## Modern Physics, solutions

### Problem 1:  (D)

The harmonic oscillator

En = (n + ½)ħω = (n + ½)hf,  n = 0. 1, 2, ...

### Problem 2:  (D)

Uncertainty principle:  ΔxΔp ~ ħ

Δp ~ ħc/(c*10-15 m) ~ (1240 eV nm)/(c*10-6 nm *2π) ~ 200 MeV/c

Tunneling:

Tunneling:

### Problem 5:  (B)

The 3D infinite square well

E = (nx2 + ny2+ nz2)2ħ2/(2mL2),  nx, ny, nz = 1, 2, 3, ...
E1 = 3π2ħ2/(2mL2),  degeneracy, = 1 (nx = ny = nz = 1)
E2 = 6π2ħ2/(2mL2),  degeneracy = 3  (ni = 2, nj = nk = 2,  i = 1, 2, 3)
E3 = 9π2ħ2/(2mL2),  degeneracy = 3  (ni = 1, nj = nk = 2,  i = 1, 2, 3)

Problem 6:  (A)

The energy operator:

Problem 7:  (E)

Commuting operators

If an operator commutes with the Hamiltonian, it is called a "constant of motion".  For each eigenvalue, the probability that a measurement will yield this eigenvalue does not change with time.

### Problem 8:  (A)

Hydrogenic atoms, scaling rule

a = a0/Z.  (Exact:  a = ħ2/(Zμe2))

### Problem 9:  (E)

Probability density:  P(x)dx = |ψ(x)|2dx

P (x2 > x > x1) = (1/d)∫-x1x2 exp(-2x/d) dx = ½∫-2x1/d2x2/d exp(-y) dy
= ½(exp(-2x1/d) -exp(-2x2/d))

### Problem 10:  (E)

Addition of angular momenta

The possible values of l range from |l1 - l1| to l1 + l2 in integer steps.  For each l, m can take on values between -l and +l in integer step.  When adding angular momenta, the m values add algebraically, m = m1 + m2.  Here m = m1 + m2 = 5,  l must be greater or equal to 5.