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 - Conduction Electrons

The mean kinetic energy of electrons in metals at room temperature is usually many times the thermal energy kT. Which of the following can best be used to explain this fact?

A. The energy-time uncertainty relation
B. The Pauli exclusion principle
C. The degeneracy of the energy levels
D. The Born approximation
E. The wave-particle duality

(GR8677 #55)

Solution:

The mean kinetic energy of electrons in metals at room temperature is usually many times the thermal energy kT due to Pauli’s exclusion principle.

See problem GR0177 #76

Answer: B

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 - Fermion Energy in 3D and 2D

Show that:

Fermion Energy in 3D: E = (3/5) E_F

Fermion Energy 2D: E = (1/2) E_F

Solution:

click image to enlarge