5. Formulation of Quantum Statistics

1. Quantum Mechanical Ensemble Theory: The Density Matrix

2. Statistics of the Various Ensembles

3. Examples

4. Systems Composed of Indistinguishable Particles

5. The Density Matrix & the Partition Function of a System of

Free Particles

Statistics Particle type Math Object

Classical Distinguishable Phase space density

Quantum Indistinguishable Density matrix

Advantage of using density matrix :

Quantum & ensemble averaging are

combined into one averaging.

Classical Statistical Mechanics

(Probability) density function ( p,q,t ) :

d ff

d

3 3N Nd d p d q

Liouville’s theorem : , Ht

Microcanonical ensemble :0

const E H E

otherwise

d

Canonical ensemble : He 3

1

!N NZ d

N h

Grand canonical ensemble : NH NN e 3

1

! N NNN

dN h

Z

Caution:Some authors, e.g., Landau-Lifshitz, use a normalized version of .

Quantum Statistical Mechanics (To be Proved)

k kk

d ff p f

d

Classical mechanics :

k kk

k

f p f Quantum mechanics :

Tr fk k

kk

p

Ensemble = phase space

Ensemble = Hilbert space

1E P

NMicrocanonical :

1 HeZ

Canonical :

1 H Ne Z

Grand canonical :

PE = projection operator onto the N-D subspace of states with energy E.

HZ Tr e

H NTr e Z

Pure State Density Operator

Expectation value of f :

f f

,n m

n n f m m

n

n n I

Orthonormal basis { | n } is complete :

Density operator for | : nm n m

n

Tr f n f n ,n m

n m m f n

,n m

n m m f n f

1

n

n n

nn

n a*

,n nm m

n m

a f a

Tr f 1Tr

r-Representation

f f ,n m

n n f m m n

n n I

d I r r r f d d f r r r r r r

*d d f r r r r r r

f is a 1-particle operator f f r r r r r

*f d f r r r r

Mixed State Density Operator

Averaged value of f :

k kk

k

f p f

, ,

k kk

k n m

p n n f m m n

n n I

Orthonormal basis { | n } is complete :

Density operator :k k

kk

p k knm k

k

p n m

n

Tr f n f n ,n m

n m m f n

, ,

k kk

k n m

p n m m f n f

Skip to ensembles Ex: Derive the quantum Liouville eq.

1 1kk

p Tr

5.1. Quantum Mechanical Ensemble Theory: The Density Matrix

, ,i ii t H tt

r r

Consider ensemble of N identical systems labelled by k = 1, 2,..., N.

Each system is described by i = 1,2,..., N

Let ,k kit t r be the wave function of the kth system in the ensemble.

Let n n i r be a set of complete orthonormal basis that spans the Hilbert space of H & satisfies the relevant B.C.s.

,k kit t r k k

n n i n nn n

a t a t r

with *k kn na t d t 3

ii

d d r

k runs through all independent solutions of this Schrodinger eq.

*k k kn na t d t n t

*k

knn t

d ai d i t

d t * k

nd H t

k kn n

n

t a t

*km n m

m

a t d H

k

knnm m

m

d ai H a t

d t where

*nm n mH d H

21kd t * *

,

1k km n m n

m n

a t a t d

*m n mnd m n 2

1kn

n

a t k

n H m

H can be t-dep

Density Operator

Density operator : *

1

k kk

k

p t t

N

1

ˆ k kk

k

p t t

N

Matrix elements : mn t m n

*

1

k kk m n

k

p a t a t

N

*m n ens

a t a t

2

nn nens

a t 21nn n

n n ens

a t

n or d ~ quantum averaging ens ~ k ~ ensemble averaging

1

k kk

k

p m t t n

N

k kn n

n

t a t

pk = weighting (or probability) factor with 1kk

p

*

1

k kmn k m n

k

p a t a t

N

k

knnm m

m

d ai H a t

d t

*

*

1

k kmn k km n

k n mk

d d a d ai p i a t a t i

d t d t d t

N

* *

1

k k k kk ml l n m l n l

k l l

p H a t a t a t H a t

N

*mn nmH H

ml l n m l l nl

H t t H

mnH H

_,d

i Hd t

,A B AB BA where

m H H n

H can be t-dep

Equilibrium Ensemble

System in equilibrium ensemble stationary :

0d

d t

_,d

i Hd t

i.e. _, 0H H and 0tH

Energy representation : n n nH E

*mn nm

mn m mnH E

H mn m mn

In a general basis , is hermitian

k

knnm m

m

d ai H a t

d t k

n nE a t expkn n n

ia t a E t

*

1

k kmn k m n

k

p a t a t

N

*

1

expk m n m nk

ip a a E E t

N

System in equilibrium 2

m mna

detailed balance

Expectation Values

*

*1

k k

k k kk

d GG p

d

N

Expectation value of a physical quantity G :

( Quantum + ensemble av. )

k kn n

n

t a t *

1 ,

k kk m n mn

k m n

G p a a G

N

*mn m nG d G

,nm mn

m n

G G *

1

k kmn k m n

k

p a t a t

N

G Tr Gi.e.

nnn

Tr 2

1

kk n

k n

p a t

N

1k

k

p

N

21k

nn

a t k normalized :1

*

1

k kk

k

G p d G

N

1k k k Assuming k normalized, i.e.,

5.2. Statistics of the Various

Ensembles

Microcanonical ensemble : Fixed N, V, E or ,2 2

E E E E

( N, V, E; ) = # of accessible microstates

Equal a priori probabilities postulate 1

0

k

k

is accessiblep

otherwise

i.e.

lnS k ( quantum statistics: no Gibbs’ paradox )

1

k kk

k

p

N

k kn n

n

a

Energy representation:

k

kn k na

n n nH E

1

0

nn

E E

otherwise

mn m mn H

k kn n

n

t a t 1

k k kk

p

N

Pure State

1 0S Only 1 state p is accessible 3rd law

k k pp

2ml l nmn

l

m p l p l p n pl

Thus m p n p mn

i.e. 2 idempotent

In another representation with basis { m } so that

mn m p n p

pl l

l

t a

p pmn m n

, normalized

( is a projector )

Energy representation :k

k

p pmn m n *

m na a

2 * *m l l nmn

l

a a a a *m na a mn 2

1ll

a

p p

Mixed State

Multiple states are accessible, i.e. > 1.

Any representation :

1 k kmn m n

k

A

1

k kk

k

p

N

1

0

k

k

is accessiblep

otherwise

Let K be the subspace spanned by the accessible k ’s.

Consider any orthonormal basis {n } such that 1, ,n

n

n

n

K

K

Since { k } is a basis of K, its completeness means

1

,

0

mnmn

m n

otherwise

= set of accessible state indices

( is diagonal w.r.t. {n } )

k k

k

I

KA

k kn n

n

a

1

1 k kmn m n

k

N

N

1,

0

mn m n

otherwise

*

1

1 k km n

k

a a

N

Nk = ensemble member index

2

1

k km ni

mnk

ae

N

N

k

nikna a e

So that 1mn

2

m nia e

Postulate of a priori random phases

Let

Canonical Ensemble 1

k kk

k

p

N

E-representation : n n nH E i.e.

kkn k na k k

n nn

t a t

Canonical ensemble : Fixed N, V, T. 1

kEkp e

Z

1kE

k kk

eZ

1 H

k kk

eZ

kE

k

Z e

1 HeZ

HTr e kE

k

Z e H

k

k e k H

k kk

e

NQ

0 !

jjH

j

e Hj

By definition

G Tr G 1 HTr e GZ

H

H

Tr e G

Tr e

Tr AB Tr BA

Grand Canonical Ensemble1

k kk

k

p

N

Grand canonical ensemble : Fixed , V, T

,

,

r s sE N

r s

e Z 1 H Ne

Z

G Tr G 1 H NTr e G Z

H NTr e Q

0

rEN

N r

e e

Z 0

N H

N

z Tr e

0

, ,N

N

z Z N V T

0

NN

N

z Q

0

rH N E NNr r

N r

Tr e G e e E N G E N

0

, ,N

NN

z Z N V T G

0

0

, ,

, ,

N

NN

N

N

z Z N V T G

Gz Z N V T

0

0

NN N

N

NN

N

z Q G

z Q

Er, s = Er (Ns )

= E of r th state of Ns p’cle sys

5.3. Examples

An Electron in a Magnetic Field

Single e with spin1

2σ & magnetic moment Bμ σ

BH σ B

2B

e

mc

B zB ˆBB z

Pauli matrices :

0 1

1 0x

0

0y

i

i

1 0

0 1z

HZ Tr e

nni j i ii j

AAA diagonal i iA

i ji je eA

B zBHe e 0

0

B

B

B

B

e

e

B BB Be e

01

0

B

B

B

B

e

Z e

0

0

B

B

B

z B

e

e

1

z zTrZ

tanh B B

agrees with § 3.9-10

signed

A Free Particle in a Box

Free particle of mass m in a cubical box of sides L.

2 22

2 2H

m m

p 2 2 2 2

2 2 22m x y z

H E

Periodic B.C : , , , , , , , ,x y z x L y z x y L z x y z L

3/2

1 iE e

L k rr with

2 2

2

kE

m

2

L

k n with , ,x y zn n nn 0, 1, 2,in

3/2

1 iE e E

L k rr r ( r - representation )

H H

E

e e E E r r r r E

E

e E E r r *EE E

E

e r r

2

2H

m

p

2 2

2

kE

m 3/2

1 iE e

L k rr with

2 2

3

1exp

2H k

e iL m

k

r r k r r

3

3

1

2V

d k

V

k

3 2 2

3 exp22

d k ki

m

k r r

3/22

2 2exp

2 2

m m

r r ( see next page )

3 2 2

3 exp22

d k kI i

m

k r r

2

2m

2

20

1

2

i k i kkd k k e e ei

r r r r

r r

12 2

20 1

1cos exp cos

2d kk d k i k

r r

22 2

1

41

2k ik yi

d k k e e d y y e

r rr r r r

2 20

0

k ik k ikd k k e d k k e

r r r r

2

2

1

2

k ikd k k ei

r r

r r

2

iy k

r r

21

43/22

ie

r r

r r

21

42 3/2

1

22I e

r r 22

3/2

222

mm

e

r r

3/22

2 2exp

2 2H m m

e

r r r r

HZ Tr e 3 Hd r e r r3/2

22

mV

2

2

1exp

2

m

V

r r

1 HeZ

r r r r r r

is symmetric

1

V r r rParticle density at r : Location

uncertainty :th

mkT

H Tr H 31 Hd r H eZ

r r

ln Z

3

2kT

1 Z

Z

31 Hd r eZ

r r

31 HH d r H eZ

r rAlternatively

, ,f fi

r r p r r

3 31 Hd r d r H eZ

r r r r

2

3 3 21

2HH d r d r e

Z m

r r r r

2

23 3 22

1exp

2 2

md r d r

V m

r r r r

2 22 22

1r re r er r r

232

12 rr e

r r

22 3 2 rr e

2 23 32 2

13 exp

2 2

m mH d r d r

V

r r r r r r

3

2kT

Uising & integrate by parts twice :

r r r r

A Simple Harmonic Oscillator

22 21

2 2

pH m q

m

1

2nE n

n = 0,1,2,...n n nH E

2

1/4

/21

2 !n nn

mq H e

n

1/4m

q

2 2n

n

n n

dH e e

d

Hermite polynomials : Rodrigues’ formula

2

1/4

/21

2 !n nn

mq H e

n

0

nEH

n

q e q q e n n q

0

nEn n

n

e q q

is real

2 2

1/2/2 1/2

0

1

2 !n

n nnn

me e H H

n

1/2

2 21 1exp tanh coth

2 sinh 4 2 2

m mq q q q

Kubo, “Stat Mech.”, p.175Mathematica

HZ d q q e q

1/22 1

exp tanh2 sinh 2

m m qd q

1/2

12 sinh tanh2

m

m

11

2sinh2

/2

1

e

e

1/2

2 21 1exp tanh coth

2 sinh 4 2 2H m m

q e q q q q q

Probability density :

1 Hq q q q e qZ

1/2 21 1tanh exp tanh

2 2

m m q

11

2sinh2

Z

q is a Gaussian with dispersion ( r.m.s. deviation ) :

2rmsq q q

2

2

2

x x

eg

1/21

coth2 2m

Classical limit :(purely thermal)

1

tanh

x

x x

2 2

2exp2 2

m mq q

kT kT

2rms

kTq

m

Quantum limit : (non-thermal)

1

tanh 1

x

x

2

expm m q

q

1/2 21 1

tanh exp tanh2 2

m m qq

2rmsqm

2

0 q = Probability density of ground state

11

2sinh2

Z

ln Z

H Tr H1 HeZ

1 1coth

2 2

1/2 21 1

tanh exp tanh2 2

m m qq

2 2

2exp2 2

m mq q

H H

2rms

Hq

m

2 21. .

2P E m q 2 21

2 rmsm q 0q 1

2H

. . . .K E H P E 1

2H

5.4. Systems Composed of Indistinguishable Particles

N non-interacting particles subject to the same 1-particle hamiltonian h.

1

, ,N

i ii

H h q p

q p h u u

E EH E q q

1

i

N

E ii

u q

q1

i

N

i

E

i = label of the eigenstate assumed by the i th particle.

Let n = # of particles occupying the th eigenstate.

N n

E n

,1

n

E L jj

u q

q

L( , j ) = label of the j th particle that occupies the th eigenstate.

,1

n

E L jj

u q

q Note: [ ... ] = 1 if n = 0.

Let P denote a permutation of the particle labels :

,1

n

E E P L jj

P P u q

q q

i P iq P q q q

1, 2, 3, 1 , 2 , 3 , ...,i N P i P P P P N

Indistinguishable particles : 1W n

Distinguishable particles :

permutations within the same counted as the same.

permutations across different ’s counted as distinct.

# of distinct microstates is !

!

NW n

n

,1

n

E L jj

u q

q Boltzmannian ( distinguishable p’cles)

Boltz q

Indistinguishable Particles

Particles indistinguishable Physical properties unchanged under particle exchange

2 2P P

symmetricP

anti symmetric

i.e. P P

2P

Anti-symmetric : P

A A BoltzP

C P q q detA iC u q

1 1 1

2 2 2

1 2

1 2

1 2N N N

N

NA

N

u q u q u q

u q u q u qC

u q u q u q

Pauli’s exclusiion principle 0A qi j

0,1n

i.e.

Symmetric : S S BoltzP

C P q q perS iC u q

21

0FD

if n NW n

otherwise

Fermi-Dirac statistics

1BEW n Bose-Einstein statistics

5.5. The Density Matrix & the Partition Function of a System of Free Particles

N non-interacting, indistinguishable particles :

1 1 1 1

1, , , , , , , ,

,H

N N N NeZ N

r r r r r r r r

31 1, , , , ,H N H

N NZ N Tr e d r e r r r r

Let i stands for ri , & i for ri .

e.g.,

1

1, , 1 , , 1, , 1 , ,,

HN N N e NZ N

Goal: To write , 1,N

Z N Z 1N

NQ Q or

1, , 1 , , 1, , 1 , ,H H

E

N e N N e E E N

1

!

1, ,!

j

j

j

NP

P j

n

N u P jN

kk

K k

*1, , 1 , ,HE E

E

e N N

Non-interacting particles

1/3

2

V

k n

2 2

2E

m

K 22

12

N

iim

k

Periodic B.C. 0, 1, 2,in 1 iu eV

k rk r

BosonsFermions

1

!

!

j

j

P j

NP

P j

n

u jN

k

k

k

1 1 1

2 2 2

1 2

1 2

1 2N N N

u u u N

u u u N

u u u N

k k k

k k k

k k k

1 2

1 2

1 2

1 1 1

2 2 2N

N

N

u u u

u u u

u N u N u N

k k k

k k k

k k k

Mathematica

2 2 / 2 *1, , 1 , , 1, , 1 , ,

mHN e N e N N

K

K KK

1

!

1, ,!

j

j

j

NP

P j

n

N u P jN

kk

K k

2 2 22

12 2

N

ii

Em m

Kk

Consider the N ! permutations among { ki } associated with a given K.

E is unchanged

1

1, ,P ji i

NP

P j

P N P u j

K kk k

1

i

P j

NP P

P j

u j

k

k

iP P P k

1, ,iP

N k

K

1

!

!

j

j

P j

NP

P j

n

u jN

k

k

k

22

12 *1

1, , 1 , , 1, , 1 , ,!

N

ii

i i

i

mHN e N e N NN

k

k kk

22

12 *

2, ,

1

!

N

ii

j P m

i

m P P

P P j m

e u P j u mN

k

k kk

nk > 1 cases neglected(measure 0)

1

!

1, ,!

j

j

j

NP

P j

n

N u P jN

kk

K k

1

!

!

j

j

P j

NP

P j

n

u jN

k

k

k

22

12 *

2, , 1

11, , 1 , ,

!

N

ii

j P m

i

Nm P PH

P P j m

N e N e u P j u mN

k

k kk

22

12 *

, 1

1

!

N

ii

j P m

i

Nm P P

P j m

e u P j u mN

k

k kk

22

12 *

1

1

!

N

ii

j j

i

Nm P

P j

e u P j u jN

k

k kk

2

2*2

1

1

!

j

j j

j

NP m

P j

e u P j u jN

k

k kk

1

1

!

NP h

jP jP j

eN

r r

arbitrary P

P = I

2

2h

m

p2-p'cle

1

11, , 1 , ,

!

NPH h

jP jP j

N e N eN

r r

3/22

2 21

1exp

! 2 2

NP

jP jP j

m m

N

r r

from § 5.3

3 /22

2 2

1exp

! 2 2

NP

jP jP j

m m

N

r r

2

3 2

1exp

!P

jP jNP jN

r r

22 2

m m k T

= thermal ( de Broglie ) wavelength

2

3 2

11, , 1 , , exp

!PH

jP jNP j

N e NN

r r

3

11, , 1 , , ,

!PH

NP j

N e N f P j jN

2

2, exp i jf i j

r r 2

2, exp 1i if i i

r rLet with

3

11, , 1, , 1 , , , , ,

!H

Ni j i j k

N e N f i j f j i f i j f j k f k iN

mean inter-particle distance =1/3

1/3Vn

N

n = particle density

1/3n , i jf i j 3

11, , 1, ,

!H

NN e N

N

3, , 1, , 1, ,N HZ N T V d r N e N 3!

N

N

V

N 1

1, ,!

NZ T V

N

Mathematica

3, ,

!

N

N

VZ N T V

N 1

1, ,!

NZ T V

N

Resolution of problems in classical statistics:

1.Gibbs correction factor ( 1 / N! ).

2. Phase space volume per state 0 h 3 33

1 N NN

d d q d ph

1/3n Classical limit :

Non-classical systems are said to be degenerate.

n 3 = degeneracy discriminant

1

1, , 1, , 1, , 1, ,, ,

HN N N e NZ N T V

3

3

! 1

!

N

N N

N

V N

1NV

Classical limitN r r ( no spatial

correlation )

1/3n

Exchange Correlation

Let N = 2 :

6

11, 2 1,2 1 1,2 2,1

2He f f

2

2, exp i jf i j

r r

2

1 26 2

1 21 exp

2

r r

23 31 2 1 26 2

1 22, , 1 exp

2Z T V d r d r

r r

3 26 2

1 21 exp

2V d r r

2 26 2

0

1 24 exp

2V V d r r r

3

11, , 1, , 1 , , , , ,

!H

Ni j i j k

N e N f i j f j i f i j f j k f k iN

2 26 2

0

1 22, , 4 exp

2Z T V V V d r r r

3/22 2

3

1 4 1 31

2 2 2 2

V

V

2 1 / 2

0

1 11

2 2nn xd x x e n

2 3

3 3/2

1 11

2 2

V

V

3 1 1

2 2 2 2

2

3

1

2

V

Classical limit

1

1, 2 1,2 1, 2 1,22, ,

HeZ T V

2

1 22 2

1 21 exp

V

r r 2

1 2

21

0

Bosons

FermionsV

r r

1, 2 1,2 1,2 1,2k k k

k

p

2

1 2,k k

k

p r r

Statistical Potential

22

21 expsv re r

2

2

2ln 1 expsv r kT r

Mathematica

2 1, 2 1,2V