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{SCP_R 1 0 23{COM_R 2 0{TEXT 3 35 "Heat Capacity of an Einstein S
olid "}}{COM_R 3 0{TEXT 1 182 "Statistical Mechanics\015Tony Scot
t.\015\012The model is composed of an ensemble of \"N\" one-dimen
sional Quantum oscillator.\012First, compute the Partition Functi
on \"q\" for a single oscillator:\012"}}{INP_R 4 0 "> "{TEXT 0 66
 "q:=exp(-(1/2)*h*nu/(k*T))*sum((exp(-h*nu/(k*T))^n),n=0..infinit
y);"}}{OUT_R 5 0 4{DAG :3n3\`q`*5(3n3\`exp`,2+3*9n3\`h`j2x0001n3\
`nu`pCn3\`k`i2x0001n3\`T`p13/3p13j2x0002pC+5pCpC(3p5,2+3p9p13p13p
13}}{COM_R 6 0{TEXT 1 33 "The N-oscillator function is just"}}
{INP_R 7 0 "> "{TEXT 0 11 "Q:=q^(3*N):"}}{COM_R 8 0{TEXT 1 23 "\0
15Em is the mean energy:"}}{INP_R 9 0 "> "{TEXT 0 35 "Em:= factor
((k*T^2)*diff(ln(Q),T));"}}{OUT_R 10 0 9{DAG :3n3\`Em`+3*B+5(3n3\
`exp`,2+3*9n3\`h`j2x0001n3\`nu`pEn3\`k`i2x0001n3\`T`p15p15pEpEpEp
Ep10pEpCpEn3\`N`pE+5p6pEp15pEp15/3i2x0003j2x0002}}{COM_R 11 0
{TEXT 1 44 "Cv is the heat capacity (at constant Volume)"}}{INP_R
 12 0 "> "{TEXT 0 23 "Cv:=factor(diff(Em,T));"}}{OUT_R 13 0 12
{DAG :3n3\`Cv`+3*Fn3\`h`j2x0002n3\`nu`p7(3n3\`exp`,2+3*9p5j2x0001
p9p13n3\`k`i2x0001n3\`T`p19p19p13n3\`N`p13+5pCp13p19p13i2x0002p17
p19p1Bp28j2x0003}}{COM_R 14 0{TEXT 1 44 "\"x\" becomes is directl
y proportional to \"T\":"}}{INP_R 15 0 "> "{TEXT 0 24 "Cvx:=subs(
T=x*h*nu/k,\");"}}{OUT_R 16 0 15{DAG :3n3\`Cvx`+3*B(3n3\`exp`,2+3
*3n3\`x`i2x0001pDj2x0001n3\`N`p10+5p5p10pDp10i2x0002n3\`k`p10pBp1
Aj2x0003}}{COM_R 17 0{TEXT 1 27 "This is the limit as T -> 0"}}
{INP_R 18 0 "> "{TEXT 0 21 "limit(Cvx,x=0,right);"}}{OUT_R 19 0 
18{DAG j2x0000}}{COM_R 20 0{TEXT 1 37 "This is the limit for T ->
 + infinity"}}{INP_R 21 0 "> "{TEXT 0 22 "limit(Cvx,x=infinity);"
}}{OUT_R 22 0 21{DAG +3*5n3\`N`j2x0001n3\`k`p4j2x0003}}{COM_R 23 
0{TEXT 1 63 "\015This is indeed the result according the Equipart
ition Theorem."}}{INP_R 24 0 "> "{TEXT 0 0 ""}}}{END}
