# Bra-ket multiplication

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

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Pengwuino
Gold Member
If it is true, it's simply a matter of doing the work.

$$(a^ - + a^ + )^2 = (a^ - a^ - + a^ - a^ + + a^ + a^ - + a^ + a^ + )$$ for example, so you do the case for the 4th power, you can work out what the result is knowing
$$\[ \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}$$

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