Solving a Piecewise Schrodinger equation

[cc94]

Homework Statement

I was trying to solve the time-independent Schrödinger's equation for this well: http://i.imgur.com/C9QrvkX.png
First I tried to look at cases where the energy of a particle is $E < V_1$.

Homework Equations

Schrödinger's equation:
$$-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi(x) - V(x)\psi(x) = E\psi(x)$$

The Attempt at a Solution

I thought the normal way to deal with these square barriers is to break the equation up piece-wise and then match the conditions at the boundary. But in region III, there's a problem. Schrödinger's equation requires an exponential solution because $V_1>E$. The wave function is

$$\psi_{III} (x) = Fe^{-\mu x} + Ge^{\mu x},$$

where $\mu = \frac{\sqrt{2m(V_1-E)}}{\hbar}$. But the boundary condition requires that at $x = a_3$, $\psi_{III} (a_3) = 0$. This isn't possible unless either F = G = 0, or the wave function is actually a sin/cos wave function here. What am I missing?

 

[DrDu]

But the boundary condition requires that at $x = a_3$, $\psi_{III} (a_3) = 0$. This isn't possible unless either F = G = 0, or the wave function is actually a sin/cos wave function here. What am I missing?

Are you sure? How about $F=\exp \mu a_3$ and $G=-\exp -\mu a_3$?

 

[cc94]

Doh, I didn't think hard enough. Thanks