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One-dimensional scattering at a potential barrier of any shape

Posted 9 years ago

I am trying to solve the problem of electron reflection at a finite potential barrier of a given functional shape (to be used as a parametrized input) in one dimension. My aim is to determine the reflection phase shift as a function of the electron energy. I am looking for a notebook to start with, and then conveniently manipulate the function parameters.

POSTED BY: Enrique Ortega
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Posted 9 years ago

Thanks to both for your answers and ideas.

I have checked Ulrich's Wolfram demostrations and look impressive. Yet I should have been more explicit in my needs, which are very special. It is the problem of a "surface state" in a metal surface.

The surface state problem can be reduced to quantum confinement within an asymmetric Quantum Well (QW), in one dimension. On one side (say, the left side) confinement is produced by reflection at the gap of a bulk crystal potential. So, there is a finite energy range within which the "surface electron" cannot penetrate the bulk crystal (if it does penetrate, it is not a surface state anymore!!). At the other side (right side of the QW) we have a vacuum barrier (image potential barrier). Traditionally, this problem is solved in the Born-Oppenheimer approach (also called phase accumulation model), because one can find a very handy description of the reflection phase in the gap of a crystal (phiC). Hence we also need the reflection phase at the other side of the quantum well, i.e., at the vacuum side, phiB. With both,one simply makes phiC+phiB=2 n Pi and finds the bound states. Note that both phiC and phiB will depend on the energy. There is abundant literature on this approach (see for example "Phase analysis of image states and surface states associated with nearly-free-electron band gaps", N. V. Smith, Phys. Rev. B 32, 3549 (1985))

One can avoid the BO approach, but this requires to solve the whole complex Hamiltonian inside the theory colleagues like that, but I am an experimentalist looking for a model to play with and test my results...My investigated systems are "modified surfaces", such as adsorbed monolayers, of any kind of materials (metals, organic molecules, etc.), and hence I have to model the "vacuum side" of my QW, but still retrieve phiB' to apply the BO sum rule phiC+phi_B'=2 n Pi.

So, you may understand now why I need the "reflection phase" for an electron wave...

POSTED BY: Enrique Ortega

Are you aware of the fact that you can simulate what happens if you let run an arbitrary wave function towards an arbitrarily formed potential barrier by making use of the time-dependent Schroedinger equation in a suitable discretized form? This works even for the relativistic electron as described by Dirac's equation. Among my contributions to Wolfram Demonstrations there are two examples of that. One only non-relativistic, and one which treates the non-relativistic and the lelativistic case in parallel. By having available dynamics of arbitrary states governed by arbitrary Hamiltonians (in 1D, though) we don't have to restrict ourselves to the specialized situations treated in the QM textbooks with the aim of working out analytic solutions. Using Mathematica simply as a means to implement the formulas generated in those textbooks for special cases does not exhaust the power of Mathematica.

POSTED BY: Ulrich Mutze

You can't solve the problem without specifying the functional form of V[x]. The Schrödinger equation is

1/2 \[Psi]''[x] + V[x] \[Psi][x] = k^2/2 \[Psi][x]

with a scattering solution of the form

\[Psi][x] = E^(I k x) + r[k, x] E^(-I k x) + t[k, x] E^(I k x)

Given V[x], you can separate the real and imaginary parts to the equation using ComplexExpand[ ], you get two rather complicated equations which determine r[k,x] and t[k,x].

POSTED BY: S M Blinder
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