Chapter 7, Problem 7F.4BE

Interpretation Introduction

Interpretation:

The orthogonality of the wavefunctions $\cos m_l\varphi$ and $\cos m_{{}_l}^{\text{'}}\varphi$ for a particle on a ring where $m_l\ne m_{{}_l}^{\text{'}}$ has to be proved.

Concept introduction:

For the orthogonality of the two different wave functions, the product of the wave functions is integrated over the entire limits.  It is expressed by the equation as given below.

  ${\displaystyle \underset{0}{\overset{\infty }{\int }}{\psi }^{\ast }{}_i{\psi }_j\text{d}\tau }=0$

Where, ${\psi }_i$ and ${\psi }_j$ are two different wave functions.  The essential condition for two wave functions to be orthogonal to each other is $i\ne j$.