Solving Bra-ket Multiplication: <n|(a-+a+)^4|n> = 5n2+5n+3

[crystalplane]

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

 

[Pengwuino]

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
\[<br /> \begin{array}{l}<br /> a^ + \left. {|n} \right\rangle = \sqrt {n + 1} \left. {|n + 1} \right\rangle \\ <br /> a^ - \left. {|n} \right\rangle = \sqrt n \left. {|n - 1} \right\rangle \\ <br /> \end{array}

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