Questions: Symmetries and Conservation Laws in Quantum Mechanics

Question 1

A Hamiltonian commutes with the translation operator T(a) for every displacement a. Which quantity is conserved?

• A. Energy only
• B. Linear momentum
• C. Angular momentum
• D. Parity

Answer: (B)

T(a) = exp(−ipa/ħ) is generated by momentum. [T(a), H] = 0 for all a means [p, H] = 0, so dp/dt = 0 in the Heisenberg picture.

Question 2

What is the parity of the n = 3 energy eigenstate of the one-dimensional harmonic oscillator?

• A. Even (+1)
• B. Odd (−1)
• C. Undefined, since oscillator states have no definite parity
• D. +3

Answer: (B)

Oscillator eigenstates have parity (−1)ⁿ. For n = 3 the wavefunction is odd, ψ(−x) = −ψ(x).

Question 3

The time-reversal operator in quantum mechanics is linear and unitary.

• T. True
• F. False

Answer: False

Reversing the direction of time in the Schrödinger equation requires complex conjugation of the wavefunction, so the operator is antilinear. Wigner showed that symmetry operators are either unitary or antiunitary; time reversal is antiunitary.

Question 4

Why must the initial and final states of an electric dipole transition have opposite parity?

Model answer: The dipole operator is proportional to the position vector, which is odd under parity. The matrix element ⟨f|r|i⟩ is the integral of an odd function unless ψ_f and ψ_i have opposite parity, so it vanishes otherwise.

For hydrogen, orbital parity is (−1)^l, so parity requires Δl to be odd. Combined with the angular momentum rule |Δl| ≤ 1 for a vector operator, this gives the selection rule Δl = ±1 for electric dipole transitions.