QM Free Particle Approaching Infinite Barrier

[leothelion]

Solved... i think. It's just Psi(x) = B sin kx

Homework Statement

Consider a free particle psi(x) = A*e^(ikx) approaching an infinite barrier from the left:
V = 0, x < 0 and V = oo, x >= 0. For this problem use only the time-independent
Schrödinger Equation.
a. Find the probability of being reflected from the barrier (recall that energy
must be conserved).
b. Find the total wavefunction Psi(x) for x < 0.

Homework Equations

Schroedinger's equation.

The Attempt at a Solution

Okay, solving the wave equation gives the general solution

psi(x, 0) = A e^(ikx) + B e^(-ikx).

We know that Psi(x, 0) = 0 when x >= 0, since E < V = oo for all values of E. Part a asks to find the reflection coefficient using conservation of energy, but I did it this way:

Psi must be continuous at x = 0. Thus A + B = 0, implying that A = -B. Thus the reflection coefficient is 1. This also reduces psi(x, 0) = A sin kx.

However, I'm a little confused as to how to find Psi(x). My original work was that Psi(x) = integral(-oo to oo) phi(k) A sin kx dk

but my professor said that due to the conservation of energy, there are only so many allowed energy states. I'm a little confused about this. Is Psi(x, 0) = A e^(ikx) - A e^(-ikx)?

Any hints would be greatly appreciated. Thanks!

 

[Avodyne]

Solved... i think. It's just Psi(x) = B sin kx

Yep, that's it!