Step potential
In quantum mechanics and scattering theory, the one-dimensional step potential is an idealized system used to model incident, reflected and transmitted matter waves.
The problem consists of solving the time-independent Schrödinger equation for a particle with a step-like potential in one dimension.
where Ĥ is the Hamiltonian, ħ is the reduced Planck constant, m is the mass, E the energy of the particle.
Once we have found the quantum-mechanical result we will return to the question of how to recover the classical limit.
To study the quantum case, consider the following situation: a particle incident on the barrier from the left side A→.
We then solve for T and R. The result is: The model is symmetric with respect to a parity transformation and at the same time interchange k1 and k2.
For incidence from the right we have therefore the amplitudes for transmission and reflection For energies E < V0, the wave function to the right of the step is exponentially decaying over a distance
In this energy range the transmission and reflection coefficient differ from the classical case.
Thus there is a finite probability for a particle with an energy larger than the step height to be reflected.
This seems superficially to violate the correspondence principle, since we obtain a finite probability of reflection regardless of the value of the Planck constant or the mass of the particle.
For example, we seem to predict that when a marble rolls to the edge of a table, there can be a large probability that it is reflected back rather than falling off.
Consistency with classical mechanics is restored by eliminating the unphysical assumption that the step potential is discontinuous.
When the step function is replaced with a ramp that spans some finite distance w, the probability of reflection approaches zero in the limit
For the case of 1/2 fermions, like electrons and neutrinos, the solutions of the Dirac equation for high energy barriers produce transmission and reflection coefficients that are not bounded.
The Heaviside step potential mainly serves as an exercise in introductory quantum mechanics, as the solution requires understanding of a variety of quantum mechanical concepts: wavefunction normalization, continuity, incident/reflection/transmission amplitudes, and probabilities.
A similar problem to the one considered appears in the physics of normal-metal superconductor interfaces.
Quasiparticles are scattered at the pair potential which in the simplest model may be assumed to have a step-like shape.
The solution of the Bogoliubov-de Gennes equation resembles that of the discussed Heaviside-step potential.
