The Final Lecture (#40): ReviewThe Final Lecture (#40): ReviewChapters 1-10, Wednesday April 23Chapters 1-10, Wednesday April 23rdrd

•Announcements

•Homework statistics

•Finish review of third exam

•Quiz (not necessarily in this order)

•Review Chapters 3 to 7

Reading: Reading: Chapters 1-10 (pages 1 - 207)Chapters 1-10 (pages 1 - 207)Final: Wed. 30th, 5:30-7:30pm in hereFinal: Wed. 30th, 5:30-7:30pm in hereExam will be cumulativeExam will be cumulative

Homework StatisticsHomework Statistics

20 40 60 80 1000

2

4

6

8

10

12

14

Mean = 81%Median = 88%

Num

ber

of s

tude

nts

Score (%)

Review of Review of Chapters 3 & 4Chapters 3 & 4

Classical and statistical probabilityClassical and statistical probability

Classical probability:

•Consider all possible outcomes (simple events) of a process (e.g. a game).

•Assign an equal probability to each outcome.

Let W = number of possible outcomes (ways)Assign probability pi to the ith outcome

1 1& 1i i

i

p p WW W

Classical and statistical probabilityClassical and statistical probability

Statistical probability:

•Probability determined by measurement (experiment).

•Measure frequency of occurrence.

•Not all outcomes necessarily have equal probability.•Make Make N N trialstrials

•Suppose Suppose iithth outcome occurs outcome occurs nnii times times

lim ii N

np

N

0 1 2 3 4 5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

N 1 0.510 0.15100 0.041000 0.013210000 0.00356100000 0.00145

log(

)

log(N)

log log

0.516

a N b

a

Statistical fluctuationsStatistical fluctuations

1/ 2N

1/ 2 1/ 2Error: , Relative error ( / )i i i i in n n n n

The axioms of probability theoryThe axioms of probability theory

1. pi ≥ 0, i.e. pi is positive or zero

2. pi ≤ 1, i.e. pi is less than or equal to 1

3. For mutually exclusive events, probabilities add, i.e.

• Compound events, (Compound events, (ii + + jj): this means either event ): this means either event ii occurs, or event occurs, or event jj occurs, or both. occurs, or both.

• Mutually exclusive: events Mutually exclusive: events ii and and jj are said to be mutually exclusive are said to be mutually exclusive if it is impossible for both outcomes (events) to occur in a single if it is impossible for both outcomes (events) to occur in a single trial.trial.

1 2 ........ rp p p p • In general, for In general, for rr mutually exclusive events, the probability that one mutually exclusive events, the probability that one

of the of the rr events occurs is given by: events occurs is given by:

Independent eventsIndependent events

Example:What is the probability of What is the probability of rolling two sixes?rolling two sixes?

Classical probabilities:Classical probabilities:

16 6p

Two sixes:Two sixes:

1 1 16,6 6 6 36p

•Truly independent events always satisfy this property.

•In general, probability of occurrence of r independent events is:1 2 ........ rp p p p

ni

xi

Statistical distributionsStatistical distributions

87 9 106

, wherei iiii

n xx N n

N Mean:

Statistical distributionsStatistical distributions

ni

xi

16

, where lim ii i ii N

nx p x p

N Mean:

N

Statistical distributionsStatistical distributions

ni

xi

16

2 2

i iix p x x Standard

deviation

2

2

1( ) exp

22

x xp x

Statistical distributionsStatistical distributions

Gaussian distribution(Bell curve)

Statistical Mechanics – ideas and Statistical Mechanics – ideas and definitionsdefinitionsA quantum state, or microstateA quantum state, or microstate

•A unique configuration.A unique configuration.•To know that it is unique, we must specify it To know that it is unique, we must specify it

as completely as possible...as completely as possible...

Classical probabilityClassical probability

•Cannot use statistical probability.Cannot use statistical probability.•Thus, we are forced to use classical Thus, we are forced to use classical

probability.probability.An ensembleAn ensemble

•A collection of separate systems prepared in A collection of separate systems prepared in precisely the same way.precisely the same way.

Statistical Mechanics – ideas and Statistical Mechanics – ideas and definitionsdefinitionsThe microcanonical ensemble:The microcanonical ensemble:

Each system has same:Each system has same: # of particles# of particlesTotal energyTotal energyVolumeVolumeShapeShapeMagnetic fieldMagnetic fieldElectric fieldElectric field

and so on....and so on....

............

These variables (parameters) specify the These variables (parameters) specify the ‘macrostate’ of the ensemble. A macrostate is ‘macrostate’ of the ensemble. A macrostate is specified by ‘an equation of state’. Many, many specified by ‘an equation of state’. Many, many different microstates might correspond to the same different microstates might correspond to the same macrostate.macrostate.

Ensembles and quantum states Ensembles and quantum states (microstates)(microstates)

Cell volume, Cell volume, VV

Volume Volume VV 10 particles, 36 cells10 particles, 36 cells

10

16

1

36

3 10

ip

Ensembles and quantum states Ensembles and quantum states (microstates)(microstates)

Cell volume, Cell volume, VV

Volume Volume VV 10 particles, 36 cells10 particles, 36 cells

10

16

1

36

3 10

ip

EntropyEntropy

Boltzmann hypothesis: the entropy of a system is related to the probability of its being in a state.

1 np S f W WW

lnBS k W

Rubber band modelRubber band model

d

! !

,! ! ! !

N NW N n

n n n N n

ln ln ln lnW N N n n N n N n

Sterling’s approximation: ln(Sterling’s approximation: ln(NN!) = !) = NNlnlnNN NN

1 1 1 1ln ln

2 2 2 2

x x x xN

Chapters 5-7Chapters 5-7

•Canonical ensemble and Boltzmann probability

•The bridge to thermodynamics through Z

•Equipartition of energy & example quantum systems

•Identical particles and quantum statistics

•Spin and symmetry

•Density of states

•The Maxwell distribution

Review of main results from lecture 15Review of main results from lecture 15

Canonical ensemble leads to Boltzmann distribution function:

exp / exp /

exp /i B i B

i

j Bj

E k T E k Tp

ZE k T

Partition function:

exp /j j BjZ g E k T

Degeneracy: gj

Entropy in the Canonical EnsembleEntropy in the Canonical Ensemble

M systemsni in state i

1 2

!

! !.. !..Mi

MW

n n n

ln lni iM B B i ii i

n nS k M k M p p

M M

lnB i iiS k p p Entropy per system:

The bridge to thermodynamics The bridge to thermodynamics through through ZZ exp / ;j B

j

Z E k T js represent different configurations

lnBF k T Z

ln lnlnB B

V V V

T Z ZFS k k Z T

T T T

2 2ln lnln lnB B B

V V

Z ZU TS F k T Z T k T Z k T

T T

2

2VV V V

U S FC T T

T T T

A single particle in a one-dimensional A single particle in a one-dimensional boxboxV(x)

V = ∞ V = 0 V = ∞

xx = L

sinn

n xA

L

0 0

2 2 22

22n

nn

mL

The three-dimensional, time-independent SchrThe three-dimensional, time-independent Schrödinger equation:ödinger equation:

2

2 , , , , , , , ,2

x y z V x y z x y z x y zm

2 2 22

2 2 2x y z

A single particle in a three-dimensional A single particle in a three-dimensional boxbox

1 2 31 2 3, , sin sin sin , ,i

n x n y n zx y z A n n n

L L L

2 2

2 2 21 2 32

, 1,2,3...2n in n n n

mL

Factorizing the partition functionFactorizing the partition function

22 231 2

1 2 3

22 231 2

1 2 3

22 231 2

trans1 1 1

1 1 1

1 2 3

0 0 0

3/3/ 2 23 3/ 2

2 3 22 2

nn n

n n n

nn n

n n n

nn n

B B

D

Z e e e

e e e

e dn e dn e dn

mk T V mk LL T

2

2 2

22mL

Equipartition theoremEquipartition theorem

22 231 2

1 2 3

trans 1 2 31 1 1

nn n

n n n

Z e e e Z Z Z

If the energy can be written as a sum of independent terms, then the partition function can be written as a product of the partition functions due to each contribution to the energy.

1 2 3 1 2 3ln lnN

N NZ Z Z Z Z N Z Z Z

1 2 3ln ln lnBF Nk T Z Z Z

free energy may be written as a sum. It is in this way that each degree of freedom ends up contributing 1/2kB to the heat capacity.

1 2 3 1 2 3ln ln ln lnB BF k T Z Z Z k T Z Z Z

Also,

Rotational energy levels for diatomic Rotational energy levels for diatomic moleculesmolecules

2

122 1

l

l

l lI

g l

I = momentof inertia

l = 0, 1, 2... is angular momentum quantum number

CO2 I2 HI HCl H2

R(K) 0.56 0.053 9.4 15.3 88

Vibrational energy levels for diatomic Vibrational energy levels for diatomic moleculesmolecules

12n n

= naturalfrequency ofvibration

n = 0, 1, 2... (harmonic quantum number)

I2 F2 HCl H2

V(K) 309 1280 4300 6330

Specific heat at constant pressure for HSpecific heat at constant pressure for H22C

P (

J.m

ol1.K

1)

5

2

R

7

2

R9

2

R

HH22 boils boils

TranslationTranslation

CCPP = = CCVV + + nRnR

Examples of degrees of freedom:Examples of degrees of freedom:

2 2 1 12 2

2 2 1 12 2

2 2 2 32

2 2 1 1, 2 2

1 1

2 21 1

2 21

21

2average, or r.m.s. value

LC B B

HO B B

trans x y z B

rot dia x y B B

E C V L i k T k T

E k x m v k T k T

E m v v v k T

E I k T k T

BosonsBosons

2,Bose 1 2 1 2 2 1 2,Bose 2 1

1, ,

2i j i jx x x x x x x x

3,Bose 1 2 2 1 2 3 2 1 3

2 3 1 3 2 1

1 1 2 1 3 2

, , i j k i j k

i j k i j k

i j k i j k

x x x x x x x x x

x x x x x x

x x x x x x

• Wavefunction symmetric with respect to exchange. There are N! terms.• Another way to describe an N particle system:

1 2 3

1 1 2 2 3 3

, , ,i

i

n n n

E n n n

• The set of numbers, ni, represent the occupation numbers associated with each single-particle state with wavefunction i.

• For bosons, occupation numbers can be zero or ANY positive integer.

FermionsFermions

2,Fermi 1 2 1 2 2 1 2,Fermi 2 1

1, ,

2i j i jx x x x x x x x

• Alternatively the N particle wavefunction can be written as the determinant of a matrix, e.g.:

1 1 1

3,Fermi 1 2 3 2 2 2

3 3 3

( ) ( ) ( )

, , ( ) ( ) ( )

( ) ( ) ( )

i j k

i j k

i j k

x x x

x x x x x x

x x x

• The determinant of such a matrix has certain crucial properties:

1. It changes sign if you switch any two labels, i.e. any two rows.

It is antisymmetric with respect to exchange

2. It is ZERO if any two columns are the same.

• Thus, you cannot put two Fermions in the same single-particle state!

FermionsFermions• As with bosons, there is another way to describe N particle system:

1 2 3

1 1 2 2 3 3

, , ,i

i

n n n

E n n n

• For Fermions, these occupation numbers can be ONLY zero or one.

0

/ 2 / 3 /Fermi

B B Bk T k T k TZ e e e

BosonsBosons

1 1 2 2 3 3iE n n n • For bosons, these occupation numbers can be zero or ANY positive

integer.

/ 2 / 3 / 4 /Bose 1 2B B B Bk T k T k T k TZ e e e e

A more general expression for A more general expression for ZZ• What if we divide by 2 (actually, 2!):

31 2

1 31 2 1 4

2 3 2 52 4

//2

1 1

2 /2 / 2 /1 1 12 2 2

// /

/ //

1

2!j Bi B

BB B

BB B

B BB

M Mk Tk T

i j

k Tk T k T

k Tk T k T

k T k Tk T

Z e e

e e e

e e e

e e e

• Terms due to double occupancy – under counted.

• Terms due to single occupancy – correctly counted.

SO: we fixed one problem, but created another. Which is worse?•Consider the relative importance of these terms....

Dense versus dilute gasesDense versus dilute gases

•Either low-density, high temperature or high mass

•de Broglie wave-length

•Low probability of multiple occupancy

•Either high-density, low temperature or low mass

•de Broglie wave-length

•High probability of multiple occupancy

Dilute: classical, particle-like Dense: quantum, wave-like

D

D (mT )1/2 D (mT )1/2

A more general expression for A more general expression for ZZ• Therefore, for N particles in a dilute gas:

1

!

N

N

ZZ

N

1ln ln 1BF Nk T Z N

and

VERY IMPORTANT:VERY IMPORTANT: this is completely incorrect if the gas is this is completely incorrect if the gas is densedense..

• If the gas is dense, then it matters whether the particles are bosonic If the gas is dense, then it matters whether the particles are bosonic or fermionic, and we must fix the error associated with the doubly or fermionic, and we must fix the error associated with the doubly occupied terms in the expression for the partition function.occupied terms in the expression for the partition function.

• Problem 8 and Chapter 10.Problem 8 and Chapter 10.

Identical particles on a latticeIdentical particles on a lattice

Localized Localized → Distinguishable→ Distinguishable

1 1and lnN

N BZ Z F Nk T Z

DeDelocalized localized → → InIndistinguishabledistinguishable

11and ln ln 1

!

N

N B

ZZ F Nk T Z N

N

SpinSpin3 51

2 2 2: , , ,....

: 0, , 2 , 3 ,....

Fermions

Bosons

12 space spin

1

2

3

4

Symmetric

Antisymmetric

}

12 1 2 1 2

12 1 2 1 2

i j j i

i j j i

x x x x

x x x x

Fermions:

LxLy

Lz

1 2 3sin sin sinix y z

n x n y n zA

L L L

2 2 2 2 21 2 32 2 22ix y z

n n n

m L L L

Particle (standing wave) in a boxParticle (standing wave) in a box

3/ 2

22B

x y z

mk TZ L L L

/i Bk T

i i

en Np N

Z

Boltzmann probability:

kkyy

kkxx

kkzz

Density of states in Density of states in kk-space-space

1

2

3

xx

yy

zz

nk

L

nk

L

nk

L

The Maxwell distributionThe Maxwell distribution

In 3D: V/3 is the density of states in k space2

2( ) ;

2

VkD k

density of states per unit k interval

D(k)dk gives the # of states in the range k to k + dk

( ) / 3

( ) /22

( )2

B

B

k k Tk k TDe

f k dk N D k dk N k e dkZ

Number of occupied states in the range k to k + dk

0

f k dk N

Distribution function f (k):

Maxwell speed distribution functionMaxwell speed distribution function

2 2

33 3 2/ 2 / 22 2

2 34

2 2B Bmu k T mu k TDm m

n u N u e N u ekT

n(

u)

u

Density of states in lower Density of states in lower dimensionsdimensionsIn 2D: A/2 is the density of states in k space

( ) ;2

AkD k

density of states per unit k interval

D(k)dk gives the # of states in the range k to k + dk

In 1D: L/ is the density of states per unit k interval

Density of states in energyDensity of states in energy

2

2( ) ( )

2

dk Vk dkD d D k d d

d d

In 3D:

2 2 2

If ( ) , then 2

k d kk

m dk m

1/ 2

2 2 2 3

2( )

2 2

Vm mVkmD

Useful relations involving Useful relations involving ff ((kk))( ) /

All 0 0

( ) ( ) 1Bk k T

k k kk

eg p D k p dk D k dk

Z

=

All 0 0

( ) ( )k k kk

N g p ND k p dk f k dk N

=

( ) /0

0

0

( ) ( )

( ) ( )

( )

Bk k Tf k A k dk

eA D k A k dk

Zf k dk

=

The molecular speed distribution The molecular speed distribution functionfunction

1/ 2 1/ 2 1/ 22 8 3

m rms

kT kT kTu u u

m m m

3/ 2

2 24 exp / 22

mn u N u mu kT

kT

n u

N

/ mu u /rms mu u / mu u

Molecular Flux Molecular Flux

2 / 2

0 0

( ) ( ) sin cos

4

dN u u n u df u du du d

dAdt V

Flux: number of molecules striking a unit area of the container walls per unit time.

2

2

/ 22 4

0 0 0

/ 23

0 0 0

( ) ( )

( ) ( )

B

B

mu k T

Bmu k T

uf u du u n u du u e du

u

f u du un u du u e du

=

is the average molecular speed in the beamBu u

The Maxwell velocity distribution The Maxwell velocity distribution functionfunction

2

1/ 2

/ 2

2x Bmv k T

xB

mN v N e

k T