Consider a particle in a piecewise constant potential. Between x1 and x2 the potential has a constant value V1, between x2 and x3 the potential has a constant value V2, etc.
The time-independent Schrödinger equation for the motion of particle in this potential is,
These matrix equations can be used to solve a number of problems where a quantum particle moves in a piecewise constant potential. A few examples are given below.
Reflection from a potential step
Consider the case where a wave approaches a potential step from the left. The potential step occurs at x = 0.
If V0 < E < V1, then the wavefunction must go to zero at + infinity and the wavefunction for x > 0 will be a decaying exponential ψ = Aexp(-k1x). Here A is the amplitude of the wavefunction at x = 0. By substituting the value of the wavefunction and the derivative of the wavefunction at x = 0 into the matrix equation above we can determine that the wavefunction to the left of the potential step is,
This is the sum of an incident wave (proportional to exp(ik0x)) and a reflected wave (proportional to exp(-ik0x)), and can be written as,
where AI is the complex amplitude of the incident wave and AR is the complex amplitude of the reflected wave.
The probability flux of the incident wave is the probability that a number of particles reach the step per unit time. The probability flux is the probablility density, AI*AI times the velocity of the particles. The reflection coefficient is the ratio of the probability flux of the reflected wave to that of the incident wave. This is the probability that a particle will be reflected.
The reflection coefficient is 1 in this case since there is no transmitted wave. There is a phase shift between the incident and reflected waves of θ = 2arctan(k1/k0).
If E > V0, V1 there will be a transmitted wave and a reflected wave.
If the wave is incident from the left, the transmitted wave will have the form ψ = Aexp(ik1x) and the wavefunction to the left of the potential step will be specified by,
The wavefunction for x < 0 can be written as the sum of an incident and a reflected wave,
The reflection and transmission coefficients in this case are,
R is the probability that a particle is reflected and T is the probability that a particle is transmitted. Since a particle must either be reflected or transmitted, R + T = 1.
Tunneling through a rectangular barrier
Wavefunctions can penetrate into a classically forbidden region. If a potential barrier is thin, some of the wave will pass right through it. This phenomena is known as tunneling. Below a potential barrier of width d divides two regions.
If V1 > E > V0, V2, then waves incident from the left will be partially reflected and partially transmitted. The transmitted wave in the region x > 0 has the form ψ = Aexp(ik2x). The wavefunction in the barrier region -d < x < 0 is specified by the matrix equation,
The wavefunction to the left of the barrier, x < -d, is specified by the matrix equation,
For x < -d, the wavefunction is the sum of an incident wave and a reflected wave,
where
and
The reflection and transmission coefficients are,
and
A further application of this method is to find the solutions of the Schrödinger equation for the Kronig - Penney potential.
