PX156 - D1 - time independent schrödinger equation

TDSE:

i ℏ \frac{\partial}{\partial t} ψ (x, t) = - \frac{ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} ψ (x, t) + V (x) ψ (x, t)

let $ψ (x, t) = ϕ (x) ρ (t) :$

\begin{aligned} i ℏ \frac{\partial}{\partial t} [ϕ (x) ρ (t)] & = - \frac{ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} [ϕ (x) ρ (t)] + V (x) ϕ (x) ρ (t) \\ i ℏ ϕ (x) \frac{\partial ρ (t)}{\partial t} & = - \frac{ℏ^{2}}{2 m} ρ (t) \frac{\partial^{2} ϕ (x)}{\partial x^{2}} + V (x) ϕ (x) ρ (t) \\ \frac{i ℏ}{ϕ (x)} \frac{\partial ρ (t)}{\partial t} & = - \frac{ℏ^{2}}{2 m ϕ (x)} \frac{\partial^{2} ϕ (x)}{\partial x^{2}} + V (x) = E \end{aligned}

$E$ is constant, hence:

\begin{aligned} \frac{i ℏ}{ρ} \frac{d ρ}{d t} = E & ⟹ i ℏ \frac{d ρ}{d t} = E ρ \\ - \frac{ℏ^{2}}{2 m ϕ} \frac{d^{2} ϕ}{d x^{2}} + V (x) = E & ⟹ - \frac{ℏ^{2}}{2 m} \frac{d^{2} ϕ}{d x^{2}} + V ϕ = E ϕ \end{aligned}

time independent schrödinger equation (TISE):

- \frac{ℏ^{2}}{2 m} \frac{d^{2} ϕ}{d x^{2}} + V ϕ = E ϕ

for $i ℏ \frac{d ρ}{d t} = E ρ$ , the solution is:

ρ (t) = \exp (- \frac{i E t}{ℏ})

solution of the TISE depends on the precise form of $V (x)$ , but the probability density is given by:

| ψ |^{2} = ϕ ϕ^{*} ρ ρ^{*} = | ϕ |^{2}

hence the probability density is independent of time as $ρ$ is essentially a phase term
solutions of this type are called stationary states
energy doesn't change with time
all components of the wave function oscillate with the same well-defined fixed frequency, therefore energy, $E = ℏ ω$ , is constant