What's the difference between them

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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).
  • #1
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<n|m> ;


|m><n| ;
 
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  • #2
<n|m> is a number.
|m><n| is an operator that would act on one wave function to give a different one.
 
  • #3
appreciate it!

i know the product.
but what the operator is?
is it pesi(m)pesi(n)*?
 
  • #4
do you mean :
when it act on one wave function,it denotes
|m><n|a> ?
 
  • #5
Think of it as *all* 1xN matrices
[tex]
|n>=A=\begin{pmatrix}
a\\b\\c\\\ldots
\end{pmatrix}
[/tex]
[tex]
<n|=A^\dag=\begin{pmatrix}
a^* & b^* & c^* & \ldots
\end{pmatrix}
[/tex]
[tex]
\psi_n(x)=<x|n>
[/tex]
Then you see that [itex]|m><n|[/itex] is a matrix ("sort of operator"). [itex]<m|n>[/itex] is a number (like the vector scalar product). Note that [itex]<n|m>\neq \psi_n^*(x)\psi_m(x)[/itex], however since [itex]\sum_x |x><x|=1[/itex]
[tex]<n|m>=\sum_x <n|x><x|m>=\sum_x \psi_n^*(x)\psi_m(x)[/tex]
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 [itex]\psi_n(x)[/itex]
 
  • #6
enricfemi said:
do you mean :
when it act on one wave function,it denotes
|m><n|a> ?
Yes...
 

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