Orthogonality in the Wave Function Representation

” Write down the orthogonality relationship”

$$\boldsymbol{\langle\varphi_m \vert \varphi_n\rangle=\delta_{mn}}$$

“in wavefunction form.”

This SWBQ is used to help the students to comfortably go between one representation of a quantum state to another.

The bra-ket form of the orthogonality relation is easy to remember. Translating it to wave function form is more difficult, but it's a useful exercise so that students can recognize orthogonality from the less obvious \[\int\limits_{ - \infty }^\infty {\varphi _n^*\left( x \right){\varphi _m}\left( x \right)dx} \]

Stress that the functions must be eigenfunctions of a linear operator, and that the integral must be over all space. It is helpful for the students to have an operational strategy for translating bra-ket notation to wave function notation. They should understand why, of course, but an operational strategy is helpful, and builds confidence. That strategy is:

1. replace bra with c.c. wave function $\left\langle \varphi \right| \to {\varphi ^*}\left( x \right) $
2. replace ket with wave function $\left| \varphi \right\rangle \to \varphi \left( x \right) $
3. replace operator with position representation $\hat A \to A\left( x \right) $ (in this case the unity operator)
4. replace bracket with integral over all space $\left\langle {} |{} \right\rangle \to \int\limits_{ - \infty }^\infty { { }dx} $

Powerpoint prompt
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