Principles of Quantum Mechanics · Problem 7.3.3

The Harmonic Oscillator — R. Shankar

Physics — Quantum Mechanics · parity-orthogonality-oscillator

Problem

If ψ(x)\psi(x) is even and ϕ(x)\phi(x) is odd under x→−xx \to -x, show that ∫−∞∞ψ(x)ϕ(x) dx=0\int_{-\infty}^{\infty} \psi(x)\phi(x)\, dx = 0 Use this to show that ψ2(x)\psi_2(x) and ψ1(x)\psi_1(x) are orthogonal. Using the values of Gaussian integrals in Appendix A.2 verify that ψ2(x)\psi_2(x) and ψ0(x)\psi_0(x) are orthogonal.

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