How Does Griffiths Derive x Using Creation and Annihilation Operators?

[kuahji]

Find the expectation value of the potential energy in the nth state of the harmonic oscillator.

This is his example 2.5 in the book, he uses a$\pm$=1/Sqrt[2hmw]($\mp$ip+mwx) to get x=Sqrt[h/2mw](a$_{+}$+a$_{-}$)

My question how does he do this? I can't seem to make the algebraic manipulations to get it into that form to follow out his example.

 

[G01]

You have two equations, one for $a_+$ and one for $a_-$.

All you have to do is solve them simultaneously for x and p. i.e. Solve one for x, solve the other for p, plug the first into the second, etc. etc.

It may help to simplify things by defining some constants A and B such that:

$$a_{\pm}=Ap\mp Bx$$

 

[kuahji]

That was easy enough, thank you for the tip!

 

[G01]

No problem!