Optics - Index of Refraction

A light source is at the bottom of a pool of water (the index of refraction of water is 1.33). At what minimum angle of incidence will a ray be totally reflected at the surface?

A. 0^o  
B. 25^o
C. 50^o 
D. 75^o
E. 90^o 

(GR9677 #56)

Solution:

Snell's Law: n_w sin θ_i = n_a sin θ_r

For total internal reflection, θ_r = 90^o
n_w sin θ_i  =  n_a
sin θ_i = n_a/n_w = 1/1.33 ≈ 3/4
Convert to degree, θ = ¾ × 180/π ≈ 45^o closer to 50^o 

Answer: C

Quantum Mechanics - Harmonic Oscillator

If v is frequency and h is Planck’s constant, the ground state energy of a one-dimensional quantum mechanical harmonic oscillator

A. 0
B. 1/3 hv
C. 1/2 hv
D. hv
E. 3/2 hv

(GR9277 #56)

Solution:

The energy of a 1-D quantum HO: E_n = (n + ½)hv

The ground state energy, n = 0 → E₀ = ½ hv

Answer: C

Quantum Mechanics - Expectation Value

If ψ is a normalized solution of the Schrodinger equation and Q is the operator corresponding to a physical observable x and the quantity ψ*Qψ may be integrated in order to obtain the 

A. Normalization constant for ψ
B. Spatial overlap of Q with ψ
C. Mean value of x
D. Uncertainty in x
E. Time derivative of x

(GR8677 #56)

Solution:

Expectation value: 

• The predicted mean value of the result of an experiment.
• The average value of the measurable quantity Q (observable x) on the state ψ

Answer: C

Fluid - Buoyancy

A balloon is to be filled with helium and used to suspend a mass of 300 kilograms in air. If the mass of the balloon is neglected, which of the following gives the approximate volume of helium required? (The density of air is 1.29 kilograms per cubic meter and the density of helium is 0.18 kilogram per cubic meter.)

A. 50 m³
B. 95 m³
C. 135 m³
D. 270 m³
E. 540 m³

(GR0177 #56)

Solution:

ΣF = 0
ρ_agV =  ρ_hgV + m₀g

(ρ_a -  ρ_h)V = m₀
V = m₀ / (ρ_a - ρ_h) = 300 / (1.29 - 0.18) = 270 m³

Answer: D