How do I compute this matrix element $$\langle 1|\delta(\hat x-b)|1\rangle$$ that models a 1-D harmonic oscillator?
I have done the same for the ground state (by seting delta function as the Fourier integral, Sakurai exercise), $\langle 0|e^{ik\bf x}|0\rangle=exp[-k^2\langle 0|x^2|0\rangle/2]$
using the BH identity, how to continue in the first excited state?
I don't know how to calculate $\langle 1|e^{ik\bf x}|1\rangle$ does anyone have an idea?
I know how to prove that $\langle n|e^{ik\bf x}|0\rangle=\left(\frac{ik}{\sqrt{n!}}\right)^n\left(\sqrt{\frac{\hbar}{2m\omega}}\right)^ne^{-k^2/2\langle 0|x^2|0\rangle}$ is it right to say $\langle 1|e^{ik\bf x}|0\rangle\langle 0|e^{ik\bf x}|1\rangle =\langle 1|e^{ik\bf x}|1\rangle$ and use the identity but conjugate?