- #1

cc94

- 19

- 2

## Homework Statement

I was trying to solve the time-independent Schrodinger'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

Schrodinger'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. Schrodinger'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?