6755 Lecture #8 —Rigoberto Hernandez

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…

6755 Lecture #8 —Rigoberto Hernandez

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 Η(qa ,qb , pa , pb ) =Η(qa , pa) +Η(qb , pb)⇒Q =QaQb

6755 Lecture #8 —Rigoberto Hernandez

Ideal Systems 3

Harmonic Oscillator, I In 1-dimension, the H-O potential:

€

V = 12 kx 2

€

H = T +V =p2

2m+12kx 2 = E

€

Q =12π!#

$ %

&

' ( dx∫ dp e−βH (x,p )∫ =

12π!#

$ %

&

' ( e

−β2kx 2

∫ dx e−β p 2

2m∫ dp

€

V = 12 kx2

The Hamiltonian:

The Canonical partition function:

6755 Lecture #8 —Rigoberto Hernandez

Ideal Systems 4

Harmonic Oscillator, II

€

Q =12π!#

$ %

&

' ( e

−β2kx 2

∫ dx e−β p 2

2m∫ dp

€

Q =12π!#

$ %

&

' (

2πβk

#

$ %

&

' (

2πmβ

#

$ %

&

' ( =

m!2β 2k

€

ω ≡km

€

⇒Q =1!βω

€

e−ax2

−∞

∞

∫ dx =πa

The Canonical partition function:

After the Gaussian integrals:

Where:

6755 Lecture #8 —Rigoberto Hernandez

Ideal Systems 5

Harmonic Oscillator, III

€

Q =12π!#

$ %

&

' ( e

−β2kx 2

∫ dx e−β p 2

2m∫ dp

€

⇒Q =1!βω

The Canonical partition function:

But transforming to action-angle variables…

€

Q =12π!

#

$ %

&

' ( dθ

0

2π

∫ e−βωI

0

∞

∫ dI

=12π!

#

$ %

&

' ( ×2π ×

1βω

#

$ %

&

' (

6755 Lecture #8 —Rigoberto Hernandez

Ideal Systems 6

Quantum Harmonic Oscillator

€

Q = e−βEnn=0

∞

∑

En = !ω n +12

'

( )

*

+ ,

⇒ Q = 2sinh(!βω / 2)( )−1

≈1!βω

as !→ 0

The Canonical partition function:

Which can be readily summed to…

6755 Lecture #8 —Rigoberto Hernandez

Ideal Systems 8

Classical Partition Function •  Note that we have a factor of Planck�s Constant, h,

in our classical partition functions:

•  The origin of the prefactors are: –  h ensures that Q is dimensionless – N! accounts for indistinguishable particles –  In the special case of the HP partition function, it

recovers the classical limit correctly •  More generally, Q remains correct even with

interactions between multipled dof�s!

€

Q =1N!

12π!#

$ %

&

' (

N

dxN∫ dpN e−βH (xN ,pN )∫

6755 Lecture #8 —Rigoberto Hernandez

Ideal Systems 9

Observables in HO

€

Q =1!βω

€

E = −∂ ln(Q)∂β

=∂∂β

ln !βω( )( ) =1β

= kBT

€

e−ax2

−∞

∞

∫ dx =πa

The Harmonic Oscillator (HO) canonical partition function:

Recall

€

V = 12 kx2

€

K.E. = 12 kBT

EquipartionTheorem

" # $

€

V = 12 kBT

6755 Lecture #8 —Rigoberto Hernandez

Ideal Systems 10

Gas Consider N particles in volume, V

€

V (! r 1,...,! r N ) = Vij

! r i −! r j( )

i< j∑

€

Q =12π!#

$ %

&

' ( 3N

d" r ∫ d" p ∫ e−βH

" r ," p ( )

with a generic two-body potential:

The Canonical partition function:

€

d! r = dr1dr2...drN

€

T( ! p 1,...,! p N ) =

! p i2

2mii∑

and kinetic energy:

6755 Lecture #8 —Rigoberto Hernandez

Ideal Systems 11

Integrating the K.E. Q in a Gas

€

Q =12π!#

$ %

&

' ( 3N

d" p ∫ e−β

pi2

2mii

N

∑d" r ∫ e−βV

" r ( )

€

Q =12π!#

$ %

&

' ( 3N 2miπ

β

#

$ %

&

' (

i

N

∏32

d" r ∫ e−βV" r ( )

May generally be written as: (Warning: this is not separability!)

€

Q =12π!#

$ %

&

' ( 3N

d" r ∫ d" p ∫ e−βH

" r ," p ( )

€

e−ax2

−∞

∞

∫ dx =πa

With the generic solution for any system

6755 Lecture #8 —Rigoberto Hernandez

Ideal Systems 12

Interacting→Ideal Gas Assume:

1.  Ideal Gas → V(r)=0 2.  Only one molecule type: mi=m

€

Q =12π!#

$ %

&

' ( 3N 2mπ

β

#

$ %

&

' (

3N2

V N

€

d! r ∫ e−βV! r ( ) = d! r ∫ = V N

€

2miπβ

$

% &

'

( )

i

N

∏32

=2mπβ

$

% &

'

( )

3N2

The ideal gas partition function:

6755 Lecture #8 —Rigoberto Hernandez

Ideal Systems 13

The Ideal Gas Law

€

Q =12π!#

$ %

&

' ( 3N 2mπ

β

#

$ %

&

' (

3N2

V N

€

P = −∂A∂V$

% &

'

( ) T ,N

€

dA = −SdT − PdV + µdN

€

A = −kBT ln Q( )

€

P = kBT∂ ln(Q)∂V

€

P = kBTNV Ideal Gas Law!

Recall:

The Pressure

6755 Lecture #8 —Rigoberto Hernandez

Ideal Systems 14

Ideal Gas: Other Observables The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.

€

E(T ,V ,N) = −∂ lnQ∂β

A(T ,V ,N) = −kT lnQ

S(T ,V ,N) =E − AT

Recall : A = E −TS

€

Δ(T ,P,N) = e−βPVQ(T ,V ,N)dV

0

∞

∫G(T , p,N) = −kT lnΔ

S(T , p,N) = k lnΔ + kT∂ lnΔ∂T

(

) *

+

, - N,P

€

Q =12π!#

$ %

&

' ( 3N 2mπ

β

#

$ %

&

' (

3N2

V N

6755 Lecture #8 —Rigoberto Hernandez

Ideal Systems 15

Noninterating Two-Level Systems •  Examples:

– Photon Gas – Phonon Gas – Magnetic Spins

•  In all cases the Hamiltonian looks something like

•  We’ll obtain the ideal Q next time.

€

Η(H ,N) = −niµH

i=1

N

∑

Completing the Square

6755 Lecture #8 —Rigoberto Hernandez

Ideal Systems 16

(x� a)2 + (b� x)2 =�x2 � 2ax + a2

⇥+

�b2 � 2bx + x2

⇥

= 2x2 � 2(a + b)x + a2 + b2

= 2�x2 � (a + b)x

⇥+ a2 + b2

= 2⇤

x2 � (a + b)x +(a + b)2

4

⌅� (a + b)2

2+ a2 + b2

= 2⇤

x� (a + b)2

⌅2

� 12

�a2 + 2ab + b2

⇥+ a2 + b2

= 2⇤

x� (a + b)2

⌅2

+12

�a2 � 2ab + b2

⇥

= 2⇤

x� (a + b)2

⌅2

+12

(a� b)2

� +⇥

�⇥dx e�2(x� (a+b)

2 )2� 12 (a�b)2 = e�

12 (a�b)2

� +⇥

�⇥dx e�2x2

� +⇥

�⇥dx e�(x�a)2�(b�x)2 = e�

12 (a�b)2

⇥�

2