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kamla alsulimi

Determine the tunneling coefficient as a function of energy

Determine the tunneling coefficient as a function of energy for the following V V(x)= 0 ,|x|

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    Daniel Lee Whitenack

    Hello,

    You can do this problem analytically without having to do approximations. Then you can visualize your results in Matlab or Mathematica or whatever works for you. Basically, you will have a piecewise wavefunction for your different regions. This will have the form:

    psi = a exp(k x) + b exp(-k x)

    where k = sqrt(2(Vo – E)) in the presence of the constant potential and sqrt(-2 E) in the region of no potential. Note, if E is postion, the scattering wavefunction we be of the form:

    psi = a exp(i k x) + b exp(-i k x)

    where k = sqrt(2 E).

    Then use your boundary conditions and normalization to solve for all the coefficients. For example, you must enforce continuity and smoothness at the boundary (+- a).

    After you have your coefficients. The transmission coefficient is just the square modulus of the outgoing coefficient over the incoming coefficient. In other words, the probability of transmission.

    The is covered very well is Griffiths “Introduction to Quantum Mechanics.” He is more in depth than I can be here.

    Best, Daniel

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    Daniel Lee Whitenack

    Hello,

    Sorry for the spelling error below. It should say, “if the energy is POSITIVE…”

    Daniel

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