Quantum particle reflection from a potential drop

[SonOfOle]

Homework Statement

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

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

Homework Equations

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

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 [tex]m[/tex] and energy [tex]E[/tex] approaches a potential drop from the [tex]-x[/tex] region, where the potential is described by:
[tex]V(x)=\left\{\stackrel{0 \textrm{if} x\leq 0}{-V_0 \textrm{if} x> 0}[/tex].

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

Homework Equations

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

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

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

Continuity Equations: [tex] \Psi_A + \Psi_B = \Psi_C [/tex] and [tex] \partial_x \Psi_A + \partial_x \Psi_B = \partial_x \Psi_C [/tex]

The Attempt at a Solution

Plug in [tex] \Psi_A[/tex], [tex] \Psi_B[/tex], and [tex] \Psi_C[/tex] into the continuity equations and get these two equations:

[tex]A + B = C [/tex]
[tex]i A k - i B k = i C \alpha [/tex]

Plug the first into the second, and get
[tex] \frac{B}{A} = \frac{k -\alpha}{k + \alpha} [/tex]

which is the probability of reflection.

Now, the math makes sense, but it doesn't make sense overall because if [tex]\alpha [/tex] is greater than [tex]k[/tex] 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 [tex]V_0[/tex] lead to large [tex]\alpha[/tex], and thus for large [tex]V_0[/tex], |B|/|A| --> 1?