Use the generating function for the Hermite polynomials to obtain the energy eigenfunction expansion of an initial wave function that has the same form as the oscillator ground state but that is centered at the coordinate $a$ rather than the coordinate origin:
$$
\psi(x, 0)=\left(\frac{m \omega}{\hbar \pi}\right)^{1 / 4} \exp \left(-\frac{m \omega(x-a)^2}{2 \hbar}\right) .
$$
(a) For this initial wave function, calculate the probability $P_n$ that the system is found to be in the $n$-th harmonic oscillator eigenstate, and check that the $P_n$ add up to unity.
(b) Plot $P_n$ for three typical values of $a$, illustrating the case where $a$ is less than, greater than, and equal to $\sqrt{\frac{\hbar}{m \omega}}$.
(c) If the particle moves in the field of the oscillator potential with angular frequency $\omega$ centered at the coordinate origin, again using the generating function derive a closed-form expression for $\psi(x, t)$.
(d) Calculate the probability density $|\psi(x, t)|^2$ and interpret the result.