Chapter 11, Problem 11.17E

Interpretation Introduction

Interpretation:

The validation of the expression $E=3/2hv$ by the substitution of the ${\Psi }_1$ into the complete Hamiltonian operator of an ideal harmonic oscillator is to be shown.

Concept introduction:

The Schrödinger equation for one-dimensional harmonic oscillator is,

$\frac{d^2\Psi }{d{x}^2}+\left(\frac{2mE}{{\hslash }^2}-{\alpha }^2{x}^2\right)\Psi =0$

Where,

• $m$ is the mass of the body.

• $\Psi$ is the wavefunction of harmonic oscillator.

• $E$ is the energy of harmonic oscillator.

• $\hslash$ has the value $\frac{h}{2\pi }$

• The value of constant $\alpha =\frac{2\pi vm}{\hslash }$