Quantum particle reflection from a potential drop

[SonOfOle]

Homework Statement

A quantum mechanical particle with mass $$m$$ and energy $$E$$ approaches a potential drop from the $$-x$$ region, where the potential is described by:
$$V(x)=\left\{\stackrel{0 textrm{if} x\leq 0}{-V_0 textrm{if} x> 0}$$.

What is the probability it will be reflected by the potential?

Homework Equations

Incident Wave: $$\Psi (x,t) = A e^{k x - \omega t} \textrm{where} k= \sqrt{2 m D} /2$$

The Attempt at a Solution

I want to say 0, but that's without doing the math on it. The continuity equations yield 3 unknowns (A, B, C

 

[SonOfOle]

Ignore above post. I posted too soon. Here's my real question.

Homework Statement

A quantum mechanical particle with mass $$m$$ and energy $$E$$ approaches a potential drop from the $$-x$$ region, where the potential is described by:
$$V(x)=\left\{\stackrel{0 \textrm{if} x\leq 0}{-V_0 \textrm{if} x> 0}$$.

What is the probability it will be reflected by the potential?

Homework Equations

Incident Wave: $$\Psi (x,t) = A e^{k x - \omega t} \textrm{where} k= \sqrt{2 m E} / \hbar$$

Reflected Wave: $$\Psi (x,t) = B e^{-k x - \omega t} \textrm{where} k= \sqrt{2 m E} / \hbar$$

Transmitted Wave: $$\Psi (x,t) = C e^{k x - \alpha t} \textrm{where} \alpha = \sqrt{2 m (E+V_0)} / \hbar$$

Continuity Equations: $$\Psi_A + \Psi_B = \Psi_C$$ and $$\partial_x \Psi_A + \partial_x \Psi_B = \partial_x \Psi_C$$

The Attempt at a Solution

Plug in $$\Psi_A$$, $$\Psi_B$$, and $$\Psi_C$$ into the continuity equations and get these two equations:

$$A + B = C$$
$$i A k - i B k = i C \alpha$$

Plug the first into the second, and get
$$\frac{B}{A} = \frac{k -\alpha}{k + \alpha}$$

which is the probability of reflection.

Now, the math makes sense, but it doesn't make sense overall because if $$\alpha$$ is greater than $$k$$ then the probability is negative. Also, reflecting from a drop in potential doesn't make sense intuitively... but that may just be QM.

Any ideas?

 

[kreil]

There should be absolute value brackets around the B/A equation, which is why alpha being greater than k doesn't produce a negative probability.

 

[SonOfOle]

Hmm... Okay. Even still, wouldn't large $$V_0$$ lead to large $$\alpha$$, and thus for large $$V_0$$, |B|/|A| --> 1?