Quantum Statistical Ensembles the Basics of Quantum Mechanics

Quantum statistical ensembles The basics of quantum mechanics

• Quantum mechanics is a more fundamental description of nature than classical mechanics. In general, a quantum microstate can be specied by a wavefunction Ψ(Ω), which is a complex function of the classical microstate Ω (with a caveat that pairs of so called canonically conjugate variables, such as position and momentum, cannot both be included in Ω). Quantum mechanics adds another layer of uncertainty to the needed statistical treatment of enormously complicated Ω. Measurement outcomes of any classical variable in Ω (e.g. the position of a certain particle) are generally random even in microscopic (non-statistical) quantum systems. The quantity |Ψ(Ω)|2, then, expresses the probability (per unit phase space volume) that the classical state Ω would be observed in a measurement. One can view a quantum state as a simultaneous realization or superposition of many classical states Ω in a single system, where the wavefunction Ψ(Ω) gives the weight of any classical state in the superposition. Since the wavefunction is complex, a quantum superposition can exhibit phenomena such as (de Broglie) waves and interference.

• For a system of N identical particles, the microstate is given by the positions of all particles Ω = (r1, r2, ··· , rN ) up to a permutation. The wavefunction is then:

Ψ(Ω) ≡ Ψ(r1, r2, ··· , rN ) Alternatively, the wavefunction can be expressed as a function of momenta of all particles by taking a Fourier transform. Since identical particles are fundamentally indistinguishable, the probability density must be symmetric under permutations P:

2 2 |Ψ(r1, r2, ··· , rN )| = |Ψ(rP(1), rP(2), ··· , rP(N))| (∀P) This implies (in three dimensions):

Ψ(r1, r2, ··· , rN ) = σP Ψ(rP(1), rP(2), ··· , rP(N))(∀P)

where σP = ±1. For a class of particles called bosons, σP = 1. Multiple bosons can occupy the same single-particle state, or position ri = rj. For a class of particles called fermions, σP = 1 if the permutation is even (an even number of pair exchanges) and σP = −1 if the permutation is odd (an odd number of pair exchanges). Two fermions cannot be in the same single-particle state due to Pauli exclusion principle:

Ψ(r1, r2) = −Ψ(r2, r1) ⇒ Ψ(r, r) = −Ψ(r, r) = 0

• Quantum measurement: An operator is a mathematical object that operates on a wavefunction and generally transforms it to a dierent function of the microstate. Examples of simple operators are multiplication by a constant, taking a derivative, etc. In quantum mechanics, an operator Aˆ is associated with every measurable physical quantity a. The possible measurement outcomes a and the corresponding quantum states are generally obtained by solving an eigen-equation:

Aψˆ a(Ω) = aψa(Ω) This is generally a dierential equation. One has to determine such values a (called eigenvalues of the operator Aˆ) and the corresponding wavefunctions ψa (called eigenfunctions of the operator Aˆ) that the action of Aˆ on ψa amounts to multiplying the same ψa by the constant a. Then, the wavefunctions ψa(Ω) are the quantum microstates in which the measurement of the quantity Aˆ would yield the value a with absolute certainty. In any other quantum state Ψ(Ω), the measurement of the same quantity Aˆ yields one of its eigenvalues randomly. The probability that a particular eigenvalue a will be measured in the quantum state Ψ(Ω) is given by: 2 ∗ P (a) = dΩ ψ (Ω)Ψ(Ω) ˆ a

43 provided that all wavefunctions are normalized:

2 2 ∗ ∗ dΩ ψ (Ω)ψa(Ω) = dΩΨ (Ω)Ψ(Ω) = 1 ˆ a ˆ

• A quantum state is stationary if its probability density |Ψ(Ω)|2 does not change in time. Since energy is always conserved, stationary states are discovered by solving the eigen-equation for energy:

Hψˆ E(Ω) = EψE(Ω) This is known as stationary Schrodinger equation; Hˆ is the Hamiltonian operator associated with energy. For a system of N interacting particles, the Hamiltonian is analogous to the one in classical mechanics: N X pˆ2 1 X Hˆ = i + U(ˆr ) + V (ˆr − ˆr ) 2m i 2 i j i=1 i6=j th but ˆri and pˆi are position and momentum operators respectively for the i particle. These operators, in turn, can be represented as: ∂ ˆri = ri , pˆi = −i~ ∂ri th i.e. the position operator simply multiplies the wavefunction by the position vector ri of the i particle, and the momentum operator takes the gradient of the wavefunction with respect to ri and multiplies it by −i~, where ~ = h/2π is a variant of the Planck's constant h (sometimes called Dirac constant). Since the Hamiltonian is an operator associated with energy, Schrodinger equation identies the possible

energy measurement outcomes (eigenvalues) E and the corresponding eigenfunctions ψE(Ω) that carry a well-dened energy E.

• A stationary quantum state described by an energy eigenfunction ψE(Ω) is generally a superposition of many classical microstates Ω, and thus hard to identify by some typical classical congurations Ω. It is always more practical to label and enumerate the stationary quantum states by quantum numbers. In the case of so called bound states (where particles cannot escape to innity: electrons inside an atom, gas inside a box), quantum numbers always take integer values. Also, there is always one quantum number per degree of freedom.

So, for example, a gas of N particles trapped in a box has quantum numbers M = (m1, m2 ... mN ), where mi = (mix, miy, miz) and mix, miy, miz are integers. It is understood that permutations among the quantum numbers of individual identical particles do not correspond to dierent quantum states (in other words, M is dened only up to a permutation of particles). The type and meaning of quantum numbers is obtained from the solution of Schrodinger equation, and depends on the given problem's context. If there are conserved quantities like angular momentum, then some of the quantum numbers label the corresponding conserved eigenstates.

For a particle trapped in a box of dimensions (Lx,Ly,Lz), stationary quantum states are standing waves ψ(r) ∝ sin(kxx) sin(kyy) sin(kzz), where the wavevectors ki, i ∈ {x, y, z} are quantized by the requirement that an integer number of wavelength halves t exactly in mi ∈ N λi = 2π/ki the box: 1 . The integers are quantum numbers. 2 miλi = Li mi Diatomic and other molecules that can rotate have two additional quantum numbers that specify a rotational state: an orbital quantum number l ∈ {0, 1, 2 ... } and a magnetic quantum number m ∈ {−l, −l + 1 . . . l − 1, l}.

Elementary particles also have spin, an internal rotation-like quantum number ms ∈ {−s, −s + 1 . . . s − 1, s}, where s is determined by the type of particle. For all fermions, s is a half-integer ( 1 for electrons, protons, neutrons, etc.). For all bosons, is an integer ( for hydrogen, s = 2 s s = 0 s = 1 for photons, etc.). Vibrational modes of linear oscillators are enumerated by an integer quantum number m ∈ {0, 1, 2 ... }.

44 • Stationary states can be degenerate, i.e. there could be multiple stationary quantum states (with dierent combinations of quantum numbers M) that have the same energy E. Therefore, quantum numbers M rather than energy E provide an unambiguous labeling scheme for stationary quantum states. One normally labels the Hamiltonian eigenfunctions ψE(Ω) by quantum numbers, ψM(Ω).

• Any wavefunction Ψ(Ω) can be expressed as a superposition of energy eigenfunctions ψN(Ω): X Ψ(Ω) = CMψM(Ω) M

The coecients CM are complex constants that can be calculated from:

C = dΩ ψ∗ (Ω)Ψ(Ω) M ˆ M if the wavefunctions are normalized. As a consequence, we can give up using the typical classical microstate specication Ω (e.g. the positions of particles) and switch to quantum numbers M as a complete enumeration scheme for quantum microstates.

Equilibrium quantum statistical ensembles

• An equilibrium quantum statistical ensemble is specied by a classical probability distribution p(M) of quantum stationary microstates labeled by quantum numbers M. By construction, p(M) depends on the microstate energy E(M), which is required in equilibrium. p(M) is the classical probability that an ensemble member would be detected in the quantum microstate M upon measurement. If indeed a system is found in the quantum state N, then measurements of various observables can still yield random outcomes due to the nature of quantum mechanics, but a measurement of energy yields the conserved value E(M) of the found microstate with certainty. • Using the same arguments as in classical physics, one denes: 1 p(M) = δ ··· in microcanonical ensemble Ω(E) E(M),E 1 p(M) = e−βE(M) ··· in canonical ensemble Z 1 −β E(M)−µN p(M) = e ··· in grand canonical ensemble Z where: X Ω(E) = δE(M),E ··· the number of quantum states with energy E M X −βE(M) Z = e ··· canonical partition function at temperature T = 1/kBβ M ∞ X X −β E(M)−µN Z = e ··· grand canonical partition function at chemical potential µ N=1 M

• The most convenient ensemble for dealing with identical particles is grand canonical. Instead of xing the total number of particles N, one species the chemical potential µ. Then, the quantum microstate of identical particles can be completely described by a set of occupation numbers : the M nm ∈ N number of particles in every possible single-particle state (identied by its quantum numbers m). The occupation numbers for bosons can be any non-negative integers. For fermions, however, the occupation numbers can take only two values, zero or one (Pauli exclusion prohibits occupying a single-particle state with more than one fermion).

45 For example, consider a two-level system where a particle can be in one of the internal states

A and B. Many body states can be labeled either by quantum numbers M = (m1 . . . mN ), or equivalently by occupation numbers N = (nA, nB). The latter rapidly becomes much more concise as the number of particles N grows. A gas having one particle has microstates:

M = (A) ⇔ N = (1, 0)

M = (B) ⇔ N = (0, 1) A gas having two identical particles has microstates:

M = (A, B) = (B,A) ⇔ N = (1, 1)

M = (A, A) ⇔ N = (2, 0) M = (B,B) ⇔ N = (0, 2) where the last two are not possible for fermions. A gas of three particles has microstates:

M = (A, A, A) ⇔ N = (3, 0)

M = (B,B,B) ⇔ N = (0, 3) M = (A, A, B) = (A, B, A) = (B, A, A) ⇔ N = (2, 1) M = (A, B, B) = (B,B,A) = (B, A, B) ⇔ N = (1, 2) none of which are possible for fermions. Note that occupation numbers are concise because they do not keep track of any information that

could distinguish between the particles. If there are nm particles in the single-particle state m, there is no way of telling any order among them. As the microstate N varies, so does the total number of particles in the system P . Then, we must rely on the chemical potential of m nm µ the grand canonical ensemble to tie the number of particles to the average hNi. Recall that a particular choice of ensemble makes no dierence for any practical purpose when the system is macroscopic.

Non-interacting identical particles

• If the energy of a particle in the single-particle state m is m, and there are nm particles in that state, then the total energy and number of all particles in the system are: X X E(N) = nmm ,N(N) = nm m m Bosons: Grand canonical partition function Z and probability p(N) are:

∞ X −β E(N)−µN(N) Y X Y 1 Z = e = e−β(m−µ)nm = 1 − e−β(m−µ) N m nm=0 m 1 −β E(N)−µN(N) Y p(N) = e = 1 − e−β(m−µ) e−β(m−µ)nm Z m Fermions: Grand canonical partition function Z and probability p(N) are:

1 X −β E(N)−µN(N) Y X Y Z = e = e−β(m−µ)nm = 1 + e−β(m−µ)

N m nm=0 m 1 −β E(N)−µN(N) Y −1 p(N) = e = 1 + e−β(m−µ) e−β(m−µ)nm Z m

46 Comments: In both cases, we began calculating Z by summing over all possible grand canonical microstates of identical particles expressed in terms of occupation numbers N = (n1, n2 ... ). This automatically includes the summation over all possible total numbers of particles N, as required in the grand canonical ensemble. Since the particles are non-interacting and independent, the number exp[−β(E − µN)] always factorizes: −β E(N)−µN(N) Y e = e−β(m−µ)nm m Then, we used the fact: ∞ ! X Y nm Y X nm Xm = Xm N m m nm=0 The right-hand-side is a product of sums, which generates many terms of the kind n1 n2 n3 X1 X2 X3 ··· that are found on the left-hand-side - one for every possible combination of (n1, n2 ... ). The sum over all microstates N on the left-hand-side is nothing but the sum over all possible combinations (n1, n2 ... ).

−β(m−µ)nm nm In the case of bosons, we also used a well-known trick to sum e ≡ x over nm ∈ {0, 1, 2, 3 ... }: M−1 X m 2 M−1 SM ≡ x = 1 + x + x + ··· + x m=0 2 3 M−1 M xSM = x + x + x + ··· + x + x 1 − xM (1 − x)S = 1 − xM ⇒ S = M M 1 − x

We must assume m − µ > 0 in order for the series SM to converge in the M → ∞ limit. This is merely a constraint on the allowed values for µ: the spectrum of single-particle energies m must −β(m−µ) be bounded from below, m ≥ 0, and it is required that µ < 0. Then x = e < 1, and xM vanishes when M → ∞: ∞ X m 1 x = lim SM = M→∞ 1 − x m=0 • The probability distribution p(N) of identical non-interacting particles is always a product of factors Pm(nm) over all possible single-particle states m. Therefore, we identify Pm(n) as the probability that a single-particle state m would be occupied by n particles:

 +1   1 − e−β(m−µ) e−β(m−µ)n , bosons n ∈ {0, 1, 2 ... }  Pm(n) = −1 −β( −µ) −β( −µ)n  1 + e m e m , fermions n ∈ {0, 1} 

The factor z = eβµ called fugacity is sometimes isolated in the formulas:

 +1   1 − ze−βm zne−βmn , bosons n ∈ {0, 1, 2 ... }  Pm(n) = −1 −β n −β n  1 + ze m z e m , fermions n ∈ {0, 1} 

• The average total number of particles in the system is: X N = f(m) m where X f(m) ≡ hnmi = n Pm(n) n

47 is the average number of particles in the single-particle state m that takes one of the following forms: 1 fBE(m) = ··· bosons eβ(m−µ) − 1 1 fFD(m) = ··· fermions eβ(m−µ) + 1

The function fFD() is known as Fermi-Dirac distribution, and the function fBE() is known as Bose- Einstein distribution. This shows that the average occupation of any single-particle state m in equilibrium depends only on its energy m. In the limit of high temperatures, both distributions converge to the classical Maxwell-Boltzmann distribution:

−β(m−µ) fMB(m) = e For bosons:

∞ ∞ n X X −β(m−µ) −β(m−µ)n −β(m−µ) X −β(m−µ) hnmi = n Pm(n) = n 1 − e e = 1 − e n e n n=0 n=0 e−β(m−µ) e−β(m−µ) = 1 − e−β(m−µ) × = 2 1 − e−β(m−µ) 1 − e−β(m−µ) 1 = eβ(m−µ) − 1

We used the trick (x = e−β(m−µ)):

∞ ∞ X d X d 1 x nxn = x xn = x = dx dx 1 − x (1 − x)2 n=0 n=0 For fermions:

1 −1 1 X −β(m−µ) −β(m−µ)n −β(m−µ) hnmi = n 1 + e e = 0 + 1 × e 1 + e−β(m−µ) n=0 1 = eβ(m−µ) + 1

At high temperatures β = 1/kBT → 0, both Fermi-Dirac and Bose-Einstein distributions become weakly dependent on energy m. Consequently, the distributions become broad in energy and even dominated by the abundant high-energy states (which have a large density of states). Specically, the total number of bosons or fermions:

X 1 N = eβ(m−µ) ± 1 m

β(m−µ) becomes dominated by contributions from high-energy states m − µ kBT where e 1, so we may approximate:

1 X X −β(m−µ) N ≈ = fMB(m) , fMB(m) = e eβ(m−µ) m m

• Internal energy is:  −1  P β(m−µ)  e − 1 m , bosons  X  m  E = mf(m) = −1 P β(m−µ) fermions m  e + 1 m ,  m

48 The ensuing heat capacity C = dE/dT (at constant volume and chemical potential) vanishes both in the T → 0 and T → ∞ limits:  −2 2  P β(m−µ) β(m−µ) m−µ bosons  kB e − 1 e k T ,  d X df(m)  m B  (E − µN) = (m − µ) = −2 2 dT dT P β(m−µ) β(m−µ) m−µ fermions m  kB e + 1 e k T ,  m B

β(m−µ) At high temperatures, β → 0 turns the exponential terms into e ≈ 1 + β(m − µ), so that 2 2 the power-law factor (m − µ) /(kBT ) → 0 wins and ensures a power-law convergence C → 0. At β(m−µ) low temperatures, β → ∞ turns the exponential terms into either e → ∞ for m − µ > 0 β(m−µ) β(m−µ) β(m−µ) 2 or e → 0 for m − µ < 0, but in either case the combination e /(e − 1) goes exponentially to zero. Therefore, the heat capacity vanishes exponentially fast at low temperatures.

The exceptions occur if the quantum-mechanical spectrum m is continuous in the vicinity of µ, so that m − µ → 0 for some quantum numbers m. Still, it turns out that heat capacity vanishes at low temperatures, but as a power-law of temperature. An example is an ideal Fermi gas, where C ∝ T at low temperatures.

• Equation of state can be derived by considering the change of grand potential

G = E − TS − µN = −kBT log(Z) due to the small change of volume:

dG = dE −d(TS)−d(µN) = (T dS −pdV +µdN)−(T dS +SdT )−(µdN +Ndµ) = −SdT −pdV −Ndµ dG p = − dV T,µ Since we know the quantum grand partition function:

Y −σ Z = 1 − σe−β(m−µ) m where σ = +1 for bosons and σ = −1 for fermions, we obtain the formula for pressure: d d X p = k T log(Z) = −σk T log 1 − σe−β(m−µ) B dV B dV m If the quantum number enumeration scheme does not make any reference to volume, then all variations

with volume are hidden in the volume-dependence of single-particle energies m:

−β(m−µ) d X −σe dm p = kBT log(Z) = −σkBT × (−β) dV 1 − σe−β(m−µ) dV m X 1 dm = − eβ(m−µ) − σ dV m X dm = f( ) × − m dV m We see that pressure is related to the adiabatic reduction of internal energy when volume increases: the lost internal energy is converted to work pdV .

Quantum ideal gas of non-relativistic particles

• The quantum states of a spinless particle inside a cubic box of volume V = L3 are enumerated by three positive integer quantum numbers, m ≡ (nx, ny, nz). The energy of a particle in the state m is shown to be: π2 2 = ~ n2 + n2 + n2 m 2mL2 x y z

49 by solving the Schrodinger equation with periodic boundary conditions. This is the energy p2/2m of a standing de Broglie wave whose momentum projections pi = h/λi are obtained from the quantization 2niλi = Li of wavelengths λi (the wave is standing because it must be stationary in an equilibrium state with a well-dened energy; a stationary wave ts an integer number ni of crests between its bounds at and ). If the particle has spin , then it can also be in one of the xi = 0 xi = L s 2s + 1 ∈ N additional internal states (corresponding to dierent spin projections on some spatial direction).

Various thermodynamic quantities are obtained from sums over all single-particle quantum numbers. The summed objects (let's call them g) depend on the quantum numbers only through the single-particle energy m in equilibrium. Therefore, these sums pick a factor of 2s + 1 if there is any spin degeneracy (all internal spin states have the same energy). The remaining

sum over (nx, ny, nz) can be converted into an integral over wavevectors k = (kx, ky, kz), where ki = πni/L > 0: X X X g(βm) = (2s + 1) g(βnx,ny ,nz ) = (2s + 1) g(βk)

m nx,ny ,nz kx,ky ,kz ∞ 2s + 1 = dkxdkydkz g(βk) ∆kx∆ky∆kz ˆ 0

The elementary increments ∆ki = π/L become innitesimal (dki) in the limit of large system size L → ∞, and the wavevector k changes negligibly inside the elementary wavevector volume 3 ∆kx∆ky∆kz = π /V . Lastly, it is more pleasant to work with unbounded values of wavevector projections, ki ∈ (−∞, ∞). Since the energy k does not change when any wavevector component ips sign (by symmetry), we may write:

∞ ∞ X V 1 d3k g(β ) = (2s + 1) × dk dk dk g(β ) = (2s + 1)V g(β ) m π3 23 ˆ x y z k ˆ (2π)3 k m −∞ −∞

where we used . Now we can simply use 2 2 for non-relativistic particles and ~ = h/2π k = ~ k /2m partially solve the integral using a spherical coordinate system k = (kx, ky, kz) = (k, θ, φ):

∞ 2π π ∞ X d3k β 2k2 (2s + 1)V β 2k2 g(β ) = (2s + 1)V g ~ = × dφ dθ sin θ dk k2g ~ m ˆ (2π)3 2m (2π)3 ˆ ˆ ˆ 2m m −∞ 0 0 0 ∞ (2s + 1)V β 2k2 = × 4π dk k2g ~ (2π)3 ˆ 2m 0 We immediately integrated out the polar and azimuthal angles θ and φ respectively. It is useful to change the integration variable from k to the dimensionless quantity ξ, where 2 2 2 : ξ = βk = β~ k /2m

3 ∞ 3 ∞ 2 2 X (2s + 1)V 2m 2 2 2m 2 2 g(βm) = × 4π dξ ξ g(ξ ) = 4π(2s + 1)V dξ ξ g(ξ ) (2π)3 β~2 ˆ βh2 ˆ m 0 0 The quantity 2m 2mk T 1 h B √ 2 = 2 = 2 , λT = βh h πλT 2πmkBT is related to the so called thermal wavelength dened here. A non-relativistic particle of energy λT √ = ~2k2/2m has an associated de Broglie wave of wavelength λ = 2π/k = h/ 2m. In a classical gas at temperature T one expects ∼ kBT , so the particles have a short thermal de Broglie

50 wavelength λ ∼ λT . Quantum eects are thermodynamically visible if the spatial separation between particles becomes comparable to λT . Using this notation, we have:

∞ X V 4 2 2 g(βm) = (2s + 1) 3 × √ dξ ξ g(ξ ) λT π ˆ m 0

• Let z = eβµ (fugacity), and σ = +1 for bosons or σ = −1 for fermions. Bose-Einstein and Fermi-Dirac distributions can be captured by a single formula:

1 1 f() = = eβ(−µ) − σ z−1eβ − σ The ideal gas of identical spin 1 particles in volume has the following properties at any nite s = 2 V temperature:

The density of states ρ() gives the number of states ρ()d in the energy interval (, + d) per unit-volume:

1 1 V 4(2s + 1) 1 4(2s + 1) βd X 2 ρ()d = δ(m − ) = × 3 √ × ξ dξ = 3 √ × β √ V V λ π ξ2=β λ π 2 β m T T

1 4(2s + 1) β3/2 √ (2m)3/2 √ √ ρ() = 3 × = 3 2π(2s + 1) λT π 2 h The number of particles:

∞ ∞ X V 4(2s + 1) V 4(2s + 1) ξ2 N = f( ) = √ dξ ξ2f(ξ2) = × √ dξ m 3 3 −1 ξ2 λT π ˆ λT π ˆ z e − σ m 0 0 Internal energy:

∞ ∞ X V 4(2s + 1) ξ2 V 1 4(2s + 1) ξ4 E = f( ) = √ dξ ξ2f(ξ2) = × √ dξ m m 3 3 −1 ξ2 λT π ˆ β λT β π ˆ z e − σ m 0 0

Pressure (equation of state): X dm X 2 m 2 X 2E p = f( ) − = f( ) = f( ) = m dV m 3 V 3V m m 3V m m m ∗ The starting formula for pressure assumes that quantum numbers do not depend on volume, so we had to initially substitute the expression for energy in terms of integers (nx, ny, nz):

π2 2 = ~ n2 + n2 + n2 m 2mL2 x y z Then, having in mind a cubic box with volume V = L3, the change of volume dV caused by the change of the cube's length is dV = 3L2dL = 3V × dL/L, we had:

π2 2 1 π2 2 2dL dL 2 dV d = ~ n2 +n2+n2 ×d = ~ n2 +n2+n2 × − = −2 = − m 2m x y z L2 2m x y z L3 L m 3 V m

d 2 m = − m dV 3 V

51 • Fermions at zero temperature: At zero temperature β → ∞, the ground state of non-interacting fermions is revealed by the Fermi-Dirac distribution function: 1 β→∞ 0 , m > µ fFD(m) = −−−−→ eβ(m−µ) + 1 1 , m < µ

All single-particle states below the chemical potential (m < µ) are occupied (to the maximum of one particle allowed in a single state by Pauli exclusion), and all states above the chemical potential

(m > µ) are empty. This many-body state is sometimes called Fermi sea, and in that context the chemical potential at zero temperature is called Fermi energy, µ ≡ f > 0. Thermodynamically, low-temperature fermions exhibit metallic behavior.

It is useful to dene a Fermi wavevector kf as the wavevector of particles whose energy equals the Fermi energy, 2 2 . Then, all single-particle states with wavevectors µ = f = ~ kf /2m k < kf are occupied, and all states with k > kf are empty. The surface |k| = kf in the wavevector (momentum) space is called Fermi surface; it separates the Fermi sea of occupied states from higher-energy empty states. Fermi velocity is the speed of particles at the Fermi vf = ~kf /m surface. The number N of fermions in a box of volume V is:

∞ kf 3 3 d k (2s + 1)V 2 (2s + 1)V kf N = (2s + 1)V fFD(β ) = × 4π dk k = × ˆ (2π)3 k (2π)3 ˆ 2π2 3 −∞ 0

The ground state energy is:

∞ kf 3 2 2 2 5 d k (2s + 1)V 2 ~ k (2s + 1)V ~ kf E = (2s + 1)V fFD(β ) = × 4π dk k = × ˆ (2π)3 k k (2π)3 ˆ 2m 2π2 2m 5 −∞ 0

Energy per particle is visibly dierent than about kBT → 0:

2 2 E 3 ~ kf 3 = = = N 5 2m 5 f

The concentration of particles n and Fermi energy f are directly related by:

3 2 3 2 2 2 3 N 2s + 1 kf 2s + 1 2mf ~ 6π n n = = = ⇒ f = V 2π2 3 6π2 ~2 2m 2s + 1

The degeneracy pressure p exists even at T = 0 because Pauli exclusion eectively pushes particles apart: 2E 2 E N 2n 2 p = = = = n 3V 3 N V 3 5 f • Bose-Einstein condensate: Non-interacting bosons at zero temperature (β → ∞) have a special ground state revealed by the Bose-Einstein distribution function: 1 β→∞ 0 , m > µ fBE(m) = −−−−→ eβ(m−µ) − 1 −1 , m < µ

The average number of particles occupying a state of energy cannot be negative, so we cannot allow the chemical potential µ to be larger than any single-particle energy m. However, if we set µ = m for some state m, then fBE(m) → ∞ indicates a macroscopic (diverging) occupation of that single- particle state. Therefore, the smallest possible energy of the many-body system is obtained when

µ = 0 coincides with the smallest energy 0 in the single-particle spectrum m. All N particles of

52 the system are placed in that state. The ground state energy of N bosons is therefore E = N0, and energy per particle is = 0 = µ. Such a macroscopic occupation of a single-particle state is called Bose-Einstein condensate (BEC). In the presence of interactions, the formation of a BEC at low temperatures leads to superuidity (or superconductivity if the bosons are charged). The particles are correlated and stick together in a single quantum state, described by a single-particle wavefunction.

• High-temperature limit: At high temperatures β → 0, so naively z = eβµ → 1. It turns out, however, that for a xed concentration of particles n = N/V fugacity can become very small, z 1. Then, it can be used as an expansion parameter. Starting from earlier formulas, we obtain an expansion for the number of particles:

∞ ∞ 2 V 4(2s + 1) ξ2 V 4(2s + 1) ξ2e−ξ N = √ dξ = √ z dξ 3 −1 ξ2 3 −ξ2 λT π ˆ z e − σ λT π ˆ 1 − σze 0 0 ∞ V 4(2s + 1) 2 −ξ2 −ξ2 2 −2ξ2 3 −3ξ2 = 3 √ z dξ ξ e 1 + σze + z e + σz e + ··· λT π ˆ 0 √ V 4(2s + 1) π σz2 z3 σz4 = √ × z + + + + ··· 3 3/2 3/2 3/2 λT π 4 2 3 4 This can be used recursively to express fugacity as an expansion in powers of concentration n = N/V : nλ3 σz2 z3 z = T − − + ··· 2s + 1 23/2 33/2 nλ3 1 nλ3 σz2 z3 2 1 nλ3 σz2 z3 3 = T + T − − + ··· − T − − + ··· + ··· 2s + 1 23/2 2s + 1 23/2 33/2 33/2 2s + 1 23/2 33/2 nλ3 1 nλ3 2 = T + T + ··· 2s + 1 23/2 2s + 1 Then, internal energy:

∞ ∞ 2 V 1 4(2s + 1) ξ4 V 1 4(2s + 1) ξ4e−ξ E = √ dξ = √ z dξ 3 −1 ξ2 3 −ξ2 λT β π ˆ z e − σ λT β π ˆ 1 − σze 0 0 ∞ V 1 4(2s + 1) 4 −ξ2 −ξ2 2 −2ξ2 3 −3ξ2 = 3 √ z dξ ξ e 1 + σze + z e + σz e + ··· λT β π ˆ 0 √ V 1 4(2s + 1) 3 π σz2 z3 σz4 = √ × z + + + + ··· 3 5/2 5/2 5/2 λT β π 8 2 3 4 can be expanded in powers of concentration as well: 3 (2s + 1) σz2 z3 σz4 E = Nk T × z + + + + ··· B 3 5/2 5/2 5/2 2 nλT 2 3 4 (" # ) 3 (2s + 1) nλ3 1 nλ3 2 σ nλ3 2 = Nk T × T + T + ··· + T + ··· + ··· B 3 3/2 5/2 2 nλT 2s + 1 2 2s + 1 2 2s + 1 3 3 2 + σ nλT = NkBT × 1 + √ + ··· 2 4 2 2s + 1 This illustrates the classical high-temperature limit 3 and quantum corrections propor- E0 = 2 NkBT tional to various powers of 3 . Classical limit is obtained when 3 , i.e. when the spatial nλT nλT 1 −1/3 separation between particles ∼ n is much larger than de Broglie wavelength λT of particles with typical thermal energy . In the opposite limit , or kBT nλT & 1 − 2 h2 N 3 nλT & 1 ⇔ T . 2πmkB V

53 we say that the gas is degenerate: quantum eects become important.

Thermodynamics of photons: blackbody radiation

• Classical electric E(r, t) (and similarly magnetic) eld can be expressed as a sum over normal modes that oscillate at well-dened frequencies:

X iωmt E(r, t) = Em(r)e m

The angular frequency ωm = 2πνm is related to the ordinary frequency νm and depends on the mode iωt label m. We use the convenient complex form e that allows a complex amplitude Em to absorb any phase shift, but impose ∗ on the amplitudes of modes and that are related by Em = Em¯ m m¯ time-reversal (this ensures that the electric eld is real). In free space, the modes are plane waves:

ikr Em(r) = E0e

where k is the wavevector whose magnitude k = 2π/λ is given by the wavelength λ = c/ν (c is the speed of light; this follows from Maxwell's equations of E&M). If electromagnetic radiation is trapped inside a metallic cube of linear size L, then electric eld must vanish at the walls and hence the modes are standing waves: πn πn πn E (r) = E sin x x sin y y sin z z m 0 L L L

labeled by three integers ni > 0. In either case, the amplitude E0 must be perpendicular to the direction of wave propagation (wavevector k). There are two independent directions in which E0 can point, so radiation has two internal states (polarizations) for every spatial mode.

• Electromagnetic elds carry energy, and there are many possible modes for them. Therefore, electromagnetic radiation (vacuum) is a thermodynamic system that can exchange energy with its environ- ment and be in equilibrium characterized by temperature, pressure, etc. Every electromagnetic mode

m is a separate degree of freedom, labeled by (nx, ny, nz) and the state of polarization in a cavity. However, by the virtue of supporting electromagnetic waves, vacuum can be viewed as a system of locally coupled oscillators, which can be represented (after a Fourier transform of the Hamiltonian) as a system of decoupled oscillators that are responsible for one mode each. Energy of an oscillator is discrete in quantum mechanics, equal to an integer multiple of the oscillator's fundamental frequency

and Planck's constant, nmm = nmhν, nm ∈ {0, 1, 2 ... }. We say that an elementary excitation of electromagnetic radiation is a photon: the state nm = 0 is unexcited, nm = 1 means one photon in the mode m, nm = 2 means two photons in the mode m, and so on. This picture is made precise in quantum electrodynamics (quantum eld theory). It regards photons as relativistic bosons of spin s = 1. A normal s = 1 particle has three internal states labeled by the magnetic quantum number −s ≤ m ≤ s. However, it turns out that the state m = 0 does not exist for photons because they are so called gauge bosons (quanta of gauge elds, related to their light speed). The single-particle state of a photon in mode m has energy:

hc hc 2π m = hνm = = = ~kmc = pmc λm 2π λm where is the wavenumber, and is the momentum of the photon in mode . km = |km| pm = ~km m For radiation in a cavity,

π hc q k = (n , n , n ) ⇒ = n2 + n2 + n2 m L x y z m 2L x y z regardless of polarization. Every single-particle state can be occupied by any integer number of

photons nm.

54 We will derive the equilibrium properties of electromagnetic radiation using the above picture of quantized E&M, specically the radiation spectrum of a blackbody. Historically, however, blackbody radiation was rst experimentally detected then explained phenomenologically by Planck, who postulated the above quantization of electromagnetic waves for that purpose. Planck's pos- tulate was subsequently found to appear in virtually all microscopic contexts, and now is deeply rooted in moder quantum theory.

• We will use grand canonical ensemble with chemical potential µ to derive thermodynamic properties of radiation. Since photons are massless bosons, their lowest single-particle energy is 0 = 0 and we must require µ ≤ 0 (or we would have an innite occupation of some nite-energy mode with photons, i.e. a bright universe). However, negative values of µ are not allowed either. Relativistic particles, such as photons, are always found to have a particle-antiparticle symmetry in quantum eld theory. Being electrically neutral, photon is its own antiparticle (indeed, it is possible to annihilate two energetic photons and create massive matter from them). A nite value of µ would set a nite energy scale that separates particles with frequency ~ω > µ from antiparticles ~ω < µ. By relativistic symmetry between particles and antiparticles, µ must be zero. • The average number of photons in a mode m is given by the Bose-Einstein distribution (at temperature T and µ = 0): 1 hnmi = fBE(m) = eβm − 1 Therefore, the internal energy density of E&M elds in thermal equilibrium is:

∞ E 1 X 2 X 2 1 = mfBE(m) = nx,ny ,nz fBE(nx,ny ,nz ) = dkxdkydkz kfBE(k) V V V V ∆kx∆ky∆kz ˆ m nx,ny ,nz 0 The factor of 2 is due to polarization, and we are applying the same tricks as before to convert the sum over mode labels to an integral over wavevectors in a large cavity ∆ki = π/L is small). Following the same steps as earlier, we nd:

∞ ∞ 3 3 E 2 L 2 L 3 = dk dk dk fBE( ) = d k fBE( ) V V π3 ˆ x y z k k V (2π)3 ˆ k k 0 −∞ ∞ 2 2 = × 4π dk k fBE( ) (2π)3 ˆ k k 0 ∞ ∞ c k3 c 1 ξ3 = ~ dk = ~ dξ π2 ˆ eβ~ck − 1 π2 (β~c)4 ˆ eξ − 1 0 0 ∞ k T 3 ξ3 = k T × B × 8π dξ B hc ˆ eξ − 1 0

The proportionality E ∝ T 4 is known as Stefan's law of blackbody radiation. Heat capacity is:

dE 3 CV = ∝ T dT V

d It is actually characteristic for relativistic bosons in d spatial dimensions to have heat capacity CV ∝ T . • Spectral energy density ρ(ν) radiation energy per unit frequency and unit volume: dE = ρ(ν)dνdV is electromagnetic energy inside volume dV stored by the modes whose frequency is in the interval (ν, ν + dν). To obtain ρ(ν), we pick from the above integral for E/V only the contribution of modes

55 within (ν, ν + dν). Since hν = ~ck, this corresponds to the wavevectors k whose magnitude k = |k| is within 2π : c (ν, ν + dν)

2π(ν+dν)/c 1 c k3 ρ(ν) = × ~ dk dν π2 ˆ eβ~ck − 1 2πν/c 1 c (2πν/c)3 2πdν = × ~ dν π2 eβ~c×2πν/c − 1 c 8πh ν3 = c3 eβhν − 1 This is known as Planck's formula for blackbody radiation. The intensity of radiation at frequency ν that escapes the cavity through a small hole is proportional to ρ(ν) and can be directly measured (after the radiation is spectrally decomposed using a prism). Planck's formula matches experiments in the entire frequency range.

At low frequencies, βhν 1, Planck's formula reproduces the Rayleigh-Jeans formula, which arises in the classical thermodynamics of radiation:

3 2 βhν1 8πh ν 8πν ρ(ν) −−−−−→ = k T c3 1 + βhν − 1 c3 B This is clearly a classical formula since it does not contain Planck's constant. The average energy

carried by a single oscillator mode is proportional to kBT in agreement with classical thermodynamics (equipartition theorem). The factor 8πν2/c3 is just the density of states at frequency ν, i.e. the number of radiation modes per unit volume and unit frequency. At high frequencies, βhν 1, Rayleigh-Jeans formula makes an unphysical prediction that the spectral density ρ(ν) can become arbitrarily large. If this were true, then the energy density of electromagnetic radiation would be innite. Instead, the correct Planck's formula predicts:

βhν1 8πh ρ(ν) −−−−−→ ν3e−βhν c3 Wien's law: the maximum spectral radiance occurs at a frequency proportional to temperature:

dρ h i = 0 ⇒ 3ν2(eβhν − 1) − ν3βheβhν = ν2 (3 − βhν)eβhν − 3 = 0 dν ⇒ (3 − βhν)eβhν = 3 ⇒ βhν = 2.82144 k T ⇒ ν = 2.82144 × B h Molecules and solids

• Small-amplitude vibrations of molecules or solids can be described by a harmonic oscillator model. In classical mechanics, an oscillator has a fundamental frequency ω = pk/m related to the oscillator stiness k and mass m. The amplitude A of oscillations is arbitrary. The classical Hamiltonian

p2 kx2 H = + 2m 2 counts as two degrees of freedom by equipartition theorem: every appearance of a microstate coordinate squared in the Hamiltonian ( or ) yields a contribution of 1 to the average thermal energy of the x p 2 kBT system. Hence, every (one-dimensional) harmonic oscillator in the system should add 1 2 × 2 kB = kB to its heat capacity. However, this is experimentally observed only at very high temperatures, while at low temperatures the oscillator's contribution to heat capacity vanishes exponentially. The experiment

56 is properly understood only using quantum mechanics. Quantum mechanics predicts that the states of a harmonic oscillator are quantized. A quantum oscillator can only have energy equal to an integer multiple of ~ω. The average energy of an oscillator at temperature T is, therefore, determined by the Bose-Einstein distribution:

− ω/k T ~ω ωe ~ B ,T → 0 BE ~ E = ~ω f (~ω) = β ω → e ~ − 1 kBT,T → ∞

( 2 ) ω − ω/kB T dE kB ~ e ~ ,T → 0 C = → kB T dT kB ,T → ∞

• Rotations of a rigid molecule are described by the Hamiltonian

L2 H = 2I where L is angular momentum and I is molecule's moment of inertia. The number of degrees of freedom that classically contribute 1 to heat capacity equal the number of axes that the molecule 2 kB can rotate about. Diatomic molecules can rotate about two axes, while non-collinear tri-atomic and other molecules can rotate about three axes. These classical expectations are, again, seen only at high temperatures. At low temperatures, the rotational contribution to heat capacity vanishes exponentially because rotational energy is quantized as ~2l(l + 1)/2I where l ∈ {0, 1, 2 ... }. • A crystal made of N atoms can transmit sound modes - waves of atom displacements. Every atom pushed by nearby atoms can oscillate about its equilibrium location, and aect neighboring atoms to oscillate as well. Therefore, a crystal is a system of many coupled oscillators. Quantized oscillations of the crystal are called phonons, and have much in common with photons (quantized oscillations of electromagnetic elds). Phonon modes in a cube-shaped crystal are enumerated the same wave as electromagnetic modes, except that there are generally three polarizations (two transversal and one longitudinal). N atomic classical oscillators with three polarization modes (three-dimensional oscillators) should classically yield CV = 3NkB at all temperatures. However, CV is exponentially suppressed at low temperatures due to quantum mechanics. At low frequencies, phonons have dispersion ν = vsλ, where ν is frequency, λ is wavelength and vs is the speed of sound. Therefore, we reproduce the low-temperature expression for internal energy that is analogous to that of photons:

3 T ,T TD CV ∝ const ,T TD where ~vs π TD = kB a is the so called Debye temperature. The high-temperature dependence is classical and deviates from

that of photons because the relativistic sound-wave dispersion ν = vsλ extends only to nite frequencies related to the cut-o k . π/a where a is the lattice spacing.