C. Cohen-Tannoudji, B. Diu, F. Laloe, Quanten mechanics 1 & 2

Stationary Solution
assumption: H time independent ⇒ separation into
time dependent and spatially dependent part:
ψ (x, t) =f (t) ψ (x)
(strange notation)
Plugging into Schrödinger eqn.
⇒
i
i ∂ ψ (x, t) =Hψ (x, t)
∂t
∂
∂t f (t) ψ (x) =f (t) Hψ (x)
⇒
i ∂
f (t) ∂t f (t)
depends only on t
= 1 Hψ (x)
ψ (x)
depends only on x
= E (constant)
⇒ i ∂ f (t) =Ef (t) ⇒ f (t) =e−iEt/
∂t
and
Hψ (x) =Eψ (x, t)
time independent Schrödinger equation
ψ (x, t) =exp (−iEt/) ψ (x) arecalledstationarystates,
since |ψ (x, t)| 2 = |ψ (x)| 2 is time independent.