The number states of the one-dimensional oscillator have definite parity,
PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin
The Structure of Quantum Mechanics, Ch Symmetries, generators and conservation laws
Problem
The number states of the one-dimensional oscillator have definite parity, Π^∣n⟩=(−1)n∣n⟩, and x^ is odd, Π^x^Π^=−x^. Without computing any integral, decide which of the following matrix elements parity alone forces to vanish.
(a) ⟨0∣x^∣1⟩
(b) ⟨1∣x^2∣2⟩
(c) ⟨0∣x^2∣2⟩
(d) ⟨0∣x^∣3⟩
(e) ⟨2∣x^3∣0⟩
One of the elements that parity allows is zero anyway. Which one, and what result from Chapter 8 shows it?