Quantum Mechanics, solutions

Problem 1:  (C)

The Bohr atom

a0 = 0.053 nm.  rn = n2a0 = 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 a0 by a0' = ħ2/(μ'Ze2) = a0(μ/μ')(1/Z), and in the eigenvalues of the Hamiltonian of the hydrogen atom we replace EI by EI' = μ'Z2e4/(2ħ2) = EI(μ'/μ)Z2.
μ = mμmp/(mμ + mp) ≈ mμ.

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 -∫0zpqe|E|dz = -qe|E|zp.
The energy of the electron in the field is ∫0zeqe|E|dz = qe|E|ze.
The potential energy of both particles is -qe|E|(zp - ze) = qe|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) = (1/21/2)α(1)[αx(2) - βx(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, L2, and Lz 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.