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Unbound particle through potential

  1. Oct 22, 2013 #1
    1. The problem statement, all variables and given/known data
    A particle with mass m moving in the positive x -direction (i.e. from left to right) is incident on a potential step of height V0 at x = 0 so that the potential experienced by the particle is;
    V(x) = 0 for x < 0 and V(x) = V0 for x ≥ 0
    2. Relevant equations

    Determine the time-independent wave function for the particle in the case where the particle energy, E, is greater than V0. This case corresponds to the solution for an ‘unbound’ particle (E > V0). Write your wave functions using complex notation; let the amplitudes of the incident, reflected and transmitted waves be CI, CR and
    CT respectively. Define the wavenumber, k, in the region x < 0 and the wavenumber
    k' in the region x ≥ 0 .

    3. The attempt at a solution
    ψ(x) = CI*eikx + CR*e-ikx for x < 0 (is probably the first part of the equation).

    My main problem is what to do with the second one, as the particle is constantly 'under the influence' of the potential V0 and at the same time I have to find CT when the wave has not been exactly transmitted so the equation can't just be CT*eikx(in my point of view)
     
  2. jcsd
  3. Oct 22, 2013 #2

    hilbert2

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    Gold Member

    The solution on the left side of the step is ##\psi(x)=Aexp\left(\frac{i\sqrt{2mE}x}{\hbar}\right)+ Bexp\left(-\frac{i\sqrt{2mE}x}{\hbar}\right)## and the solution on the right side of the step is ##\psi(x)=Cexp\left(\frac{i\sqrt{2m(E-V_{0})}x}{\hbar}\right)+ Dexp\left(-\frac{i\sqrt{2m(E-V_{0})}x}{\hbar}\right)##. Start by determining the conditions that the constants ##A,B,C,D## must satisfy so that ##\psi## and its first derivative are continuous at ##x=0##.
     
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