Quantum Mechanics - Schrodinger Equation

The solution to the Schrödinger equation for a particle bound in a one-dimensional, infinitely deep potential well, indexed by quantum number n, indicates that in the middle of the well the probability density vanishes for

A. The ground state (n = 1) only
B. States of even n (n = 2, 4, ...)
C. States of odd n (n = 1, 3, ...)
D. All states (n = 1, 2, 3, ...)
E. All states except the ground state

(GR9677 #51)

Solution:

Wavefunctions for the first 5 states for a particle bound in a 1-D, infinitely deep potential well

Pic: ecee.colorado.edu

The even wave functions (n = 2, 4, ...) always have nodes in the middle.

→ probability density for states of even n vanishes.

Answer: B

Quantum Mechanics - Infinite Potential Well

Questions 51-53

A particle of mass m is confined to an infinitely deep square-well potential:

V(x) = ∞, x ≤ 0, x ≥ a
V(x) = 0, 0  x  a

The normalized eigenfunction, labeled by the quantum number n, are



For any state n, the expectation value of the momentum of the particle is

A. 0

B. 

C. 

D. 

E. 

(GR9277 #51)

Solution:

Infinitely deep square-well potential
→ there is zero probability for particle to be outside the well
→ 〈〉= 0

If 〈〉 ≠ 0 the particle would tend to go to the right or left and leave the well, which is impossible for infinitely deep square-well potential.

Answer: A


Alternative Answer #1:

Eigen function,  ψ_n = λψ_n
The eigenvalue, λ is associated with expectation value:〈Â〉= 〈ψ | Â | ψ〉

Since  is imaginer and ψ_n is real →  the eigenvalue, λ is imaginer = not real = not observable
→ 〈〉= 0


Alternative Answer #2:

since sine and cosine are orthogonal the whole period.

Thermal Physics - Einstein solid

One feature common to both the Debye theory and the Einstein theory of the specific heat of a crystal composed of N identical atoms is that the

1. Average energy of each atom is 3kT
2. Vibrational energy of the crystal is equivalent to the energy of 3N independent harmonic oscillators
3. Crystal is assumed to be continuous for all elastic waves
4. Speed of the longitudinal elastic waves is less than the speed of the transverse elastic waves
5. Upper cutoff frequency of the elastic waves is the same

(GR8677 #51)

Solution:

Einstein solid:

• Thermal properties of crystal
• Highly idealized model focusing only the vibrational modes of crystal
• Assumption: Each of N atoms is free to vibrate around its equilibrium position in any of 3 coordinate directions by a harmonic force with a natural frequency ω₀.

Einstein solid equipartition energy:

• Number of oscillator N = N₀,
• 3-D Harmonic Oscillator, f = 6
• U = (6/2) N₀kT = 3N₀kT

Answer: B

Optics - Polarization

Unpolarized light of intensity I₀ is incident on a series of three polarizing filters. The axis of the second filter is oriented at 45^o to that of the first filter, while the axis of the third filter is oriented at 90^o to that of the first filter. What is the intensity of the light transmitted through the third filter?

A. 0
B. I₀/8
C. I₀/4
D. I₀/2
E. I₀/√2

(GR0177 #51)

Solution:

Polarizer:
The first filter always reduces the intensity of the light to half, I₁=½ I₀
The next filter reduces the intensity by I_n = I_(n-1) cos² θ
θ is the angle with respect to the n^th filter.

Thus,

I₁ = ½ I₀

I₂ = I₁ (cos 45)² = I₁ (½ √2)² = ½ I₁ = ¼ I₀

I₃=I₂ (cos 45)² = ½ I₂ = I₀/8

Answer: B