Term 2

Question 1

Please define Density of States (DOS)

Answer 1

The density of states, g(E), describes how the availiable energy states are arranged.

g(k)dk = Number of allowed states with wave-vector magnitude between k and k+dk, for a wavefunction of the for Ψ_x = Ae^ik_xx

Question 2

Please give the DOS equation

Answer 2

• In 3D: g(k)dk = vol of spherical shell of thickness dk
  vol of one k-state
• g(k)dk = Vk²/2π² dk

Question 3

Please give the single particle partition function equation

Answer 3

z₁ = ∫₀^∞e^-βE(k)g(k)dk

where e^-βE(k) is the Boltzmann factor

Question 4

Please give the single particle partition function for a given momentum and energy

Answer 4

As p(k) = hk/2π, E(k) = (hk)²/8π²m:
z₁ = V(2πmk_BT/h)^3/2

Question 5

Please define the quantum concentration, n_Q, along with the regimes it defines

Answer 5

• n_Q = (2πmk_BT/h)^3/2
• n_Q = 1/λ_th³
• if n &laquo_space;n_Q: Classical Gas
• if n&raquo_space; n_Q: Quantum Gas

Question 6

Please define the many particle partition function for distinguishable and indistinguishable particles

Answer 6

• Distinguishable: z_N = z₁^N
• Indistinguishable: z_N = z₁^N/N!

Question 7

Please give the conclusion of the solution to the Gibbs Paradox

Answer 7

Indistinguishability is a fundamental property, not an experimental inadequacy.

Question 8

Please define the chemical potential, μ

Answer 8

• μ = change in energy on adding a particle to the system
• μ = G/N : Gibbs free energy per particle

Question 9

Please give the Gibbs distribution and the Grand Partition function equations

Answer 9

• P_i = e^{-β(E_i - μN_i)/Z}
• Z = Σ_ie^{-β(E_i - μN_i)}

Question 10

Please give the Grand Potential equation

Answer 10

• Φ_G = -K_BTlnZ
• Hence, Z = e^-βΦ_G

Question 11

Please give the energy density equation for photons and the DOS in terms of k

Answer 11

• u = U/V = 1/V∫₀^∞g(ω)⟨E(ω)⟩dω
• g(k)dk = Vk²dk/2π x2

where the x2 comes from the two independent polarisations of light

Question 12

Please give the dispersion relation and wavelength-vector relation

Answer 12

• ω = ck
• λ = 2π/k

Question 13

Please define Phonons

Answer 13

Lattice vibration modes in a crystalline solid

Question 14

Please state the two assumptions of the Einsteing model of phonons

Answer 14

1. Ions are independent SHOs, with 3N independent modes in a crystal of N atoms
2. All atoms oscillate with the same frequency, ω_E

Question 15

What does the Einstein model of phonons work well for?

Answer 15

High and mid temps, but not low.

Question 16

What are the assumptions of the DeBye model for phonons?

Answer 16

1. Lattice vibrations are waves with dispersion relation ω = v_sq
2. There is a cutoff frequency, ω_D

where q is the wavevector and v_s is the speed of sound

Question 17

What is the D.O.S equation for the DeBye model in terms of q and ω

Answer 17

• g(q)dq = Vq²/2π² dq x3
• g(ω)dω = 3Vω²/2π²v_s³dω

the x3 is from the polarisations: 2 transverse and 1 longitudinal

Question 18

What is the DeBye Frequency equation?

Answer 18

ω_D = (6Nπ²V_s^3/V)1/3

Question 19

Please give the equation defining the DeBye Temperature

Answer 19

Θ_D = hω_D/k_B

Question 20

Please define Boson and Fermions

Answer 20

From |Ψ(r₁,r₂)|² = |Ψ(r₂,r₁)|², Ψ(r₁,r₂) = ±Ψ(r₂,r₁)
- If Ψ is symmetric under particle exchange it is a Boson with integer spin
- If Ψ is antisymmetric under particle exchange it is a Fermion with half integer spin

Question 21

Please state the Pauli Exclusion principle

Answer 21

Two identical fermions cannot coexist in the same quantum state

Question 22

Please define the Grand Partition Function and the mean occupation number of a many particle system with a single state at energy, E.

Answer 22

• Z = Σ_n(e^{-nβ(E - μ)}
• ⟨n⟩ = Σ_n(e^{-nβ(E - μ)/Z) = -1/βZ dZ/dE = - 1/β d(ln(Z))/dE}

where n is the occupation number: number of particles in the state

Question 23

Please give the Fermion Grand Partition function and mean occupation number for a single state

Answer 23

• Z = 1 + e^-β(E-μ)
• ⟨n⟩ = 1/(e^β(E-μ)+1)

Question 24

Please give the Boson Grand Partition function and mean occupation number for a single state

Answer 24

• Z = 1 / (1- e^-β(E-μ))
• ⟨n⟩ = 1 / (e^β(E-μ) - 1)

Question 25

Please give the equation for the Fermi-Dirac distribution and its range.

Answer 25

- f(E) = 1/(e^β(E-μ)+1) - Range: 0 - 1

Question 26

Please give the equation for the Bose-Einstein distribution and its range.

Answer 26

- f(E) = 1/(e^β(E-μ)-1) - Range: 0 - ∞

Question 27

Please give the equations for N and U for fermions and bosons

Answer 27

- N = ∫₀^∞g(E)f(E)dE - U = ∫₀^∞Eg(E)f(E)dE ## Footnote where N is the number of particles and U is the total internal energy of the system

Question 28

Please define the Fermi energy, and state its equations

Answer 28

- Energy of the highest single-particle state in a quantum system of non-interacting fermions at absolute zero. - E_F = h_bar² / 2m (3π²n)^2/3 = h_bar²k_F² / 2m ## Footnote where n = N/V and k_F is the fermi wavevector

Question 29

Please define Degeneracy Pressure

Answer 29

Pressure due to electrons at T = 0K - P = -(dU/dV)_S = 2/5 nE_F ## Footnote where n = N/V

Question 30

What is a Bose-Einstein condensate? Why does this state of matter form?

Answer 30

- A state of matter where a macroscopic number of particles occupy the same quantum state. - At low temperatures, the discreteness of energy levels annot be ignored as the assumption was that the thermal energy >> seperation between energy levels but this assumption no longer holds.

Question 31

What is the equation for the fraction of particles in the ground state (Bose-Einstein condensate)?

Answer 31

n₀ / n = 1 - (T_c / T) ^3/2