Schrödinger Equation

Hψ(r,t)=[−ℏ22mΔ+U(r)]ψ(r,t)=iℏ∂∂tψ(r,t)H\bm\psi(\bm r, t) = \left[-\frac{\hbar^2}{2m}\Delta + \mathcal{U}(\bm r)\right]\bm\psi(\bm r, t) = i\hbar \frac\partial{\partial t}\bm\psi(\bm r, t)

Where

HH

is a hamiltonian operator. If

HH

is time independent separation of variables gives:

ψ(r,t)=Φ(r)⋅e−iℏEt\bm\psi(\bm r, t) = \bm\Phi(\bm r)\cdot e^{-\frac i\hbar Et}
[−ℏ22mΔ+U(r)]Φ(r)=EΦ(r)\left[-\frac{\hbar^2}{2m}\Delta + \mathcal{U}(\bm r)\right]\bm\Phi(\bm r) = E\bm\Phi(\bm r)

The general time dependent solution is:

ψ(r,t)=∑nan⋅Φ(r)e−iℏEt\bm\psi(\bm r, t) = \sum_n a_n\cdot\bm\Phi(\bm r)e^{-\frac i\hbar Et}

Where

ana_n

are found through the boundary conditions (

t=0t = 0

):

an=∫Φn∗(r)⋅ψ(r,t=0) d3ra_n = \int\bm\Phi_n*(\bm r) \cdot\bm\psi(\bm r, t=0) \: d^3r