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

$