📝 Exercise

Write the Time-Dependent Schrödinger equation for one particle, in three dimensions, confined by the time-dependent potential \(V(x,y,z,t)\).

The time-dependent Schrödinger equation is:

\[ \hat{H} \Psi(x,y,z,t) = i \hbar \frac{d \Psi(x,y,z,t)}{dt} \]

Substituting in the specific value of the Hamiltonian,

\[ \left(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}-\frac{\hbar^2}{2m}\frac{d^2}{dy^2}-\frac{\hbar^2}{2m}\frac{d^2}{dz^2} + V(x,y,z,t) \right) \Psi(x,y,z,t) = i \hbar \frac{d \Psi(x,y,z,t)}{dt} \]