ω(E) is the volume of the phase space in the microcanonical ensemble.
Z
1
Ω(E) = N.d
dN.d p dN.d q
h N ! Γ(E)
(1)
(2)
where Γ(E) is a thin shell of thickness ∆ around a surface of constant energy E and d is the dimension of space and
N is the total number of particles. We wish to generalize the above correspondence to quantum systems. To do this
we have to realize that in quantum systems, the energy level are discrete (there are some exceptions but typically if
one imposes periodic boundary conditions on the wavefunctions, they will always be discrete). Thus we have to make
the correspondence,
Z
X
dE
E → En ; Ω(E) → gn = degeneracy of the energy level n ;
=
∆
n
and the integration is replaced by a summation. Thus in quantum mechanics we must have,
X
ZC,quant =
gn e−βEn
(3)
n
Now we wish to take some specific examples and evaluate the partition function for the classical and quantum harmonic
oscillator in one dimension.
Classical Harmonic Oscillator:
Let us assume that the harmonic oscillator is described by the hamiltonian,
H=
p2
1
+ k q2
2m 2
We wish to calculate the canonical partition function of this system. First we calculate the microcanonical function
Ω(E).
Z
1
Ω(E) = d.N
dN.d q dN.d p
h N ! Γ(E)
Here the number of particles N = 1 and we are working in one dimension d = 1. hence,
Z Z
1
Ω(E) =
dq dp
p2
h
E−∆/2≤ 2m
+ 12 kq 2 ≤E+∆/2
2
To evaluate this it is better to transform to polar coordinates.
r
√
2
p = 2m r cosθ ; q =
r sinθ
k
¯
¯
dq dp ≡ ¯¯
r
¯
√
¯
¯ dr dθ = 2m 2 rdrdθ
¯
k
∂q
∂r
∂q
∂θ
∂p
∂r
∂p
∂θ
and,
H(q, p) =
p2
1
+ kq 2 = r2
2m 2
Therefore,
1
Ω(E) =
h
Z
√
r
2m
E−∆/2≤r 2 ≤E+∆/2
2
r dr
k
Z
r
2π
dθ =
0
4m π∆
k h
Therefore,
Z
∞
ZC,class =
0
dE −βE
e
Ω(E) =
∆
Z
∞
0
dE −βE
e
∆
r
4m π∆
=
k h
r
4m π
1
=
k βh
β~ω
(4)
q
k
where ω = m
. Now that we have evaluated the partition function of the classical harmonic oscillator we now wish
to evaluate the same quantity for the quantum harmonic oscillator. In other words, we wish to answer the question :
‘What is the canonical partition function if the mass attached to the spring obeys Schrodinger’s equation instead of
Newton’s laws ?’.
Quantum Harmonic Oscillator:
We know that the energy levels of the quantum harmonic oscillator are given by,
En = ~ω (n + 1/2)
q
where ω =
k
m.
ZC,quant =
The degeneracy of each level gn = 1. Hence the canonical partition function in the quantum case is,
X
n
gn e−βEn =
∞
X
e−β
~ω (n+1/2)
n=0
=
∞
X
e−β
~ω (n+1/2)
= e−β
~ω
2
∞
X
n=0
n=0
~ω
e−β
~ω n
=
e−β 2
1 − e−β ~ω
(5)
Now we wish to see if the classical partition function in Eq.(4) may be obtained as a limiting case of the quantum
partition function in Eq.(5). We know that quantum mechanics tends to classical mechanics as Planck’s constant
tends to zero. To see this explicitly we examine,
~ω
lim~→0 ZC,quant = lim~→0
e−β 2
1
≈
= ZC,class
1 − e−β ~ω
β ~ω
(6)
Alternatively and more importantly, the classical result is obtained at temperatures large ( kB T ) compared to the
typical energy scale in the system namely ~ω, or kB T = β1 À ~ω.
System of Harmonic oscillators
Consider a system of N distinguishable harmonic oscillators but with the same k and m (they may be distinguished
by some other non-mechanical attribute such as color). In this case the hamiltonian would be,
H=
¶
N µ 2
X
pi
1
+ kqi2
2m 2
i=1
3
Instead of evaluating the canonical partition function in the manner of the previous section, we now wish to evaluate
it differently. We substitute Eq.(2) into Eq.(1) to arrive at,
Z
Z
dE
1
ZC,class,indisting. =
dN.d p dN.d q e−βE
(7)
∆ hN.d N ! Γ(E)
Since we have assumed that the oscillators are distinguishable we should not have the N ! in the denominator. Hence
for this specific example,
ZC,class,distng. = N ! ZC,class,indisting.
(8)
Here,
1
∆
Z
dN.d p dN.d q =
Γ(E)
V olume of shell of thickness ∆
= Surf ace area of radius E
∆
Therefore,
Ã
Z
dE
1
∆
!
Z
d
N.d
pd
N.d
q
Z
=
Z
dE (Surf ace area of radius E) =
(Entire volume of phase space)
Γ(E)
Or,
ZC,class,indisting. =
Z
1
dN.d p dN.d q e−βH(q,p)
hN.d N !
ZC,class,disting. = N ! ZC,class,indisting.
(9)
(10)
This formula is much simpler for evaluation of the partition function. Thus for the system of harmonic oscillators
PN
p2i
+ 12 kqi2 .
above we may write H(q, p) = i=1 h(qi , pi ) and h(qi , pi ) = 2m
QN ≡ ZC,class,disting.
1
= N
h
µ Z
¶N
1
−βh(q1 ,p1 )
=
dp1 dq1 e
= (β~ω)−N
h
Z
N
N
d pd qe
−βH(q,p)
(11)
The Helmholtz free energy of the system is given by,
µ
A(V, N, T ) = −kB T lnQN = N kB T ln
~ω
kB T
¶
(12)
whence,
µ
P ≡−
∂A
∂V
¶
=0
(13)
N,T
since the above A does not depend on V .
µ
S=−
∂A
∂T
¶
³
But, A = U − T S, or U = A + T S = N kB T ln
·
kB T
~ω
¶
¸
ln
+ T N kB
£ ¡ kB T ¢
¤
ln ~ω + 1 = N kB T and CV = CP = N kB .
N,V
~ω
kB T
µ
= N kB
´
+1
(14)
Homework : Calculate U and CV for the quantum case with distinguishable subsystems assuming that QN =
QN
N =1 .