While we introduced the variational method and the power method both as a way to approximate the ground state of some system, they both can be used to approximate excited states as well, with appropriate modification. For concreteness, in this problem we will attempt to use the variational method to identify the first excited state of the infinite square well, extending what we did in Example 10.2. As in that example, we will take the width of the well to be \(a=1\).

(a) A fundamental property of the first excited state \((n=2)\) with respect to the ground state \((n=1)\) of the infinite square well is that they are orthogonal. So, our ansatz for the first excited state should be orthogonal to our ansatz of the ground state, Eq. (10.49), over the well. With this in mind, let's make the ansatz that the first excited-state wavefunction is

\[\begin{equation*}\psi_{2}(x ; \beta)=N x^{\beta}(1-x)^{\beta}(1-2 x) \text {, } \tag{10.117}\end{equation*}\]

for some parameter \(\beta\) and normalization constant \(N\). Show that this is indeed orthogonal to the ansatz we used for the ground state in Eq. (10.49).

(b) Determine the normalization constant \(N\) such that \(\psi_{2}(x ; \beta)\) is normalized over the well. You will likely need to use the definition of the Beta function from Eq. (10.51). Also, the Gamma function satisfies the identities \[\begin{equation*}\Gamma(x+1)=x \Gamma(x), \quad \Gamma(1)=1 \tag{10.118}\end{equation*}\]

(c) Now, determine the expectation value of the infinite square well's Hamiltonian on this ansatz wavefunction, as a function of the parameter \(\beta\). You should find \[\begin{equation*}\left\langle\psi_{2}|\hat{H}| \psi_{2}\rightangle=\frac{\hbar^{2}}{2 m} \frac{6 \beta(3+4 \beta)}{2 \beta-1} . \tag{10.119}\end{equation*}\]

(d) Now, minimize over \(\beta\) to establish an estimate for the energy of the first excited state of the infinite square well. How does this compare to the exact result?

(e) Can you extend this procedure to the second excited state? What do you find in that case for an estimate of its energy?