Physics 112 Formula Sheet
Chitraang Murdia
May 3, 2023
1
Thermodynamics
1.1
Systems
1. Isolated system: No exchange or interaction with the environment.
2. Closed system: Exchanges energy with the environment.
3. Open system: Exchanges energy and particles with the environment.
1.2
1.3
Equilibrium
Type of Equilibrium
Exchanged Quantity
Equalized Quantity
Mechanical
Thermal
Diffusive
Volume
Energy (Heat)
Particle Number
Pressure
Temperature
Chemical Potential
State vs. Process Variables
State Variable: Properties of the equilibrium system. Change depends only on initial and final states, and not
on the path/process.
E.g.: P, V, T, N, U, µ, S
Process Variable: Change depends on the path/process
E.g.: W, Q
The infinitesimal form of a process variable is an inexact differential, whereas the infinitesimal form of a state
variable is an exact differential.
ˆ ˙
ˆ ˙
Bq
Bq
Exact differential:
dq “
dx `
dx.
(1.1)
Bx y
By x
1.4
Extensive and Intensive Properties
Extensive Property: Scale with the size of the system (double when the system is doubled or half when the
system is halved).
E.g.: V, N, U, S, m pmassq
1
Intensive Property: Independent of the size of the system (do not change when the system is doubled or halved).
E.g.: P, T, µ
The ratio of two extensive properties is intensive.
E.g.: n “
1.5
N
V
pconcentrationq, ρ “
m
V
pdensityq
Zeroth Law of Thermodynamics
If system A is in thermal equilibrium with system B and system B is in thermal equilibrium with system C then
system A is in thermal equilibrium with system C.
1.6
Ideal Gas
Gas particles are approximated as point particles with no inter-particle interactions.
Ideal Gas Equation:
P V “ N kB T.
(1.2)
N
,
V
ρ “ nM,
P “ nkB T,
n“
P M “ ρkB T,
(1.3)
(1.4)
where M is molecular mass.
1.7
Kinetic Theory of Gases
Assumptions:
1.
2.
3.
4.
V
.
Gas consists of small particles i.e., Vparticle ! Vcell “ n1 “ N
Gas consists of a large number of particles i.e., N " 1.
All collisions are elastic.
Other than collisions there are no inter-particle interactions.
xvx2 y “ xvy2 y “ xvz2 y “
1.8
kB T
M
(1.5)
Equipartition Theorem
For a classical system in thermal equilibrium at temperature T , each quadratic degree of freedom on average has
the same amount of energy 21 kB T .
Quadratic degrees of freedom (f): Individual terms in the Hamiltonian that are quadratic in phase-space variables.
2
Substance
ftranslation
frotation
fvibration
ftotal
Monoatmoic gas
Diatomic gas
Linear triatomic gas
Non-linear triatomic gas
Solid
3
3
3
3
0
0
2
2
3
0
0
2
8
6
6
3
5 or 7
5 or 13
6 or 12
6
For a system with N particles and f d.o.f.’s per particle,
U“
1.9
f
N kB T.
2
(1.6)
Heat and Work
Heat (Q): Process variable that is the flow of energy into a system due to temperature difference.
Work (W): Process variable that is the flow of energy into a system due to something else (e.g., mechanical force).
Mechanical work:
δW “ ´P dV.
(1.7)
Note that the pressure here is the external pressure at the piston which equals the internal pressure of the system
if the process is reversible.
1.10
First Law of Thermodynamics
For a closed system,
∆U “ Q ` W,
(1.8)
dU “ δQ ` δW.
(1.9)
or equivalently,
This is a statement of the conservation of energy.
1.11
Heat Capacity
Heat needed per unit temperature change
ˆ
C“
δQ
BT
˙
.
(1.10)
CpT q dT.
(1.11)
process
This is a process-dependent quantity.
Equivalently,
ż Tf
Q“
Ti
If C is temperature-independent
Q “ C∆T.
3
(1.12)
Two common paths are
ˆ
Isochoric (constant V ):
CV “
ˆ
Isobaric (constant P ):
CP “
δQ
BT
˙
δQ
BT
˙
ˆ
“
V,N
ˆ
“
P,N
BU
BT
˙
BU
BT
˙
.
(1.13)
V,N
ˆ
`P
P,N
BV
BT
˙
.
(1.14)
P,N
Specific heat is the heat capacity per unit mass (intensive quantity),
c“
C
.
m
(1.15)
γ“
CP
.
CV
(1.16)
Adiabatic coefficient is
1.12
Heat Capacity for Ideal Gas
For an ideal gas satisfying equipartition theorem,
f
N kB ,
2
f `2
CP “
N kB .
2
CV “
γ “1`
1.13
2
.
f
(1.17)
(1.18)
(1.19)
Reversible vs. Irreversible Transformation
Reversible Transformation: Transformation where the system stays nearly in equilibrium at all times, so the
transformation is quasi-statical (slow). Note that a slow transformation is not necessarily reversible.
Irreversible Transformation: Transformation that is not reversible. A spontaneous transformation is always
irreversible.
1.14
Entropy
Entropy is a state variable defined infinitesimally as
dS “
δQreversible
.
T
(1.20)
The first law of thermodynamics is
dU “ T dS ´ P dV,
(assuming N is constant).
4
(1.21)
1.15
Second Law of Thermodynamics
The entropy of the universe must increase.
∆Suniv “ ∆Ssys ` ∆Ssurr ě 0,
(1.22)
with equality iff the process is reversible.
1.16
1.16.1
Common Reversible Processes
Isochoric
V is constant.
δW “ 0,
Ideal gas:
1.16.2
dU “ δQ “ Cv dT.
∆U “ Q “
(1.23)
f
N kB ∆T.
2
(1.24)
Isobaric
P is constant.
δQ “ CP dT.
Ideal gas:
1.16.3
(1.25)
W “ ´P ∆V “ ´N kB ∆T.
Isothermal
T is constant.
Ideal gas:
1.16.4
(1.26)
∆U “ 0,
W “ ´Q “ N kB T ln
Vi
.
Vf
(1.27)
Isentropic
S is constant (no heat exchanged).
δQ “ 0,
dS “ 0
Ideal gas:
∆U “ W “
dU “ δW.
(1.28)
f
N kB ∆T.
2
(1.29)
For an ideal gas, P V γ remains constant (equivalently, T V γ´1 remains constant).
1.17
Cyclic Process
A process Γ where the system goes back to its initial state. The net change in any state variable is zero.
∆U “ QΓ ` WΓ “ 0,
(1.30)
∆Ssys “ 0.
(1.31)
∆Suniv “ ∆Ssurr ě 0
(1.32)
The second law of thermodynamics tells us
5
This is related to the Clausius inequality as
¿
´∆Ssurr “
δQ
ď 0,
T
(1.33)
with equality iff the process is reversible.
1.18
Heat Engine
A heat engine is a setup where the system is taken through a cyclic process such that net heat is absorbed by
the system and net work is done by it. In general, the engine absorbs heat |Qin | from a high-temperature bath
which is partly used to do work |Wnet | and partly supplied as heat |Qout | to the low-temperature bath,
|Qin | “ |Wnet | ` |Qout |
(1.34)
|Qin |
|Wnet |
“1´
|Qout |
|Qout |
(1.35)
The efficiency is
η“
A heat engine can be run in reverse by doing work and it acts as a refrigerator (to cool the low-temperature
bath) or a heat pump (to cool the high-temperature bath).
1.19
Reversible Carnot Engine
Reversible engine where Qin only comes in from a high-temperature bath at TH and Qout only goes out to a
low-temperature bath at TL . The cycle consists of two isothermal and two adiabatic processes.
Carnot Theorem: The efficiency of a Carnot engine only depends on the temperatures TH and TL ,
ηC “ 1 ´
1.20
TL
.
TH
(1.36)
Third Law of Thermodynamics
The entropy of a system approaches a constant value at absolute zero temperature,
lim S “ S0 ě 0.
T Ñ0
(1.37)
This S0 is independent of all other intensive variables.
Consequences:
1. The temperature of a closed system cannot be reduced to absolute zero in a finite number of finite steps.
2. Specific heat must vanish at absolute zero temperature,
lim C “ 0
T Ñ0
3. Ideal gas law cannot be extended to absolute zero temperature.
6
(1.38)
1.21
Fundamental Thermodynamic Relation
The first law of thermodynamics tells us
ÿ
dU “ T dS ´ P dV `
µi dNi ,
(1.39)
i P species
where µi dNi is chemical work (due to particles entering or leaving the system). U pS, V, tNi uq is a natural function
of these variables.
ˆ
˙
BU
,
BS V,tNi u
ˆ
˙
BU
P “´
,
BV S,tNi u
˙
ˆ
BU
.
µi “
BNi S,V,tNj‰i u
T “
(1.40)
(1.41)
(1.42)
This can be rearranged to get entropy as SpU, V, N q.
dS “
1
“
T
1.22
ˆ
BS
BU
˙
,
V,tNi u
ÿ µi
1
P
dU ` dV ´
dNi ,
T
T
T
i
P
“
T
ˆ
BS
BV
˙
,
S,tNi u
µi
“´
T
(1.43)
ˆ
BS
BNi
˙
.
(1.44)
S,V,tNj‰i u
Conjugate Pairs
A pair of thermodynamic variables that consists of:
1. A generalized displacement (extensive)
2. A corresponding generalized force (intensive)
Conjugate Pair: (Displacement, Force)
Associated Energy
(Volume, Pressure)
(Entropy, Temperature)
(Particle Number, Chemical Potential)
Mechanical Work, δWmech “ ´P dV
Heat, δQ “ T dS
Chemical Work, δWchem “ µdN
Legendre transform is used to switch from one variable in a conjugate pair to the other.
1.23
Entropy and Equilibrium
Principle of Maximum Entropy: For an isolated system (i.e., fixed U, V, tNi u), the entropy S is maximized at
equilibrium.
Principle of Maximum Entropy: For a closed system at fixed entropy (i.e., fixed S, V, tNi u), the internal energy
U is minimized at equilibrium.
7
1.24
Enthalpy
Obtained from U by Legendre transforming V Ñ P
H “ U ` P V,
(1.45)
dH “ T dS ` V dP `
ÿ
µi dNi .
(1.46)
i
HpS, P, tNi uq is natural.
ˆ
T “
BH
BS
˙
ˆ
,
V “
P,tNi u
BH
BP
˙
ˆ
,
µi “
S,tNi u
BH
BNi
˙
.
(1.47)
S,P,tNj‰i u
∆H is the maximum amount of thermal energy (i.e., heat + work) that can be extracted at constant P .
H is the capacity to do non-mechanical work and release heat.
Related to isobaric heat capacity
ˆ
CP “
1.25
δQ
BT
˙
ˆ
“
P,N
BH
BT
˙
.
(1.48)
P,N
Helmholtz Free Energy
Obtained from U by Legendre transforming S Ñ T
F “ U ´ T S,
(1.49)
dF “ ´SdT ´ P dV `
ÿ
µi dNi .
(1.50)
i
F pT, V, tNi uq is natural.
ˆ
S“´
BF
BT
˙
ˆ
,
P “´
V,tNi u
BF
BV
˙
ˆ
,
µi “
T,tNi u
BF
BNi
˙
.
(1.51)
T,V,tNj‰i u
∆F is the maximum amount of work that can be performed at constant T .
F is the capacity to work.
1.26
Gibbs Free Energy
Obtained from H by Legendre transforming S Ñ T or from F by Legendre transforming V Ñ P
G “ H ´ T S “ F ` P V “ U ´ T S ` P V,
ÿ
dG “ ´SdT ` V dP ` µi dNi .
(1.52)
(1.53)
i
GpT, P, tNi uq is natural.
ˆ
S“´
BG
BT
˙
ˆ
,
P,tNi u
V “
BG
BP
˙
ˆ
,
T,tNi u
8
µi “
BG
BNi
˙
.
T,P,tNj‰i u
(1.54)
∆G is the maximum amount of non-mechanical work that can be performed at constant T, P .
G is the capacity to do non-mechanical work.
At constant T, P , for the chemical reaction
m1 A ` m2 B é n1 C ` n2 D
(1.55)
m1 µA ` m2 µB “ n1 µC ` n2 µD .
(1.56)
to be in equilibrium,
1.27
Maxwell Relations
Relationships between partial derivatives that follow from (A.3).
E.g.: The Maxwell relations for S, T, V, P are
ˆ
˙
ˆ
˙
BT
BP
“´
BV S
BS V
ˆ
˙
ˆ
˙
BT
BV
“
BP S
BS P
˙
ˆ
˙
ˆ
BP
BS
“
BV T
BT V
ˆ
˙
ˆ
˙
BS
BV
´
“
BP T
BT P
1.28
from U,
(1.57)
from H,
(1.58)
from F,
(1.59)
from G.
(1.60)
Euler Equation
Euler Equation:
U “ TS ´ PV `
ÿ
µi Ni ,
(1.61)
i
This follows from scaling the system by a factor λ, in which case extensitivity tells us U pλS, λV, tλNi uq “
λU pS, V, tNi uq.
H “ TS `
ÿ
µi Ni ,
(1.62)
i
F “ ´P V `
ÿ
µi Ni ,
(1.63)
i
G“
ÿ
µi Ni .
(1.64)
i
The Gibbs-Duhem equation relating the intensive differentials is
ÿ
SdT ´ V dP ` Ni dµi “ 0.
i
These three intensive variables are not independent.
9
(1.65)
2
Transport Processes
2.1
Heat Conduction
Heat flux density is heat exchanged per unit area per unit time,
δQ
.
dAdt
qx “
Fourier’s law of heat conduction is
qx “ ´kt
(2.1)
BT
,
Bx
(2.2)
where kt is the thermal conductivity.
The heat equation in one-dimension is
BT
B2 T
“K 2,
Bt
Bx
K“
kt
.
ρc
(2.3)
This equation explicitly breaks time-reversal symmetry i.e., forward and backward time evolutions are different.
The corresponding results in three-dimensions are
⃗
⃗q “ ´kt ∇T,
(2.4)
BT
“ K∇2 T.
Bt
(2.5)
Steady-state is given by Laplace equation ∇2 T “ 0.
2.2
Diffusion
Diffusion is particle flow due to a difference in concentration.
Particle flux density is particles exchanged per unit area per unit time,
δN
.
dAdt
(2.6)
Bn
,
Bx
(2.7)
B2 n
Bn
“ D 2.
Bt
Bx
(2.8)
Jx “
Fick’s Law of Diffusion is
Jx “ ´D
where D is the diffusion coefficient.
The diffusion equation in one-dimension is
This equation explicitly breaks time-reversal symmetry i.e., forward and backward time evolutions are different.
The corresponding results in three-dimensions are
⃗
J⃗ “ ´D∇n,
(2.9)
Bn
“ D∇2 n.
Bt
(2.10)
Steady-state is given by the Laplace equation ∇2 n “ 0.
10
2.3
Mean Free Path and Mean Free Time
Interactions are described via the collision cross-section σ. Any particle passing within this collision cross-section
will collide (and randomize momentum).
E.g.: For the hard sphere model, σ “ 4πr2 where r is the radius of the hard sphere.
The mean free path is the average distance traveled by a particle between two collisions.
ℓ“
1
.
σn
(2.11)
The mean free time is the average time spent by a particle between two collisions.
∆t “
3
3.1
ℓ
vrms
“
1
.
σnvrms
(2.12)
Classical Statistical Mechanics
Microstates and Macrotates
Microstate: A specification of all possible aspects of the system
Macrostate: A specification of select bulk/net properties
ÿ
ppmacrostateq “
ppmicrostateq
(3.1)
microstates in
the macrostate
3.2
One-dimensional Random Walk
At any step, go to right with probability pR and left with probability pL .
pR ` pL “ 1.
(3.2)
Microstate is described by a sequence of steps
ppmicrostateq “
ź
R NL
ppstepiq “ pN
R pL .
(3.3)
stepi
Macrostate is described by the number of right steps NR or net displacement
X “ pNR ´ NL qℓ “ p2NR ´ N qℓ.
(3.4)
The multiplicity of a macrostate is
˜
ΩN pNR q “
N
NR
¸
“
N!
.
NR !NL !
(3.5)
The probability of a macrostate is
R NL
pN pNR q “ ΩN pNR qpN
R pL .
11
(3.6)
xNR y “ N pR ,
(3.7)
xXy “ ℓN ppR ´ pL q “ ℓN p2pR ´ 1q,
a
σX “ 2ℓ N pR pL
3.3
(3.8)
(3.9)
Paramagnet
A binary model system of spins in a magnetic field B.
Microstate is described by the spin state (Ò or Ó) of each spin.
The magnetization is given by
N
ÿ
M“
µσi ,
(3.10)
i“1
and the energy is
U “ ´BM “ ´
N
ÿ
µBσi .
(3.11)
i“1
Here σi “ 1 for spin-up (Ò) and σi “ ´1 for spin-down (Ó).
Macrostate is described by the number of up-spins NÒ . Equivalent macrostate labels are the energy or magnetization
U “ ´µB p2NÒ ´ N q ,
M “ µ p2NÒ ´ N q .
(3.12)
The multiplicity is
˜
ΩpU, N q “
3.4
N
NÒ
¸
“
N!
.
NÒ !NÓ !
(3.13)
Einstein Solid
A system of N quantum harmonic oscillators with identical frequency ω. The energy levels of a quantum harmonic
oscillator are
ˆ
˙
1
En “ n `
ℏω,
(3.14)
2
where n “ 0, 1, 2, . . . .
Microstate is described by the energy level of each oscillator tni u.
The total energy is
˜
¸
˙
N ˆ
N
ÿ
ÿ
1
N
Ei “
ni `
E“
ℏω “
ni ℏω ` ℏω.
2
2
i“1
i“1
i“1
N
ÿ
We can define the energy above the ground state as
˜
¸
N
ÿ
U“
ni ℏω “ ℏωq
i“1
12
(3.15)
(3.16)
where q “
řN
i“1
ni .
Macrostate is described by the total energy, or equivalently by U or q. The multiplicity is
˜
¸ ˜
¸
q`N ´1
q`N ´1
ΩpU, N q “
“
.
N ´1
q
3.5
(3.17)
Ideal Gas
A system of N particles of gas in volume V .
Microstate is described by the position and velocity of all N particles. The phase space is the 6N -dimensional
space of all positions and momenta.
Macrostate is described by total energy U
U“
N
ÿ
|⃗
pi |2
.
2m
i“1
(3.18)
The multiplicity is
1
1
p2πmU q3N {2
ˆ 3N ˆ
ˆVN
N! h
p3N {2q!
ˆ
˙3N {2 ˆ ˙N
4πm U
V
«
e5N {2 .
3h2 N
N
ΩpU, V, N q “
3.6
(3.19)
Microcanonical Ensemble
The ensemble associated with an isolated system at given energy U and other extensive properties like V and N .
Fundamental assumption: Given a macrostate for an isolated system, each microstate in the microcanonical
ensemble consistent with the known macrostate values occurs with the same probability
ppmicrostateq “
1
.
Ωpmacrostateq
(3.20)
A partition function gives all the statistical properties of an equilibrium system. The microcanonical partition
function is Ωpmacrostateq.
3.7
Entropy in Microcanonical Ensemble
Entropy is given by Boltzmann’s equation
S “ kB ln Ω.
(3.21)
σ “ ln Ω.
(3.22)
Dimensionless entropy is given by
Thermodynamic beta is defined as
ˆ
β“
Bσ
BU
˙
“
N,...
13
1
.
kB T
(3.23)
This is constant across systems in thermal equilibrium.
The intensive thermodynamic quantities are related to the entropy via
ˆ
˙
ˆ
˙
ˆ
˙
1
BS
P
BS
µi
BS
“
,
“
,
“´
.
T
BU V,tNi u
T
BV S,tNi u
T
BNi S,V
(3.24)
The lowest value of multiplicity is Ω “ 1, so
S ě kB ln 1 “ 0.
(3.25)
Stotal “ SA ` SB
(3.26)
Entropy is an extensive quantity,
where A and B are subsystems that make up the full system.
Maximum entropy principle: Any large system will be found in the macrostate that maximizes the total isolatedsystem entropy.
3.8
Ideal Gas in Microcanonical Ensemble
The entropy is
„
SpU, V, N q “ N kB
3
ln
2
ˆ
4πm U
3h2 N
˙
ˆ
` ln
V
N
˙
`
ȷ
5
.
2
(3.27)
The intensive thermodynamic quantities are given by
3 N kB
1
“
,
T
2 U
P
N kB
“
,
T
V „
ˆ
˙
ˆ ˙ȷ
µ
3
4πm U
V
“ ´kB
ln
` ln
.
2
T
2
3h N
N
(3.28)
(3.29)
(3.30)
The thermal wavelength is defined as
λth “ ?
h
,
2πmkB T
(3.31)
and the quantum concentration is defined as
1
nQ “ 3 “
λth
ˆ
2πmkB T
h2
˙3{2
.
(3.32)
The chemical potential is
ˆ
µ “ kB T ln
14
n
nQ
˙
.
(3.33)
3.9
Canonical Ensemble
The ensemble associated with a closed system at a fixed temperature T .
The probability factor is given by the Boltzmann factor
p̃p q “ e´Eps q{kB T .
s
(3.34)
The relative probabilities between two microstates is
pp 1 q
e´Eps 1 q{kB T
“ ´Eps q{k T “ e´∆E{kB T .
2
B
pp 2 q
e
s
s
3.10
(3.35)
Canonical Partition Function
The canonical partition function is defined as
e´Eps q{kB T .
ÿ
Z“
microstate
(3.36)
s
This acts as the normalization factor for probabilities
s
pp q “
1 ´Eps q{kB T
e
.
Z
(3.37)
For an N -element system with energy
s
Ep q “
N
ÿ
s
Ej p j q,
(3.38)
Zj ,
(3.39)
j“1
the partition function factorizes
Z“
N
ź
j“1
assuming the elements are distinguishable.
If the N elements are the same, then the partition function for distinguishable elements is
Z “ Z1N ,
(3.40)
and for indistinguishable elements is
Z“
1 N
Z ,
N! 1
(3.41)
where Z1 is the single element partition function.
If gpEq is the degeneracy/multiplicity of energy E,
ÿ
Z“
gpEq e´E{kB T ,
(3.42)
E
and the probability the system having energy E is
ppEq “
1
gpEq e´E{kB T .
Z
(3.43)
The expectation value is given by
ÿ
xQy “
s
s s
Qp qpp q “
1 ÿ
Qp qe´Eps q{kB T .
Z s
15
s
(3.44)
3.11
Partition Function for Classical Systems
The canonical partition function given f position-momentum pairs is
ż ż f f
d xd p ´Hptxu,tpuq{kB T
e
Z“
.
hf
(3.45)
The probability density function is
ρptxu, tpuq “
1 ´Hptxu,tpuq{kB T
e
.
Zhf
(3.46)
For a one-dimensional classical harmonic oscillator,
Z“
kB T
.
ℏω
(3.47)
If we want to focus on a classical variable q that appears in the energy via
H “ Qpqq ` Hother ,
(3.48)
the normalized probability density function is
ρq pqq “ ş
e´Qpqq{kB T
dq e´Qpqq{kB T
(3.49)
2
For a quadratic term Qpqq “ c pq ´ q0 q which represents a degree of freedom,
xQy “
kB T
.
2
(3.50)
This is the equipartition theorem.
3.12
Some Examples of Partition Function
For a paramagnet,
ˆ
˙
µB
,
kB T
„
ˆ
˙ȷN
µB
.
Z “ 2 cosh
kB T
Z1 “ 2 cosh
(3.51)
(3.52)
For an Einstein solid,
„
Z1 “ 2 cosh
ˆ
ℏω
2kB T
˙ȷ´1
,
˙ȷ´N
„
ˆ
ℏω
Z “ 2 cosh
.
2kB T
Note that Z1 is the partition function for a single quantum harmonic oscillator.
16
(3.53)
(3.54)
For an ideal gas,
1
V
3{2
p2πmkB T q V “ 3 “ nQ V,
h
λth
ˆ
˙3N {2
1 2πmkB T
1
N
pnQ V q .
Z“
VN “
2
N!
h
N!
Z1 “
(3.55)
(3.56)
For a quantum rotor,
Z1 “
8
ÿ
p2j ` 1qeℏ
2
jpj`1q{2IkB T
.
(3.57)
j“0
At high temperature, we can approximate the sum as an integral to get
Z1 «
2IkB T
.
ℏ2
(3.58)
Note that the states of the quantum rotor are described by two qunatum number j, mj which are given by
L2 “ ℏ2 jpj ` 1q,
j “ 0, 1, 2, . . . ,
(3.59)
mj “ ´j, ´j ` 1, . . . j ´ 1, j.
(3.60)
ℏ2
2I
(3.61)
gpEj q “ 2j ` 1.
(3.62)
Lz “ ℏmj ,
The energy levels are
Ej “ jpj ` 1q
with degeneracy
3.13
Energy and Heat Capacity
The energy cannot be specified exactly in the canonical ensemble. The average energy is
xEy “ ´
Bpln Zq
1 BZ
“´
.
Z Bβ
Bβ
(3.63)
This corresponds to the internal energy U in thermodynamics.
Energy fluctuations are characterized by
2
σE
“
B 2 pln Zq
.
Bβ 2
(3.64)
2
σE
.
kB T 2
(3.65)
The heat capacity is related to energy fluctuations via
CV “
Note that
B
B
“ ´kB T 2
Bβ
BT
17
(3.66)
3.14
Helmholtz Free Energy and Entropy
The Helmholtz free energy is canonical ensemble is
F “ ´kB T ln Z.
(3.67)
Entropy is given by
ˆ
S“´
BF
BT
In this case F “ U ´ T S becomes,
ˆ
F “ xEy ` T
3.15
˙
.
(3.68)
V,N
BF
BT
˙
.
(3.69)
V,N
Maxwell-Boltzmann Distribution
The distribution of speeds in an ideal gas is given by the probability density function
˙3{2
ˆ
2
m
v 2 e´mv {2kB T .
ρv pvq “ 4π
2πkB T
This distribution has a peak at the most probable speed
c
vmp “
2kB T
.
m
(3.70)
(3.71)
The energy distribution is
c
ρE pEq “ 2
3.16
E
π
ˆ
1
kB T
˙3{2
v 2 e´E{kB T .
(3.72)
Chemical Potential
In the microcanonical ensemble,
ˆ
µi “ ´T
BS
BNi
˙
,
S “ kB ln Ω.
(3.73)
F “ ´kB T ln Z.
(3.74)
S,V,tNj‰i u
In the canonical ensemble,
ˆ
µi “
BF
BNi
˙
,
S,V,tNj‰i u
For a classical ideal gas with internal structure captured by Z1,int ,
ˆ
˙
n
µ “ kB T ln
.
nQ Z1,int
(3.75)
For a chemical reaction
R1 ` R2 ` . . . Rm é P1 ` P2 ` . . . Pn
(3.76)
in equilibrium, the sum of the chemical potentials of the reactants equals the sum of the chemical potentials of
the products,
µR1 ` µR2 ` . . . µRm “ µP1 ` µP2 ` . . . µPn .
(3.77)
18
3.17
Grand Canonical Ensemble
The ensemble associated with an open system at a fixed temperature T and chemical potential µ.
The probability factor is given by the Gibbs factor
p̃p q “ e´βpEps q´µN ps qq .
s
3.18
(3.78)
Grand Partition Function
The canonical partition function is defined as
e´βpEps q´µN ps qq .
ÿ
Ξ“
microstate
(3.79)
s
This acts as the normalization factor for probabilities
s
pp q “
1 ´βpEps q´µN ps qq
e
.
Ξ
(3.80)
The expectation value is given by
ÿ
xQy “
s s
Qp qpp q “
s
1ÿ
Qp qe´βpEps q´µN ps qq .
Ξ s
s
(3.81)
The average particle number is
kB T
xN y “
Ξ
ˆ
BΞ
Bµ
˙
ˆ
“ kB T
T,V
Bpln Ξq
Bµ
˙
.
(3.82)
T,V
Fluctuations are characterized by
ˆ
2
σN
2
“ pkB T q
B 2 pln Ξq
Bµ2
˙
ˆ
“ kB T
T,V
B xN y
Bµ
˙
.
(3.83)
T,V
The average energy is
ˆ
xEy “ ´
Bpln Ξq
Bβ
˙
` µ xN y .
(3.84)
V,µ
The grand canonical partition function can be expressed as a sum involving the canonical partition function
ÿ
ΞpT, V, µq “
eβµN ZpT, V, N q.
(3.85)
N
3.19
Grand Potential
Obtained from F by Legendre transforming N Ñ µ
Φ “ F ´ µN
“ U ´ T S ´ µN
“ G ´ P V ´ µN
“ ´P V.
19
(3.86)
ΦpT, V, µq is natural
ˆ
S“´
dΦ “ ´SdT ´ P dV ´ N dµ,
ˆ
˙
BΦ
BΦ
,
P “´
,
BT V,µ
BV T,µ
(3.87)
˙
ˆ
N“
BΦ
Bµ
˙
.
(3.88)
T,V
The grand potential is related to the grand canonical partition function via
Φ “ ´kB T ln Ξ.
3.20
(3.89)
Entropy
The Gibbs entropy formula is
ÿ
S “ x´kB ln py “ ´kB
where we sum over the microstates
s
s
pp q ln pp q,
microstate
s
(3.90)
s depending on the ensemble.
For microcanonical ensemble, we get
S “ kB ln Ω.
(3.91)
For canonical ensemble, we get
xEy
p´F ` xEyq
“
.
T
T
(3.92)
p´Φ ` xEy ´ µ xN yq
pxEy ´ µ xN yq
“
.
T
T
(3.93)
S “ kB ln Z `
For grand canonical ensemble, we get
S “ kB ln Ξ `
4
4.1
Quantum Statistical Mechanics
Common Quantum Systems
Energy levels for a particle in a 1D box of length L,
En “
n 2 π 2 ℏ2
,
2mL2
n P Z` .
(4.1)
Energy levels for a particle in a 3D cubical box of side length L,
En “
pn2x ` n2y ` n2z qπ 2 ℏ2
|⃗n|2 π 2 ℏ2
“
,
2mL2
2mL2
nx , ny , nz P Z` .
Energy levels for a simple harmonic oscillator with frequency ω,
ˆ
˙
1
En “ n `
ℏω,
n P Zě0 .
2
(4.2)
(4.3)
Energy levels for a Hydrogen like atom with nuclear charge `Ze,
En “ ´
Z2
E0 ,
n2
20
n P Z`
(4.4)
with states labeled by |n, l, ml , ms y and degeneracy gpEn q “ 2n2 .
The orbital angular momentum states are given by
L2 “ lpl ` 1qℏ2 ,
Lz “ ml ℏ,
l P t0, 1, 2 . . . u
ms P t´l, ´l ` 1, . . . l ´ 1, lu.
(4.5)
(4.6)
The states labeled as |l, ml y.
A particle can have an integer or half-integer spin s. The spin angular momentum states are given by
S 2 “ sps ` 1qℏ2 ,
Sz “ ms ℏ,
(4.7)
ms P t´s, ´s ` 1, . . . s ´ 1, su.
(4.8)
The states labeled as |ms y and degeneracy gs “ 2s ` 1.
4.2
Multiparticle States and Statistics
Let N be the number of particles and A be the number of one-particle states. For distinguishable particles,
Number of N -particle states “ AN .
(4.9)
These distinguishable particles correspond to Maxwell-Boltzmann statistics.
ˆ
Number of permutation classes “
˙
A`N ´1
.
N
There is exactly one bosonic (totally symmetric) state for each permutation class.
ˆ
˙
A`N ´1
Number of bosonic states “
.
N
(4.10)
(4.11)
These correspond to Bose-Einstein statistics.
A permutation class admits a fermionic (totally antisymmetric) state only if it no one-particle state has more
than one particle.
ˆ ˙
A
Number of fermionic states “
.
(4.12)
N
These correspond to Fermi-Dirac statistics.
The classical limit of Bose-Einstein and Fermi-Dirac statistics is given by
ˆ
˙ ˆ ˙
A
AN
A`N ´1
.
,
Ñ
N!
N
N
This corresponds to classically indistinguishable particles with Maxwell-Boltzmann statistics.
21
(4.13)
4.3
Distribution Functions
For fermions, the grand canonical partition function for a single state |αy is
ΞF “ 1 ` e´βpEα ´µq .
(4.14)
The average particle number is the Fermi-Dirac distribution function
n̄F “
1
.
eβpEα ´µq ` 1
(4.15)
In the zero temperature limit, we get
$
&1
lim n̄F “
T Ñ0
%0
E ă EF
,
(4.16)
E ą EF
where EF “ µpT “ 0q is the Fermi energy.
For bosons, the grand canonical partition function for a single state |αy is
1
.
1 ´ e´βpEα ´µq
The average particle number is the Bose-Einstein distribution function
ΞB “
1
.
eβpEα ´µq ´ 1
The chemical potential for bosons is smaller than the ground state energy at all temperatures
n̄B “
µ ă Eground state .
(4.17)
(4.18)
(4.19)
Taking the classical limit βpEα ´ µq " 1, we get the classical limit or Boltzmann distribution function
n̄C “
1
.
(4.20)
N
.
eβpEα ´µq
(4.21)
eβpEα ´µq
For indistinguishable particles,
n̄D “
4.4
Density of States
The number of one-particle states with energy between E and E ` dE is
dΩpEq “ gpEqdE,
(4.22)
where gpEq is the density of states.
The density of states for non-relativistic particles in a 3D box is
ˆ
˙3{2
?
gs
2m
gpEq “
V
E.
4π 2 ℏ2
(4.23)
The average particle number and energy are
ż8
xN y “
dE gpEqn̄pEq,
(4.24)
dE EgpEqn̄pEq.
(4.25)
Eg
ż8
xEy “
Eg
22
4.5
Photon Gas
For photons in a 3D cubical box of side length L, the modes are characterized by the wavevector
⃗k “ π pnx , ny , nz q “ π ⃗n,
L
L
nx , ny , nz P Z` .
(4.26)
and polarization. There are two polarizations for every wavevector ⃗k corresponding to the two orthogonal
directions. The frequency and wavelength are
πc
|⃗n|,
ω “ c|⃗k| “
L
λ“
2π
2L
.
“
⃗
|⃗n|
|k|
(4.27)
The density of oscillators in a 3D cubical box for n-space and frequency are
gn pnq “ πn2 ,
(4.28)
2
gω pωq “
Vω
.
π 2 c3
(4.29)
The Planck distribution function is
n̄P “
xEy
1
1
“ ℏω{k T
“ βE
.
B
ℏω
e ´1
e
´1
(4.30)
Photons are bosons with chemical potential µγ “ 0.
The spectral energy density (energy density per unit frequency or wavelength) is
1
ℏω 3
,
π 2 c3 eℏω{kB T ´ 1
8πhc
1
uλ pλq “
,
5
hc{λk
BT ´ 1
λ e
uω pωq “
(4.31)
(4.32)
This is Planck’s blackbody radiation law.
The spectral density uλ has a peak given by Wein’s displacement law
λpeak T “
1 hc
.
4.965 kB
(4.33)
The total energy density is
xuy “
Etotal
π pkB T q4
“
.
V
15 pℏcq3
(4.34)
The radiation pressure is
1
xuy
3
(4.35)
J “ σT 4 ,
(4.36)
P “
4.6
Blackbody Radiation
The energy flux emitted by a blackbody is
23
where
σ“
4
π 2 kB
60 ℏ3 c2
(4.37)
is the Stefan-Boltzmann constant.
The power emitted is
P “ JA “ σAT 4 .
4.7
(4.38)
Debye Model
Model of solids that modifies by the Einstein model by including a spectrum of oscillation frequencies.
The modes are labeled by the wavevector
⃗k “ π pnx , ny , nz q “ π ⃗n,
L
L
nx , ny , nz P Z` .
(4.39)
and polarization. There are three polarizations, two transverse and one longitudinal, for every wavevector ⃗k.
The frequency is
πcs
|⃗n|,
(4.40)
ω “ cs |⃗k| “
L
where cs is the speed of sound.
Acoustic modes have |⃗k| ! 1{a, where a is the lattice spacing. For these modes, the resolution of the lattice does
not matter. Optical modes have |⃗k| „ 1{a.
The density of acoustic modes in a 3D cubical box for n-space and frequency are
3 2
πn ,
2
3V ω 2
.
gω pωq “
2π 2 c3s
gn pnq “
(4.41)
(4.42)
The Debye frequency ωD is the effective maximum frequency obtained via
ż ωD
3N “
dω gω pωq.
(4.43)
0
For a 3D cubical box,
ˆ
ωD “
6π 2 N
V
˙1{3
cs .
(4.44)
The modes can be interpreted as particles called phonons. Phonons are bosons with µ “ 0 and their number is
given by the Planck distribution function
1
n̄P “ ℏω{k T
.
(4.45)
B
e
´1
The total energy is
ż ωD
dω ℏωgω pωqn̄P pωq.
xU y “
0
24
(4.46)
For a 3D cubical box,
ˆ
xU y “ 9N kB TD
T
TD
˙4 ż TD {T
x3
,
´1
dx
(4.47)
ex
0
where we define the Debye temperature
ℏωD
kB
limit, we reproduce the equipartition result
(4.48)
TD “
In the T " TD
xU y “ 3N kB T,
CV “ 3N kB .
(4.49)
In the T ! TD limit,
3
xU y “ π 4 N kB TD
5
4.8
ˆ
T
TD
˙4
12 4
CV “
π N kB
5
,
ˆ
T
TD
˙3
.
(4.50)
Fermi Gas
The Fermi energy EF “ µpT “ 0q is obtained via
ż EF
N“
dE gpEq.
(4.51)
0
For a 3D non-relativistic Fermi gas at T “ 0,
ˆ
˙2{3
h2 3N
EF “
,
8m πV
3
U “ N EF ,
5
3U
.
P “
5V
(4.52)
(4.53)
(4.54)
Sommerfeld expansion for kB T ! EF gives
«
µ « EF
π2
1´
12
ˆ
kB T
EF
π2
3
N EF `
N EF
5
4
ˆ
˙
π2
kB T
CV «
N kB
.
2
EF
U«
4.9
ff
˙2
` ... ,
ˆ
kB T
EF
(4.55)
˙2
` ...,
(4.56)
(4.57)
Bose-Einstein Condensation
The occupation number of the ground state is
xn0 y “
«
1
eβpE0 ´µq
(4.58)
´1
kB T
E0 ´ µ
for E0 ´ µ ! kB T.
We set the ground state energy E0 “ 0, then the condition to find T for which µ “ 0 is
ż8
gpEq
N“
dE E{k T
.
e B ´1
0
25
(4.59)
(4.60)
For a 3D non-relativistic Bose gas, the critical temperature is the Einstein condensation temperature
TC “
ℏ2
2{3 2mk
B
pgs ζp3{2qq
1
ˆ
N
V
˙2{3
.
(4.61)
For T ď TC , µpT q “ 0 and xn0 y become macroscopic. This is a Bose-Einstein condensate (BEC). We need to
treat the xn0 y separately as
ż8
gpEq
N “ xn0 y `
dE E{k T
.
(4.62)
B
´1
e
0
For a 3D non-relativistic BEC with T ď TC ,
«
xNBEC y “ xn0 y “ 1 ´
ˆ
xNexcited y “ N ´ xn0 y “
26
T
TC
˙3{2 ff
T
TC
˙3{2
ˆ
N,
(4.63)
N.
A
Partial Derivative Identities
By
Bx
˙´1
˙ ˆ
˙
ˆ
Bx
“
By
ˆ
Bx
By
˙ ˆ
z
ˆ
B
By
B
B.1
By
Bz
Bf
Bx
x
˙
(A.1)
Bz
Bx
B
“
Bx
(A.2)
“ ´1
y
ˆ
Bf
By
˙
(A.3)
Some Important Integrals
Gamma Function
The gamma function is defined by the integral
ż8
Γpsq “
dx xs´1 e´x
s ě 1.
(B.1)
0
The gamma function satisfies the relation
Γps ` 1q “ sΓpsq.
(B.2)
This relation is used to define the Gamma function for s ă 1.
Some special values of the Gamma function are
n P Z` ,
Γpnq “ pn ´ 1q!
Γp0q “ 1,
ˆ ˙
?
1
“ π.
Γ
2
B.2
(B.3)
(B.4)
(B.5)
Gaussian Integrals
ż8
´x2
dx e
´8
ż8
dx x2n e´x
´8
2
ˆ ˙
?
1
“Γ
“ π.
2
ˆ
˙
1
“Γ n`
,
2
ż8
(B.6)
n P Z` .
c
´ax2
dx e
´8
27
“
π
.
a
(B.7)
(B.8)
B.3
Related to Quantum Statistical Mechanics
The integral related to the bosonic distribution function is
ż8
xn
dx x
“ Γpn ` 1qζpn ` 1q.
e ´1
0
The zeta function is defined as
ζpsq “
8
ÿ
1
,
s
n
n“1
(B.9)
(B.10)
with particular values
π2
,
6
π4
ζp4q “
,
90
6
π
ζp6q “
.
945
ζp2q “
The fermionic version is
ż8
xn
“
ex ` 1
dx
0
ˆ
1´
1
2n
(B.11)
˙
Γpn ` 1qζpn ` 1q.
(B.12)
The integral related to the Sommerfeld expansion is
$
’
’
&0
ż8
xn
“ 1
dx x
e ´1 ’
´8
’
% π2
3
C
C.1
n is odd
n“0
.
(B.13)
n“2
Probability
Discrete Distribution
A random variable X takes value in a discrete set txi u, then we have a discrete distribution pi “ ppxi q. These
probabilities satisfy 0 ď pi ď 1.
Normalization is
ÿ
pi “ 1.
(C.1)
i
Probability of finding X in the interval pa, bq is
ppa ă X ă bq “
ÿ
pi .
(C.2)
aăxi ăb
C.2
Continuous Distribution
A random variable X takes value in a continuous interval. The distribution is described by the probability density
function ρpxq, which means that
ppx ă X ă x ` dxq “ ρpxqdx.
(C.3)
28
Normalization is
ż
dx ρpxq “ 1.
(C.4)
R
where R is the range for X.
Probability of finding X in the interval pa, bq is
żb
ppa ă X ă bq “
dx ρpxq.
(C.5)
a
C.3
Expectation Values
The expectation value for a discrete random variable is defined as
ÿ
xf pXqy “
pi f pxi q
(C.6)
i
The expectation value for a continuous random variable is defined as
ż
xf pXqy “
dx ρpxqf pxq.
(C.7)
R
The expectation value is linear
xaf pXq ` bgpxqy “ a xf pXqy ` b xgpxqy .
(C.8)
The rms-value is defined as
Xrms “
a
xX 2 y.
(C.9)
The standard deviation is defined as
σX “
cA
E b
2
2
pX ´ xXyq “ xX 2 y ´ xXy .
29
(C.10)