## Modern Physics

• Photons and matter waves
The photoelectric effect, the deBroglie relations, the uncertainty principle
• Quantum Mechanics
The Schroedinger equation, eigenfunctions and eigenvalues, momentum and energy operators, incompatible observables, the Pauli exclusion principle
• Quantum-mechanical systems
Square potential, the harmonic oscillator, hydrogenic atoms, multi-electron atoms, tunneling
• Angular momentum and spin
Quantization of angular momentum, spin ½ particles

### Problem 1:

The third lowest energy of level of a one-dimensional quantum-mechanical harmonic oscillator of frequency f has an energy of

(A)  0    (B)  (3/2)hf    (C)  2hf    (D)  (5/2)hf   (E)  3hf

### Problem 2:

If an electron were confined to nuclear dimensions, the uncertainty in its momentum would be

(A)  0.2 eV/c    (B)  200 eV/c    (C)  200 keV/c    (D) 200 MeV/c   (E)  200 GeV/c

### Problem 3:

According to quantum mechanics, which of the energies shown represents a possible permanent bound state for a particle?

(A)  E1 only   (B)  E2 only    (C)  E3 only    (D)  E1 or E2   (E)  E2 or E3

### Problem 4:

Which of the energies shown in the figure represents a possible meta-stable state, from which a confined particle would eventually tunnel out?

(A)  E1 only   (B)  E2 only    (C)  E3 only    (D)  E1 or E2   (E)  E2 or E3

### Problem 5:

A spinless particle is confined to a cubical box of side length L, for which the potential is
V = 0 for 0 ≤ x, y , z, ≤ L,  V = ∞ otherwise.
What is the degeneracy of the third quantum level in the box?

(A) 1   (B)  3    (C)  6    (D)  9   (E)  12

### Problem 6:

In quantum mechanics, the operator for the total energy is

(A)  iħ ∂/∂t   (B)  iħ2 ∂2/∂t2    (C)  -iħ    (D)  iħ    (E)  ħ·

### Problem 7:

Assume that the thre operators (Lx, Ly, Lz) for the components of the orbital angular momentum commute with the Hamiltonian.  Therefore the angular momentum is

(A)  equal to zero
(B)  equal to energy in magnitude
(C)  a unit vector
(D)  proportional to sinθ
(E)  a constant of motion

Problem 8:

If a0 is the Bohr radius of the first Bohr orbit in hydrogen, then the radius of the first bohr orbit in doubly ionized lithium is

(A)  a0/3   (B)  a0/√3    (C)  a0    (D)  √3 a0    (E)  3 a0

Problem 9:

A free particle moving in one dimension has a wave function ψ(x) = d exp(-|x|/d) at t = 0.
What is the probability that a measurement of the position of the particle at t = 0 will yield a result between x1 and x2 (x2 > x1 > 0)?

(A)  0   (B)  (x2 - x1)/d   (C)  (x2 - x1)2/d2
(D)  ½(exp(-x1/d) -exp(-x2/d))    (E)  ½(exp(-2x1/d) -exp(-2x2/d))

### Problem 10:

Two particles have angular momentum quantum numbers l1 = l2 = 4, m1 = 3 and m2 = 2.
Which of the following is an allowed value for l corresponding to L = L1 + L2?

(A) 0    (B)  1    (C)  2    (D)  4    (E)  6