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 - Fermi-Dirac Statistics

Questions 71-73

A system in thermal equilibrium at temperature T consists of a large number N₀ of subsystems, each of which can exist only in two states of energy E₁ and E₂, where . In the expressions that follow, k is the Boltzmann constant.

For a system at temperature T, the average number of subsystems in the state of energy E₁ is given by

A. 

B. 

C.

D.

E.

(GR9277 #71)

Solution:

For only 2 states of energy → Fermi-Dirac distribution

Number of states: 

Since , take E₂ = 2ϵ and E₁ = ϵ

For E₁ = ϵ, number of states:

Answer: B

Nuclear & Particle Physics - Spin

The hypothesis that an electron possesses spin is qualitatively significant for the explanation of all the following topics EXCEPT the

A. Structure of the periodic table
B. Specific heat of metals
C. Anomalous Zeeman effect
D. Deflection of moving electron by a uniform magnetic field
E. Fine structure of atomic spectra

(GR8677 #27)

Solution:

(A) TRUE
Spin → Pauli exclusion principle → electron configuration → Structure of the periodic table

(B) TRUE
Specific heat for Fermions (½-integer spin) is different from Bosons (integer spin).

(C) TRUE
Zeeman Effect: the splitting of spectral lines when an external magnetic field is applied.
"Normal" Zeeman effect → This type of splitting is observed for spin 0 states since the spin does not contribute to the angular momentum.
"Anomalous" Zeeman effect → When electron spin is included, there is a greater variety of splitting patterns.

(D) FALSE
Deflection of moving electron by a uniform magnetic field does not depend on spin

(E) TRUE
Fine structure = the splitting of the spectral lines of atoms due to quantum mechanical (electron spin) and relativistic corrections.

Answer: D

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