## Quantum Mechanics, solutions

### Problem 1: (C)

The Bohr atom

a_{0} = 0.053 nm. r_{n} = n^{2}a_{0} = 0.5 mm.

### Problem 2: (B)

Electron Screening

All the electron wave functions with non-zero angular momentum are zero
at the origin. The greater the angular momentum, the farther the electron is
kept away from the nucleus.

### Problem 3: (C)

Hydrogenic atoms

To find the eigenfunctions and eigenvalues of the
Hamiltonian of a hydrogenic atom we replace in the eigenfunctions of the
Hamiltonian of the hydrogen atom a_{0} by a_{0}' = ħ^{2}/(μ'Ze^{2})
= a_{0}(μ/μ')(1/Z), and in the eigenvalues of the Hamiltonian of the
hydrogen atom we replace E_{I} by E_{I}' = μ'Z^{2}e^{4}/(2ħ^{2})
= E_{I}(μ'/μ)Z^{2}.

μ = m_{μ}m_{p}/(m_{μ}
+ m_{p}) ≈ m_{μ}.

### Problem 4: (D)

Addition of angular momenta

L = 2, S = 1/2 , J = 3/2, 5/2 are possible. From the given choices, only (D) is possible.

### Problem 5: (A)

Stark effect

The energy of the proton in the field is -∫_{0}^{zp}q_{e}|**E**|dz
= -q_{e}|**E**|z_{p}.

The energy of the electron in the field is ∫_{0}^{ze}q_{e}|**E**|dz
= q_{e}|**E**|z_{e}.

The potential energy of both particles is -q_{e}|**E**|(z_{p}
- z_{e}) = q_{e}|**E**|z = -**p∙E.**

### Problem 6: (B)

Entangled particles

After the measurement ψ(1,2) = α(1)β(2).

### Problem 7: (D)

Entangled particles, spin

After the measurement of particle 1 ψ(1,2) = α(1)β(2) = (1/2^{1/2})α(1)[α_{x}(2)
- β_{x}(2)]

### Problem 8: (B)

One-dimensional potential wells

The bottom of the well is shifted up. In one dimension, there is no degeneracy.

### Problem 9: (E)

Spherically symmetric potentials, postulates of QM

For a spherically symmetric potential H, L^{2}, and L_{z}
commute, their values can be known simultaneously, separation of variables is
possible.

Eigenfunctions of any observable with different eigenvalues are orthogonal.

### Problem 10: (D)

Selection rules

For optically allowed transitions we need
Δl = ±1.