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Concept
Solution of Schrödinger equation for a step potential
Summary
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. Typically, the potential is modeled as a Heaviside step function.
The time-independent Schrödinger equation for the wave function is
where Ĥ is the Hamiltonian, ħ is the reduced Planck constant, m is the mass, E the energy of the particle. The step potential is simply the product of V0, the height of the barrier, and the Heaviside step function:
The barrier is positioned at x = 0, though any position x0 may be chosen without changing the results, simply by shifting position of the step by −x0.
The first term in the Hamiltonian, is the kinetic energy of the particle.
The step divides space in two parts: x < 0 and x > 0. In any of these parts the potential is constant, meaning the particle is quasi-free, and the solution of the Schrödinger equation can be written as a superposition of left and right moving waves (see free particle)
where subscripts 1 and 2 denote the regions x < 0 and x > 0 respectively, the subscripts (→) and (←) on the amplitudes A and B denote the direction of the particle's velocity vector: right and left respectively.
The wave vectors in the respective regions being
both of which have the same form as the De Broglie relation (in one dimension)
The coefficients A, B have to be found from the boundary conditions of the wave function at x = 0. The wave function and its derivative have to be continuous everywhere, so:
Inserting the wave functions, the boundary conditions give the following restrictions on the coefficients
It is useful to compare the situation to the classical case. In both cases, the particle behaves as a free particle outside of the barrier region. A classical particle with energy E larger than the barrier height V0 will be slowed down but never reflected by the barrier, while a classical particle with E < V0 incident on the barrier from the left would always be reflected.
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