Solving 1D Quantum Mechanics Homework for Square Well w/ Infinite Wall

[loobloke]

Homework Statement

Given a square well,
Infinite wall at x=0
Wall height U for x>L

For E<U, find solutions to the schrondinger equation inside the well, and beyond x>L which satisy boundary conditions for x=0 and x=\infty

Taking conditions at x=L, find the allowable energies of the system.

Homework Equations

Schrondinger equation

The Attempt at a Solution

Know U=0 inside the well, 0<x<L.

Conditions we get are, (I can't find the wavefunction symbol, Y looks the closest.)
x is continuous at 0, hence Y(0)=0

What condition do I need at x=L? Or am I completely missing the plot.

Do I need to do another condition at x=\infty?
Ie, there are two solutions? One inside the well and one outside?

 

[vela]

Yes, there are two solutions. You need to solve the SE in both regions and then join the solutions smoothly at x=L, i.e., ψ(L^-)=ψ(L⁺) and ψ'(L^-)=ψ'(L⁺).