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Homework Help: Quantum particle reflection from a potential drop

  1. Jul 23, 2008 #1
    1. The problem statement, all variables and given/known data
    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?


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


    3. 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
     
  2. jcsd
  3. Jul 23, 2008 #2
    Ignore above post. I posted too soon. Here's my real question.

    1. The problem statement, all variables and given/known data
    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?


    2. Relevant 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]


    3. 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?
     
  4. Jul 23, 2008 #3

    kreil

    User Avatar
    Gold Member

    There should be absolute value brackets around the B/A equation, which is why alpha being greater than k doesnt produce a negative probability.
     
  5. Jul 23, 2008 #4
    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?
     
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