The chapter asserts

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Wave mechanics in one dimension

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The harmonic oscillator

Problem

The chapter asserts, without carrying out the integral, that

\int|a_+\psi_0|^2dx = 1

, confirming

a_+\psi_0 = \sqrt1\,\psi_1

. Carry out that integral explicitly, using

\psi_0 = (m\omega/\pi\hbar)^{1/4}e^{-m\omega x^2/2\hbar}

and the standard result

\int_{-\infty}^{\infty}x^2e^{-\beta x^2}dx = \dfrac{1}{2\beta}\sqrt{\pi/\beta}