# Solving a Piecewise Schrodinger equation

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1. Jun 7, 2017

### cc94

1. The problem statement, all variables and given/known data
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 $E < V_1$.

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

3. 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?

2. Jun 7, 2017

### DrDu

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

3. Jun 7, 2017

### cc94

Doh, I didn't think hard enough. Thanks