Consider a system with a real Hamiltonian that occupies a state having a real wave function both at time $t=0$ and at a later time $t=t_{1}$. Thus, we have
$$\psi^{}(x, 0)=\psi(x, 0), \quad \psi^{}\left(x, t_{1}\right)=\psi\left(x, t_{1}\right)$$
Show that the system is periodic, namely, that there exists a time $T$ for which
$$\psi(x, t)=\psi(x, t+T)$$
In addition, show that for such a system the eigenvalues of the energy have to be integer multiples of $2 \pi \hbar / T$.

Solution:、

If we consider the complex conjugate of the evolution equation of the wave function for time $t_{1}$, we get
$$\psi\left(x, t_{1}\right)=e^{-i t_{1} H / \hbar} \psi(x, 0) \quad \Longrightarrow \quad \psi\left(x, t_{1}\right)=e^{i t_{1} H / \hbar} \psi(x, 0)$$
$$\psi(x, 0)=e^{i t_{1} H / \hbar} \psi\left(x, t_{1}\right)=e^{2 i t_{1} H / \hbar} \psi(x, 0)$$
Also, owing to reality,
$$\psi(x, 0)=e^{-2 i t_{1} H / \hbar} \psi(x, 0)$$
Thus, for any time $t$ we can write
$$\psi(x, t)=e^{-i t H / \hbar} \psi(x, 0)=e^{-i t H / \hbar} e^{-2 i t_{1} H / \hbar} \psi(x, 0)=\psi\left(x, t+2 t_{1}\right)$$
It is, therefore, clear that the system is periodic with period $T=2 t_{1}$.

Expanding the wave function in energy eigenstates, we obtain
$$\psi(x, t)=\sum_{n} C_{n} e^{-i E_{n} t / \hbar} \psi_{n}(x)$$
The periodicity of the system immediately implies that the exponentials $\exp \left(-i T E_{n} / \hbar\right)$ must be equal to unity. This is only possible if the eigenvalues $E_{n}$ are integer multiples of $2 \pi \hbar / T$.

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