The Hamiltonian for a one dimensional simple harmonic oscillator is given by \(H=\frac{p^2}{2 m}+\frac{1}{2} m \omega^2 x^2\). The harmonic oscillator is in the state \(|\psi〉=\frac{1}{\sqrt{1+\lambda^2}}\left(|1〉+\lambda e^{i \vartheta}|2〉\right)\), where |1〉 and |2〉 are the normalised first and second excited states of the oscillator and \(\lambda, \vartheta\) are positive real constants. If the expectation value \(\langle\psi| x|\psi〉=β \sqrt{\frac{\hbar}{m \omega}}\), the value of β is
1
\(\frac{1}{\sqrt{2}\left(1+\lambda^2\right)}\)
2
\(\frac{\sqrt{2} \lambda \cos \vartheta}{1+\lambda^2}\)
3
\(\frac{2 \lambda \cos \vartheta}{1+\lambda^2}\)
4
\(\frac{\lambda^2 \cos \vartheta}{1+\lambda^2}\)