Bra-ket multiplication

  • #1
hey guys,

<n|(a-+a+)^4|n> = 5n2+5n+3

I am wondering if some one can show me why left and right side are equal

    Thanks in advance
 

Answers and Replies

  • #2
Pengwuino
Gold Member
4,989
16
If it is true, it's simply a matter of doing the work.

[tex](a^ - + a^ + )^2 = (a^ - a^ - + a^ - a^ + + a^ + a^ - + a^ + a^ + )[/tex] for example, so you do the case for the 4th power, you can work out what the result is knowing
[tex]\[
\begin{array}{l}
a^ + \left. {|n} \right\rangle = \sqrt {n + 1} \left. {|n + 1} \right\rangle \\
a^ - \left. {|n} \right\rangle = \sqrt n \left. {|n - 1} \right\rangle \\
\end{array}[/tex]

and any term that doesn't return the ket to |n> can be ignored since it'll go away.
 

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