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Concept
Rectangular potential barrier
Summary
In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. The problem consists of solving the one-dimensional time-independent Schrödinger equation for a particle encountering a rectangular potential energy barrier. It is usually assumed, as here, that a free particle impinges on the barrier from the left.
Although classically a particle behaving as a point mass would be reflected if its energy is less than , a particle actually behaving as a matter wave has a non-zero probability of penetrating the barrier and continuing its travel as a wave on the other side. In classical wave-physics, this effect is known as evanescent wave coupling. The likelihood that the particle will pass through the barrier is given by the transmission coefficient, whereas the likelihood that it is reflected is given by the reflection coefficient. Schrödinger's wave-equation allows these coefficients to be calculated.
The time-independent Schrödinger equation for the wave function reads
where is the Hamiltonian, is the (reduced)
Planck constant, is the mass, the energy of the particle and
is the barrier potential with height and width .
is the Heaviside step function, i.e.,
The barrier is positioned between and . The barrier can be shifted to any position without changing the results. The first term in the Hamiltonian, is the kinetic energy.
The barrier divides the space in three parts (). In any of these parts, the potential is constant, meaning that 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). If
where the wave numbers are related to the energy via
The index on the coefficients and denotes the direction of the velocity vector. Note that, if the energy of the particle is below the barrier height, becomes imaginary and the wave function is exponentially decaying within the barrier.
Official source
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