Quantum Mechanics - Fermion

The wave function for identical fermions is antisymmetric under particle interchange. Which of the following is a consequence of this property?

A. Pauli exclusion principle
B. Bohr correspondence principle
C. Heisenberg uncertainty principle
D. Bose-Einstein condensation
E. Fermi's golden rule

(GR9677 #35)

Solution:

Pauli Exclusion Principle: no two electrons can have exactly the same quantum number.

m_s1 =  ½  and m_s2 =  −½
Total spin quantum number, s = ½ + (−½) = 0
Multiplicity, 2 · 0 + 1 = 1 → Singlet state (antisymmetric)

Answer: A

Notes:

Singlet: 2s + 1 = 1, s = 0
Singlet state is anti-symmetric: ψ(1,2) = −ψ(2,1)
Obeys Fermi-Dirac statistics → fermion

Triplet: 2s + 1 = 3, s = 1
Triplet state is symmetric: ψ(1,2) = ψ(2,1)
Obeys Bose-Einstein statistics → bosons

Thermal Physics - Maxwell-Boltzmann Statistics

Consider a system of N non-interacting particles confined in a volume V at a temperature such that the particles obey classical Boltzmann statistics. If the temperature is lowered to the point at which quantum effects become important, the pressure of the gas may differ depending on whether the particles are fermions or bosons. Let P_F be the pressure exerted by the particles if they are fermions. P_B be the pressure if they are bosons and, and P_C be the pressure the particles would exert if quantum effects are ignored. Which of the following is true?

A.
B.
C.
D.
E.

(GR8677 #88)

Solution:

Ideal gas:

Classical (Maxwell-Boltzmann Statistics):




For Bosons (photon):



→

For Fermion:
= Fermi Energy,

For the same temperature:  

Answer: B

Quantum Mechanics - Wave Function

A system containing two identical particles is described by a wave function of the form

Where x₁ and x₂ represent the spatial coordinates of the particles and α and β represent all the quantum numbers, including spin, of the states that they occupy. The particles might be 

A. Electrons
B. Positrons
C. Protons
D. Neutrons
E. Deuterons

(GR8677 #89)

Solution:

Wave function: 

Symmetric function (+ sign):

→ obey Bose-Einstein statistics → bosons

Anti-symmetric function (−sign):
→ obey Fermi-Dirac statistics → fermion

Electrons and positron (anti-elecron) → fermion

Protons and neutron → fermionic hadrons

• composite particles (hadron)
• composed of 3 fermionic quarks

Deuterons → bosons

• nucleus of deuterium, an isotope of hydrogen
• composed of a neutron (spin 1/2) and a proton (spin 1/2), total spin = 1

Answer: E

Thermal Physics - Pressure of photon gas

Compute the pressure exerted by gas of photons.

Solution:

According to kinetic theory analysis, pressure:



Momentum: p = mv
For photon, v = c
Energy of photon: E = pc

→ Pressure:



Ideal Gas: PV = NkT

→ Pressure:



→ Energy: