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Homework Help: Quantum Mechanics: Tunneling, Help!

  1. Feb 22, 2010 #1
    I have no idea how to solve these problems.
    Picture is attached.

    An electron with energy E is incident on a barrier of height U0 and total width L. It enters the barrier at x = 0, from the left.

    1) For an electron with energy E = 6 eV and a barrier of height U0 = 12 eV, at what penetration depth into the barrier will the probability density fall to 1/4 of its value at the surface x = 0 ? Assume the barrier width L is sufficiently large that the in-barrier wavefunction is well approximated by a single exponential, i.e., negligible reflections from the x = L interface. [For an electron, mec2 = 0.511 MeV.]

    0.055 nm
    0.110 nm
    0.282 nm
    0.159 nm
    0.319 nm
    2) What is the probability density for this electron at the depth 3 times as large as the answer to the previous question, compared to its value at x = 0 ? Assume that the electron is still inside the barrier at this depth.

    (1/4) 2
    (1/4) 3

    5) For the same barrier width L, if the height of the barrier and the energy of the electron are both reduced by a factor of 4, U1 = 0.25 U0 and E1 = 0.25 E, what is the transmission coefficient T1?

    T1 = 10-8/4= 0.25 x 10-8
    T1 = 10-8
    T1 = 4 x 10-8
    T1 = 2 x 10-4
    The problem does not provide enough information to answer this question.

    Attached Files:

    Last edited: Feb 22, 2010
  2. jcsd
  3. Feb 23, 2010 #2
    The general approach is to solve Schrodinger equation for 3 regions:
    x < 0;
    0 < x < L;
    x > L;
    and then build the solution for whole space, using wave function and it's derivative continuity. This allows to answer 5).

    For large L this procedure gives:

    \left|\frac{\psi(x)}{\psi(0)}\right|^2 = \exp(-2\alpha x).

    That's enough to answer the question 2). For 1) you need to know [tex] \alpha [/tex].

    The procedure described above is not convenient for complex barriers. The transfer-matrix method is more effective and gives the transmission coefficient explicitly.
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