In almost every introductory QM book they treat the QM harmonic oscillator. As a result, one finds out that the eigenfunctions in position space are given in terms of Hermite polynomials (in natural units)
$\psi_n(x) = \frac{1}{\pi^{\frac{1}{4}}\sqrt{2^n (n!)}} e^{-\frac{x^2}{2}} H_n(x)$.
In 'algebraic' derivations one uses $|n\rangle$ to denote the $n$-th eigenstate $\psi_n$, consequently $\langle x|n\rangle = \psi_n(x)$ is the representation of the wave function in the position space. Therefore, we have
$\langle x| n\rangle= \frac{1}{\pi^{\frac{1}{4}}\sqrt{2^n (n!)}} e^{-\frac{x^2}{2}} H_n(x)$.
I tried to find a similar representation for the wave function in position space, $\langle p | n \rangle$ but couldn't find anything in my QM literature. I know that I can apply a Fourier transform to obtain the representation in position space (which I did for the ground state). This is quite tedious and becomes infeasible once going to higher $n$.
For symmetry reasons, I would expect some quite similar expression as for $\langle x | n \rangle$, maybe with some additional complex phase ($i$?), coming from the rotation in phase space.
Does anyone know a reference, where they give an expression (and or derivation) for $\langle p |n \rangle$ explicitly?
Or is it an easy calculation and I have overseen something, making life too complicated?
Thank you for your help!
Best Answer
With $$ \varphi_n(x)=\frac 1{\sqrt{2^nn! \sqrt{ \pi}}}H_n(x)e^{-x^2/2} $$ being the normalized oscillator wavefunctions, and defining the Fourier transform by
$$ {\mathcal F}[f](p) =\frac 1{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{ipx} f(x)dx $$ you have $$ {\mathcal F}[\varphi_n](p)= i^n \varphi_n(p). $$
This is most easily derived from the generating function for Hermite polynomials:
We evaluate the Fourier transform of the generating function
$$ \exp\left\{2xt -t^2 -\frac 12 x^2\right\}= \sum_{n=0}^{\infty} \frac {t^n}{n!} H_n(x)e^{-x^2/2} $$ to get
$$ \sum_{n=1}^\infty \frac {t^n}{n!} \int_{-\infty}^\infty e^{ipx} H_n(x) e^{-x^2/2}\,dx =\frac 1 {\sqrt{2\pi}} \int_{-\infty}^\infty e^{ipx} e^{2xt -t^2 -x^2/2}\, dx\nonumber\\ = \exp\left \{-t^2 + \frac 12 (2t+ip)^2\right\} \nonumber\\ = \exp\left\{t^2 +2itp - \frac 12 s^2\right\}\nonumber\\ = \exp\left\{-(it)^2 +2itp - \frac 12 s^2\right\}\nonumber\\ = \sum_{n=0}^{\infty} \frac {t^n}{n!} i^n H_n(p)e^{-p^2/2}. $$
Identifying coefficients of $t^n$ then gives the desired result.