# Finding $\langle \psi_n | x | \psi_m \rangle$ for the harmonic oscillator

I need to prove the following for the quantum harmonic oscillator, whose potential is given by $$V(x) = \frac{1}{2}m\omega^2 x^2$$:

$$\langle x\rangle_{n m}=\left\langle\psi_{n}|x| \psi_{m}\right\rangle=\left\{\begin{array}{cl} \frac{1}{\alpha}\left(\frac{n+1}{2}\right)^{1 / 2} & m=n+1 \\ \frac{1}{\alpha}\left(\frac{n}{2}\right)^{1 / 2} & m=n-1 \\ 0 & \text { otherwise } \end{array}\right.$$

And I think I am on the right path but I am not very sure how to proceed. I know that the wavefunction obtained as a solution for Schrödinger's equation and the mentioned potential is:

$$\psi_n(x) = \sqrt{\frac{\alpha}{\sqrt{\pi}2^n n!}} H_n(\alpha x)e^{-a^2x^2/2}$$

Where $$H_n(x)$$ is the $$n$$-th order Hermite polynomial and $$\alpha = \sqrt{m\omega / \hbar}$$. I know that the following holds for Hermite polynomials: $$H_{n+1} = 2 x H_n - 2 n H_{n-1}$$ And therefore: $$x H_n = \frac{1}{2}H_{n+1} + n H_{n-1}$$

I also know that these are a set of orthogonal polynomials, since:

$$\int_{-\infty}^\infty e^{-x^2}H_n(x) H_m(x) dx = \sqrt{\pi} 2^n n! \delta_{nm}$$ Where $$\delta_{nm}$$ is the Kronecker delta. With these things in mind, this is what I have tried to do: $$\langle x \rangle_{nm} = \int_{-\infty}^\infty \psi_n(x)\cdot x \cdot \psi_m(x) dx$$

Now some constants come out of the integral, and the integral which I am having trouble solving is: $$I = \int_{-\infty}^\infty H_n(\alpha x) H_m(\alpha x) e^{-\alpha^2 x^2} \cdot x \cdot dx$$

And taking $$\mu = \alpha x$$ we get:

$$I = \frac{1}{\alpha^2}\int_{-\infty}^\infty H_n(\mu) H_m(\mu) e^{-\mu^2} \cdot \mu \cdot d\mu$$

It is clear that the orthogonality of the Hermite polynomials is going to be useful somehow, but I don't know how to keep going because the simpler integrals I obtain always diverge or turn out to be more difficult to solve. Any tips would be appreciated.

• Tip: work in Fock space, so you never have to do an integral. Dec 18, 2021 at 19:21
• This is a problem from my introductory class to quantum theory. I have no idea what a Fock space is. Dec 18, 2021 at 19:24
• Have you learned about creation and annihilation operators and their action on the vacuum state? Equivalently, have you learned about the orthonormality of Hermite functions, not Hermite polynomials? Dec 18, 2021 at 19:25
• The operators you mention are addressed briefly in this class. Hermite functions as a general group are not mentioned, sorry. Would it be helpful to go over those? Dec 18, 2021 at 19:29
• here. Dec 18, 2021 at 19:58

You already wrote down everything you need.

First you wrote: $$x H_n = \frac{1}{2}H_{n+1} + n H_{n-1}$$

Then you wrote: $$I = \frac{1}{\alpha^2}\int_{-\infty}^\infty H_n(\mu) H_m(\mu) e^{-\mu^2} \cdot \mu \cdot d\mu$$

So.. now you can substitute $$\mu H_n$$: $$I = \frac{1}{\alpha^2}\int_{-\infty}^\infty e^{-\mu^2} H_m(\mu) \left[\frac{1}{2}H_{n+1}(\mu) + n H_{n-1}(\mu)\right] d\mu$$

Where you can use now something you wrote already: $$\int_{-\infty}^\infty e^{-x^2}H_n(x) H_m(x) dx = \sqrt{\pi} 2^n n! \delta_{nm}$$

Does this make it easier?

• Thanks! This was really helpful. I think I was obfuscated trying to use the tools I already knew I had in a certain way which wasn't yielding very good results. I have been able to prove the required statement. Thanks again! Dec 18, 2021 at 21:34