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

Nuclear & Particle Physics - Spin

The ground state of the helium atom is a spin

A. singlet
B. doublet
C. triplet
D. quartet
E. quintuplet

(GR9277 #59)

Solution:

Spectroscopic notation: N^2s+1 L_j

2s + 1 = multiplicity

Singlet: 2s + 1 = 1, s = 0
Doublet: 2s + 1 = 2, s = 1/2
Triplet: 2s + 1 = 3, s = 1

Electron configuration of Helium: 1s²

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

Answer: A 

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 – Conduction Electrons

The mean kinetic energy of the conduction electrons in metals is ordinarily much higher than kT because

A. electrons have many more degrees of freedom than atoms do
B. the electrons and the lattice are not in thermal equilibrium
C. the electrons form a degenerate Fermi gas
D. electrons in metals are highly relativistic
E. electron interact strongly with phonons

(GR0177 #76)

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.

According to classical physics, the mean thermal energy of the electrons is ³⁄₂kT.

According to quantum physics, the mean energy of electrons (fermion energy) is  ³⁄₅E_F
where

n = N/a³→ the number of electrons per unit volume

If E_F ≪ kT : electrons are hot (high T) → electrons are non-degenerate → classic (Maxwell-Boltzman) statistic.

In reality, E_F ≫ kT: electrons are cold (room T) → electrons are degenerate → quantum (Fermi-Dirac) statistic.

Answer: C