Question 705025

The Hamiltonian for a one-dimensional simple harmonic oscillator is given by $H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2$. The harmonic oscillator is in the state $|\psi\rangle = \frac{1}{\sqrt{1+\lambda^2}}(|1\rangle + \lambda e^{i\theta}|2\rangle)$, where $|1\rangle$ and $|2\rangle$ are the normalised first and second excited states of the oscillator and $\lambda, \theta$ are positive real constants. If the expectation value $\langle\psi|x|\psi\rangle = \beta \sqrt{\frac{\hbar}{m\omega}}$, the value of $\beta$ is: