{{page>wiki:headers:hheader}}

====== Orthogonality in the Wave Function Representation ======

===== Prompt =====

**" Write down the orthogonality relationship" ** 

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

** "in wavefunction form." **



===== Context =====

This [[strategy:smallwhiteboard:|SWBQ]] is used to help the students to
comfortably go between one representation of a quantum state to another.

===== Wrap Up =====

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:

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

{{wvswortho.ppt|Powerpoint prompt}}
\\\\
{{wvswortho.pdf|PDF prompt}}

{{page>wiki:footers:courses:wvfooter}}
{{page>wiki:footers:topics:qmfooter}}