# What's the difference between them

• enricfemi
In summary, <n|m> is a number and |m><n| is an operator that acts on one wave function to give a different one. It can be thought of as a matrix or "sort of operator". <n|m> is a number, while <m|n> is a vector scalar product. It should not be rearranged, and only numbers commute. When substituted with <x|n>, it can be replaced with the wave function \psi_n(x).

#### enricfemi

<n|m> ;

|m><n| ;

<n|m> is a number.
|m><n| is an operator that would act on one wave function to give a different one.

appreciate it!

i know the product.
but what the operator is?
is it pesi(m)pesi(n)*?

do you mean :
when it act on one wave function,it denotes
|m><n|a> ?

Think of it as *all* 1xN matrices
$$|n>=A=\begin{pmatrix} a\\b\\c\\\ldots \end{pmatrix}$$
$$<n|=A^\dag=\begin{pmatrix} a^* & b^* & c^* & \ldots \end{pmatrix}$$
$$\psi_n(x)=<x|n>$$
Then you see that $|m><n|$ is a matrix ("sort of operator"). $<m|n>$ is a number (like the vector scalar product). Note that $<n|m>\neq \psi_n^*(x)\psi_m(x)$, however since $\sum_x |x><x|=1$
$$<n|m>=\sum_x <n|x><x|m>=\sum_x \psi_n^*(x)\psi_m(x)$$
So
1. don't rearrange |n> expressions
2. only numbers <n|m> commute
3. if you ever get to <x|n>, you can substitute with $\psi_n(x)$

enricfemi said:
do you mean :
when it act on one wave function,it denotes
|m><n|a> ?
Yes...

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