1. The problem statement, all variables and given/known data

This isn’t exactly a homework question, but I figured this would be the best subforum for this sort of thing. For the sake of a concrete example, let’s just say my question is:

Express the position operator’s eigenstates in terms of the number operator’s eigenstates.

2. Relevant equations

The number operator is given by
$$\hat{N} = \hat{a}^\dagger\hat{a}.$$
It has eigenvalues and eigenstates
$$\hat{N}|n\rangle = n|n\rangle.$$
The position operator is given by
$$\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}\Big(\hat{a}^\dagger+\hat{a}\Big) \equiv \gamma\Big(\hat{a}^\dagger+\hat{a}\Big).$$
The action of the position operator on one of the number eigenstates is
$$\hat{x}|n\rangle = \gamma\Big(\hat{a}^\dagger|n\rangle + \hat{a}|n\rangle\Big) =\gamma\Big(\sqrt{n+1}|n+1\rangle + \sqrt{n}|n-1\rangle\Big).$$

3. The attempt at a solution

We’d like to find eigenstates of $\hat x$, that is, states $|x\rangle$ satisfying
$$\hat x|x\rangle = x|x\rangle.$$
The number basis is complete, so whatever these states we’re looking for might be, they are representable as a linear combination of the number states:
$$|x\rangle = \sum_n |n\rangle\langle n|x\rangle;$$
I just need to know the coefficients $C_n \equiv \langle n|x\rangle$.

If I look again at the action of $\hat x$,
$$\hat x|x\rangle = \sum_n \hat x|n\rangle\langle n|x\rangle \\ \qquad = \sum_n \gamma\Big(\sqrt{n+1}|n+1\rangle + \sqrt{n}|n-1\rangle\Big) \langle n|x\rangle.$$
Suppose I attack this equation from the left with a particular $\langle n’|$. Then I get
$$x\langle n’|x\rangle = \sum_n \gamma\Big(\sqrt{n+1}\langle n’|n+1\rangle + \sqrt{n}\langle n’|n-1\rangle\Big) \langle n|x\rangle \\ \qquad = \gamma\Big(\sqrt{n’}\langle n’-1|x\rangle + \sqrt{n’+1}\langle n’+1|x\rangle\Big),$$
which allows me to develop a recurrence relationship
$$C_{n+1} = \frac{x}{\gamma\sqrt{n+1}}C_n-\frac{\sqrt{n}}{\sqrt{n+1}}C_{n-1}.$$

But I don’t understand what this says. What is that $x$ doing in there? What does that even mean? And can I use this recurrence relation to get a closed-form answer? (At the very least I’d need to calculate $C_0$ and $C_1$ explicitly, which I don’t know how to do.)

Also, I think at some point the position-space wavefunctions ought to come into it:
$$\langle x’|n\rangle = \sqrt[4]{\frac{m\omega}{\pi\hbar}}\frac{1}{\sqrt{2^nn!}}H_n\Big(\sqrt{\frac{1}{ \sqrt{2}\gamma} x}\Big)e^{-x^2/2\gamma^2}$$
and
$$\langle x’|x\rangle = \delta(x-x’).$$

http://ift.tt/1bNgoNy