1D problems
Time-independent Schroedinger equation:
(-ħ2/(2m))∂2ψ(x)/∂x2 + U(x)ψ(x) - Eψ(x) = 0
∂2ψ(x)/∂x2 + k(x)2ψ(x) = 0,
with
k(x)2 = 2m(E - U(x))/ħ2
The solutions are stationary states.
Infinite square well: k2 is independent of x inside the well, ψ(x) is zero outside the well.
∂2ψ(x)/∂x2 + k2ψ(x) = 0
Solutions: ψn(x) = (2/L)1/2sin(nπx/L), En = n2π2ħ2/(2mL2).
What do the wave functions look like?
How many points of zero probability?
How many points of maximum probability?