Lecture #7 Ideal Systems 1
Major Concepts• Calculating observables using Statistical Mechanics• Noninteracting Systems
– Separable approximations– Transformations to separable Hamiltonians– Harmonic Oscillator– Ideal Gas– Other examples?
• Statistical Mechanics of Gases– Classical Mechanical Systems
• The Kinetic Energy is “Separable”• Molecules
– Translations may satisfy ideal gas condition– Rotations and vibrations are nearly separable
• In non-interacting limit, recover ideal gas law– In general, must use numerics or approximate theories…
Lecture #7 Ideal Systems 2
Noninteracting Systems• Separable Approximation
• Note: lnQ is extensive!• Thus noninteracting (ideal) systems are reduced to
the calculation of one-particle systems!• Strategy: Given any system, use CT’s to construct
a non-interacting representation!– Warning: Integrable Hamitonians may not be separable
!
if "(qa ,qb , pa , pb ) ="(qa , pa) +"(qb , pb)
#Q =QaQb
Lecture #7 Ideal Systems 3
Harmonic OscillatorIn 1-dimension, the H-O potential:
!
V = 12 kx
2
!
H = T +V =p2
2m+1
2kx
2= E
!
Q =1
2"h
#
$ %
&
' ( dx) dp e
*+H (x,p )) =1
2"h
#
$ %
&
' ( e
*+
2kx2
) dx e*+ p 2
2m) dp
!
V = 12kx
2
The Hamiltonian:
The Canonical partition function:
Lecture #7 Ideal Systems 4
Gaussian Integrals
!
x = r cos"
y = r sin"
!
r2
= x2
+ y2
dxdy = rdrd"
!
e"ax2
"#
#
$ dx = e"ay2
dy
"#
#
$ e"ax2
"#
#
$ dx
%
&
' ' '
(
)
* * *
12
= dx
"#
#
$ dy e"a(x
2+y
2)
"#
#
$%
&
' ' '
(
)
* * *
12
= d+
0
2,
$ re"ar2
0
#
$ dr
%
&
' ' '
(
)
* * *
12
= 2, - 12
e"au
0
#
$ du
%
&
' ' '
(
)
* * *
12
where u = r2
and du = 2rdr
= , 0"1
a
%
& '
(
) *
%
& '
(
) *
12
=,
a
%
& '
(
) *
12
!
e"ax 2
"#
#
$ dx =%
a
Lecture #7 Ideal Systems 5
Harmonic Oscillator
!
Q =1
2"h
#
$ %
&
' ( e
)*
2kx2
+ dx e)* p 2
2m+ dp
!
Q =1
2"h
#
$ %
&
' (
2"
)k
#
$ %
&
' (
2"m
)
#
$ %
&
' ( =
m
h2) 2k
!
" #k
m
!
"Q =1
h#$
!
e"ax 2
"#
#
$ dx =%
a
The Canonical partition function:
After the Gaussian integrals:
Where:
Lecture #7 Ideal Systems 6
Harmonic Oscillator
!
Q =1
2"h
#
$ %
&
' ( e
)*
2kx2
+ dx e)* p 2
2m+ dp
!
"Q =1
h#$
The Canonical partition function:
But transforming to action-angle vailables…
!
Q =1
2"h
#
$ %
&
' ( d)
0
2"
* e+,-I
0
.
* dI
=1
2"h
#
$ %
&
' ( /2" /
1
,-
#
$ %
&
' (
Lecture #7 Ideal Systems 7
Classical Partition Function• Note that we have a factor of Planck’s
Constant, h, in our classical partitionfunctions:
• This comes out for two reasons:– To ensure that Q is dimensionless– To connect to the classical limit of the
quantum HO partition function…
!
Q =1
2"h
#
$ %
&
' (
N
dxN) dp
Ne*+H (xN ,pN ))
Lecture #7 Ideal Systems 8
Harmonic Oscillator
!
Q =1
h"#
!
E = "# ln(Q)
#$=#
#$ln h$%( )( ) =
1
$= kBT
!
e"ax 2
"#
#
$ dx =%
a
The Canonical partition function:
Recall
!
V = 12kx
2
!
V = 12kBT
Lecture #7 Ideal Systems 9
GasConsider N particles in volume, V
!
V (r r 1,...,
r r N ) = Vij
r r i "
r r j( )
i< j
#
!
Q =1
2"h
#
$ %
&
' (
3N
dr r ) d
r p ) e
*+Hr r ,
r p ( )
with a generic two-body potential:
The Canonical partition function:
!
dr r = dr
1dr2...dr
N
!
T(r p 1,...,
r p N ) =
r p i2
2mii
"
and kinetic energy:
Lecture #7 Ideal Systems 10
Integrating the K.E. Q in a Gas
!
Q =1
2"h
#
$ %
&
' (
3N
dr p ) e
*+pi2
2mii
N
,dr r ) e
*+Vr r ( )
!
Q =1
2"h
#
$ %
&
' (
3N2mi"
)
#
$ %
&
' (
i
N
*32
dr r + e
,)Vr r ( )
May generally be written as: (Warning:this is not separability!)
!
Q =1
2"h
#
$ %
&
' (
3N
dr r ) d
r p ) e
*+Hr r ,
r p ( )
!
e"ax 2
"#
#
$ dx =%
a
With the generic solution for any system
Lecture #7 Ideal Systems 11
InteractingIdeal GasAssume:
1. Ideal Gas V(r)=02. Only one molecule type: mi=m
!
Q =1
2"h
#
$ %
&
' (
3N2m"
)
#
$ %
&
' (
3N2
VN
!
dr r " e
#$Vr r ( )
= dr r " = V
N
!
2mi"
#
$
% &
'
( )
i
N
*32
=2m"
#
$
% &
'
( )
3N2
The ideal gas partition function:
Lecture #7 Ideal Systems 12
The Ideal Gas Law
!
Q =1
2"h
#
$ %
&
' (
3N2m"
)
#
$ %
&
' (
3N2
VN
!
P = "#A
#V
$
% &
'
( ) T ,N
!
dA = "SdT " PdV + µdN
!
A = "kBT ln Q( )
!
P = kBT" ln(Q)
"V
!
P = kBTN
VIdeal Gas Law!
Recall:
The Pressure
Lecture #7 Ideal Systems 13
Ideal Gas: Other Observables
!
E(T ,V ,N) = "# lnQ
#$
A(T ,V ,N) = "kT lnQ
S(T ,V ,N) =E " A
T Recall : A = E "TS
!
"(T ,P,N) = e#$PV
Q(T ,V ,N)dV
0
%
&G(T , p,N) = #kT ln"
S(T , p,N) = k ln" + kT' ln"
'T
(
) *
+
, - N,P
!
Q =1
2"h
#
$ %
&
' (
3N2m"
)
#
$ %
&
' (
3N2
VN
Lecture #7 Ideal Systems 14
Noninterating Two-Level Systems• Examples:
– Photon Gas– Phonon Gas– Magnetic Spins
• In all cases the Hamiltonian lookssomething like
!
"(H ,N) = #niµH
i=1
N
$
