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

Optics - Diffraction Grating

Light of wavelength 5200 angstroms is incident normally on a transmission diffraction grating with 2000 lines per centimeter. The first-order diffraction maximum is at an angle, with respect to the incident beam, that is most nearly

A. 3^o
B. 6^o
C. 9^o
D. 12^o
E. 15^o

(GR9277 #35)

Solution:

d sin θ = mλ 

λ = 5200 Angstrom = 5200  × 10⁻¹⁰ m = 5.2 × 10⁻⁷ m
d = 1 cm / 2000 = 5 × 10⁻⁴ cm =  5 × 10⁻⁶ m
m = 1

sin θ = mλ / d  =  (5.2 × 10⁻⁷) / (5 × 10⁻⁶) ≈ 0.1

→ arcsin θ = 6^o 

Or, since sin θ ≪ 1 → sin θ ≈ θ 

Convert the angle from radians to degrees: 0.1  × 180/π  ≈ 18/3 =  6^o


Answer: B

Classical Mechanics - Hamiltonian

Question 34-36

The potential energy of a body constrained to move on a straight line is kx⁴ where k is a constant. The position of the body is x, its speed v, its linear momentum p, and its mass m.

The Hamiltonian function for this system is

A. (p²/2m) + kx⁴
B. (p²/2m) − kx⁴
C. kx⁴
D. ½mv² − kx⁴
E. ½mv²

(GR8677 #35)

Solution:

Hamiltonian: H = T + U
U = kx⁴
T = ½mv² = p²/2m
H = (p²/2m) + kx⁴

Answer: A

Thermal Physics - Blackbody Radiation

If the absolute temperature of a blackbody is increased by a factor of 3, the energy radiated per second per unit area does which of the following

A. Decreases by a factor of 81
B. Decreases by a factor of 9
C. Increases by a factor of 9
D. Increases by a factor of 27
E. Increases by a factor of 81

(GR0177 #35)

Solution:

Blackbody radiation: u = σT⁴

Thus, if T increases then u (energy) increases
(A) and (B) are FALSE.

The ratio of energy radiated is



Answer: E