timeCanonical partition function fixed T.vn

Q zTefEv dlnQ.lt n.u LEE

BA finde

µSEZ

knotCv

Computable for non interacting systems

E E 1 EatEst En pas ptv par plur

Q IT e BE g singleparticlepartition function

qa qN

Translatonal motion E Emv2

aim E Emf aim Eml

fk E that

DX'sv him

Classically Erns and PNS are statistically met

E U t kept Q QµQepotential Tey

configurational

Consider N particles interacting w potential U

all Wl mass m

E UK E ring zpi4m

Q htsnffarfdri fdinef.HN lfapifdpi fdpiefEm

1Qe 3N

2am342

TqmT

phi Qe if a o Qc WN

N ysn when thermal wavelength

kkzumb.AT

kLE7dY.QpI i thfpI

LuS ZNhgT 3N p's

plE pculpik gmane gemally

PIA B pcAlplB if AsBare Ind

AB A B

Harmonic oscillator in id

E Emp mw2 2

Classically v x Px

e fzmw2x2 pEmpx

g gfjpf.dff.IR

wxye.pex4zmI9Ix fmtw.Yapf2Tpm k

2 quadvariablesx p

l E 1 hot 2x test

EquipartinthommAmy indent classical Gaussian

degree of freedom contribute that to avg E

Quantum mechanically

E

1 Eiiii O

g EI e Huh Ieftw fhwy

eP'hook 1

geometric series

w

2 sink Bank

Doesn't look like the classical result in high T limit

p o O ewk

I fkwk

qQM I gemphew p w map

again if sxsp h both agree K htt

how T limit p a

paw Pawe few t

ePA te ca l g e

each term is smaller

q e ftw ePaw then thrust

just like a two level system emm

at tout

e i

q It e Post ED se l post

0 as T so B so

SEZ diff DEZ ePHefty 2

fBEne e as f ow

Graphically E jclassicalresult

it

QM iam many ne fghotmatters when

WBTE BE thwCv

Cim hrs

Qmresultt1 histtw

Harmonic Oscillators in the wild

1 Intramolecular vibrations

it

Uhh Umint Mw Ir rm t

Usually very stiff ne s host

LBT 210 cmlegOH stretch AE 3000cmsoon

most covalent bonds are in their groundstates

2 Low T solids

000 O O

88880 080888000

E i n

Xtc amorphous solid

glass

Either case low T E doesn't vary much

Fi Fil fry c small

energy min

urn WITH I 2T Sri E Serjis e

complicated

but Fe is symmetric

can be diagonalized into a set of independent modes

linear combinations of ISE Ea

E Ema's Emei si

LE 3N hist classically from equiparhkin

be df 3Nhrs

At low T 2 LS

3Nh.rsEgg pose ePdt

hot otcmw.dewYd'twhI

For whole solidCE Jawg phew e f

Einstein V Debeye final showdown

Einstein 0,0 every particle crib in a cage

0 0 of its neighbors w Cy Woto gut 3N's braes delta

ftp 3N phono e f vanishes too quickly

Debye collective modes w low freq

00000888 88

phonons standing waves of acoustic modes

2Tg ka w

gch a gcwa w

wTh Idw jcpawpe fhwa p.tw

da padw

P2 Idu 1pm p a en

a p3 integral denpends

weakly onta 1

3