Bosons

Question 1

Boson

Definition

Answer 1

• subatomic particles
• many can occupy the same state
• have zero or integer spin (in units of ħ)
  e. g. photons, He 4 atoms, Rb 87 atoms

Question 2

Classical Boltzmann Statistics Recap

Answer 2

• Maxwell-Boltzmann distribution
• many more particles than states
• there is either 1 or 0 particles per state

Question 3

Multiplicity of Bosons

Derivation

Answer 3

• any number of particles per state
• consider n particles in m states as n particles in m boxes
• replace boxes by partitions
• there are m-1 partitions between the m states and n particles which together can be arranged in (m-1+n)! ways
• but there are (m-1) identical partitions and swapping them with each other doesn’t produce a different distribution so divide by (m-1)!
• the n particles are also identical so divide by n!

Question 4

Multiplicity of Bosons

Equation

Answer 4

Ω = (n+m-1)! / n!(m-1)!

Question 5

Lagrange’s Method of Undetermined Multipliers

Answer 5

• consider your function
• define you constraint with a separate formula and rearrange so you have an expression equal to zero
• multiply each zero constraint expression by a different unknown Lagrange factor
• add these terms to the original function
• partially differentiate your function with respect to each variable separately
• set each of these expressions equal to zero to find the conditions for maxima/minima
• these expressions will contain your unknown Lagrange factor(s), to find out what they are, substitute your original constraint equations back in

Question 6

Boltzmann Distribution Function

Answer 6

f = ni / N = exp(-Ei/kT) / Σexp(-EI/kT)

Question 7

Bose-Einstein Distribution Function

Answer 7

f = ni / ωi = 1 / [exp(α + Ei/kT) - 1]
more commonly written:
f =

Question 8

Bose-Einstein Function

ni

Answer 8

ni = 1 / [exp(Ei-μ / kT) - 1]

Question 9

Bose-Einstein Condensation

Answer 9

• at T~0K all bosons crowd into the ground state
• this begins at T=Tc, a critical temperature
• there are particles in the ground state above Tc but not very many, and there are particles in excited states below Tc

Question 10

Normal Condensation vs Bose-Einstein Condensation

Answer 10

• normal condensation, the vapour-to-liquid transition, is due to interparticle attraction
• Bose-Einstein condensation is driven by exchange interactions
• each particle in the BE condensate has a eave function that fills the entire volume of the container

Question 11

Bose-Einstein Condensation

Critical Temprature Equation

Answer 11

Tcc = 0.53/k * (h²/2πm) * (N/V)^(2/3)

Question 12

Photon Gas

Answer 12

-consider a cavity filled with radiation where the walls emit and absorb em waves
-at equilibrium the walls and radiation have the same temperature T
-the energy of radiation is spread over a range of frequencies, energy density:
u(T) = ∫ us(ν, T) dν
-where us(ν, t) is the spectral energy density, and ν is frequency

Question 13

Ideal Gas vs Photon Gas

Isothermal Expansion

Answer 13

• for an ideal gas, an isothermal expansion conserves the energy of the gas
• whereas for a photon gas, it is the energy density that is conserved

Question 14

Absorptivity

Answer 14

• a real surface absorbs only a fraction of the radiation falling on it
• the absorptivity α is a function of ν and T

Question 15

Blackbody

Definition

Answer 15

-a surface for which α(ν) = 1 for all frequencies is called a blackbody

Question 16

Wien’s Displacement Law

Answer 16

h νmax / kT ≈ 2.8

-where νmax is the most likely frequency of blackbody radiation with temperature T

Question 17

Stefan-Boltzmann Law of Radiation

Answer 17

P = Aeσ (T^4 - To^4)

Question 18

Heat Capacity of a Photon Gas at Constant Volume

Answer 18

Cv = ∂u(T)/∂T = 16σ/c * VT²

Question 19

Entropy of a Photon Gas

Answer 19

S(T) = ∫ Cv(T)/T dT
= 16σV/c * ∫ T² dT
-integrate between 0 and T:
S(T) = 16σ/3c * VT^3

Question 20

Pressure of a Photon Gas

Radiation Pressure

Answer 20

P = - ∂F/∂V = 4σ/3c * T^4

= 1/3 * u