Consider a quantum mechanical particle moving in an upside-down harmonic oscillator potential. Its wavefunction $\Psi(x, t)$ evolves according to the time-dependent Schrödinger equation,

$$i \hbar \frac{\partial \Psi}{\partial t}=-\frac{\hbar^{2}}{2} \frac{\partial^{2} \Psi}{\partial x^{2}}-\frac{1}{2} x^{2} \Psi$$

(a) Verify that

$$\Psi(x, t)=A(t) e^{-B(t) x^{2}}$$

is a solution of equation (1), provided that

$$\frac{d A}{d t}=-i \hbar A B$$

and

$$\frac{d B}{d t}=-\frac{i}{2 \hbar}-2 i \hbar B^{2}$$

(b) Verify that $B=\frac{1}{2 \hbar} \tan (\phi-i t)$ provides a solution to (3), where $\phi$ is an arbitrary real constant.

(c) The expectation value of an operator $\mathcal{O}$ at time $t$ is

$$\langle\mathcal{O}\rangle(t) \equiv \int_{-\infty}^{\infty} d x \Psi^{*}(x, t) \mathcal{O} \Psi(x, t),$$

where $\Psi(x, t)$ is the normalised wave function. Show that for $\Psi(x, t)$ given by (2),

$$\left\langle x^{2}\right\rangle=\frac{1}{4 \operatorname{Re}(B)}, \quad\left\langle p^{2}\right\rangle=4 \hbar^{2}|B|^{2}\left\langle x^{2}\right\rangle$$

Hence show that as $t \rightarrow \infty$,

$$\left\langle x^{2}\right\rangle \approx\left\langle p^{2}\right\rangle \approx \frac{\hbar}{4 \sin 2 \phi} e^{2 t}$$

[Hint: You may use

$$\frac{\int_{-\infty}^{\infty} d x e^{-C x^{2}} x^{2}}{\int_{-\infty}^{\infty} d x e^{-C x^{2}}}=\frac{1}{2 C}$$