QM solutions to the Schrodinger Equation

[Maybe_Memorie]

Homework Statement

Here's something that's confusing me. Say we have a potential

V(x) = V_o if x < 0, x > a
and
V(x) = 0 if 0 < x < a

(yes I know the notation with greater than/equals etc isn't totally correct, but you know what I'm talking about.)

In the middle section, ψ'' + k²ψ = 0

Whenever I see the solutions it's always ψ = Asin(kx) + Bcos(kx) in the middle section where k² = -2mE/h²

In the right/left sections, ψ'' + f²ψ = 0, with f² = -2m(E - V₀)/h²

The solutions here always seem to be complex exponentials.

Can someone please explain the difference to me in the solutions?

 

[Goddar]

You must know the identities:
exp(±iø) = cosø ± i*sinø
So the two sets of solutions you're referring to are basically equivalent once you take the real part, only the exponential form turns out to be more practical in most cases because easier to manipulate.