The motion of a particle in one dimension is described by the time-independent hermitian Hamiltonian operator $H$ whose normalized eigenstates $\psi_{n}(x), n=0,1,2, \ldots$, satisfy the Schrödinger equation

$$H \psi_{n}=E_{n} \psi_{n},$$

with $E_{0}<E_{1}<E_{2}<\cdots<E_{n}<\cdots$. Show that

$$\int_{-\infty}^{\infty} \psi_{m}^{*} \psi_{n} d x=\delta_{m n}$$

The particle is in a state represented by the wavefunction $\Psi(x, t)$ which, at time $t=0$, is given by

$$\Psi(x, 0)=\sum_{n=0}^{\infty}\left(\frac{1}{\sqrt{2}}\right)^{n+1} \psi_{n}(x) .$$

Write down an expression for $\Psi(x, t)$ and show that it is normalized to unity.

Derive an expression for the expectation value of the energy for this state and show that it is independent of time.

Calculate the probability that the particle has energy $E_{m}$ for a given integer $m \geqslant 0$, and show that this also is time-independent.