Consider a quantum system with Hamiltonian $\widehat{H}$and energy levels

$$E_{0}<E_{1}<E_{2}<\ldots$$

For any state $|\psi\rangle$ define the Rayleigh-Ritz quotient $R[\psi]$ and show the following:

(i) the ground state energy $E_{0}$ is the minimum value of $R[\psi]$;

(ii) all energy eigenstates are stationary points of $R[\psi]$ with respect to variations of $|\psi\rangle$.

Under what conditions can the value of $R\left[\psi_{\alpha}\right]$ for a trial wavefunction $\psi_{\alpha}$ (depending on some parameter $\alpha$ ) be used as an estimate of the energy $E_{1}$ of the first excited state? Explain your answer.

For a suitably chosen trial wavefunction which is the product of a polynomial and a Gaussian, use the Rayleigh-Ritz quotient to estimate $E_{1}$ for a particle of mass $m$ moving in a potential $V(x)=g|x|$, where $g$ is a constant.

[You may use the integral formulae,

$$\begin{aligned} \int_{0}^{\infty} x^{2 n} \exp \left(-p x^{2}\right) d x &=\frac{(2 n-1) ! !}{2(2 p)^{n}} \sqrt{\frac{\pi}{p}} \\ \int_{0}^{\infty} x^{2 n+1} \exp \left(-p x^{2}\right) d x &=\frac{n !}{2 p^{n+1}} \end{aligned}$$

where $n$ is a non-negative integer and $p$ is a constant. ]