📝 Exercise

📝 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} Ψ (x, y, z, t) = i ℏ \frac{d Ψ (x, y, z, t)}{d t}

Substituting in the specific value of the Hamiltonian,

(- \frac{ℏ^{2}}{2 m} \frac{d^{2}}{d x^{2}} - \frac{ℏ^{2}}{2 m} \frac{d^{2}}{d y^{2}} - \frac{ℏ^{2}}{2 m} \frac{d^{2}}{d z^{2}} + V (x, y, z, t)) Ψ (x, y, z, t) = i ℏ \frac{d Ψ (x, y, z, t)}{d t}