Modern Physics: Mastering Physics: Normalizing the Wave Functions for the Harmonic Oscillator

The wave function for the ground state of the harmonic oscillator is

$\psi_0(x)=Ce^{-[m\omega/(2\hbar)]x^2}$ ,

where

is an arbitrary constant, hbar

is Planck's constant divided by $2\pi$ ,

is the mass of the particle, $\omega=\sqrt{k/m}$ , and

is the "spring constant" for the harmonic oscillator.

Part A
Normalize this wave function. What is the (positive) value of once this wave function is normalized? You will need the formula $\int _{-\infty}^\infty e^{-ax^2}=\sqrt{\frac{\pi}{a}}$ .

Express your answer in terms of omega

, and

ANSWER:

$\left(\frac{m{\omega}}{{\hbar}{\pi}}\right)^{\frac{1}{4}}$

Modern Physics