- #1

- 11

- 0

<n|(a

^{-}+a

^{+})^4|n> = 5n

^{2}+5n+3

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

Thanks in advance

- Thread starter crystalplane
- Start date

- #1

- 11

- 0

<n|(a

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

Thanks in advance

- #2

Pengwuino

Gold Member

- 4,989

- 16

[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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