- #1

- 555

- 0

## Homework Statement

A flow of particles collides with a potential step. The potential is zero for x < 0 and a certain value [tex]V_0[/tex] for x > 0.

The flow of particles comes in from the left (x<0 area).

One of the questions is to give the general form of the wavefunction in the area x > 0 and to find an expression for the wavenumber k.

## Homework Equations

[tex]-\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V_0 \psi = E \psi[/tex]

(For x > 0)

## The Attempt at a Solution

I first write the schrodinger eq. into a more general form:

[tex]-\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + (V_0 - E) \psi = 0[/tex]

[tex]- \frac{d^2 \psi}{dx^2} + \frac{2m (V_0-E)}{\hbar^2} \psi = 0[/tex]

[tex]\frac{d^2 \psi}{dx^2} - \frac{2m (V_0-E)}{\hbar^2} \psi = 0[/tex]

[tex]\frac{d^2 \psi}{dx^2} - k^2 \psi = 0[/tex]

So

[tex]k^2 = \frac{2m (V_0-E)}{\hbar^2}[/tex]

And the solution are complex exponents due to the minus-sign (they would be real exponents if it was a plus sign):

[tex]\psi (x) = A e^{ikx} + B e^{-ikx}[/tex]

Now, when I look at the answer, I noticed that they also use complex exponents, but their k was different:

[tex]k^2 = \frac{2m (E-V_0)}{\hbar^2}[/tex]

(E-V instead of V-E)

This is a pretty important error because:

In my case: If E > V, the particle is 'above the potential step' and thus it should be a complex exp. (sin/cos). But, if E > V then k becomes complex, and the exponents become

*real*!

If E < V, the particle is 'inside the potential step' and thus it should be a real exponent (as usual with penetration in classically forbidden area). But, if E < V then k is real and the exponents stay complex!...

Obviously I made an error with a minus sign somewhere, but I honestly can't find it..!?

I also tried switching the minus sign for a plus sign (and swapping the E and V again obviously) to get:

[tex]\frac{d^2 \psi}{dx^2} + \frac{2m (E-V_0)}{\hbar^2} \psi = 0[/tex]

So now, my k is in accordance with the answer (E-V), but because of the + I get REAL exponents, and if E > V, k is real and the exponents stay real (wrong), and if E < V, k is complex and the exponents become complex (wrong)...???

I can't figure it out...