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Solving Schroedinger Equation for a Step Potential

  1. Mar 13, 2013 #1
    Undergraduate Quantum Mechanics problem. However the course hasn't gone as far to include R or T so I'm assuming there must be a way to solve this without needing to know about those.
    1. The problem statement, all variables and given/known data

    Asked to show that [tex]\psi(x)=A\sin(kx-\phi_0)[/tex] is a solution to the 1D-time-independent Schroedinger Equation for [itex]x<0[/itex].
    Then asked to show that the general solution for [itex]x>0[/itex] is [tex]\psi(x)=Be^{{-x}/{\eta}}+Ce^{{x}/{\eta}}[/tex].
    Question then is, by considering how the wave function must behave at [itex]x=0[/itex], show that [tex]\phi_0=arctan(\eta k)[/tex]

    2. Relevant equations

    [tex]\psi(x)=A\sin(kx-\phi_0)\quad x<0[/tex]
    [tex]\psi(x)=Be^{{-x}/{\eta}}+Ce^{{x}/{\eta}}\quad x>0[/tex]
    [tex]\phi_0=arctan(\eta k)\quad x=0[/tex]

    3. The attempt at a solution
    I've tried various ways, attempting to put the first solution into exponential form, then attempting to put the second solution into trigonometric form. Neither of these seem to give the desired result, I just end up with [itex]i[/itex] everywhere. I also can't see how you get [itex]\eta k[/itex] out.
    Feel like I'm missing a step or something in order to be able to solve this, any help would be greatly appreciated.
     
  2. jcsd
  3. Mar 13, 2013 #2

    DrClaude

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    Staff: Mentor

    So how does the wave function behave at [itex]x=0[/itex]?
     
  4. Mar 13, 2013 #3

    vela

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    And as x goes to +∞?
     
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