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.

Calculation

is the Hamiltonian is the (reduced) Planck constant is the mass is the Heaviside step function

is the kinetic energy.

). 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 where the wave numbers

is treated below.

have to be found from the boundary conditions. The wave function and its derivative have to be continuous

Inserting the wave functions, the boundary conditions give the following restrictions on the coefficients

Transmission and reflection

would always incident on the barrier would always get reflected.

To study the quantum case, consider the following situation: a particle incident on the barrier from the left side It may be reflected or transmitted

and

The result is:

Due to the mirror symmetry of the model, the amplitudes for incidence from the right are the same as those from the left. Note that these expressions hold for any energy If then so there is a singularity in both of these expressions.

Analysis of the obtained expressions

E < V0

for the particle to be transmitted through the barrier, with This effect, which differs from the classical case, is called quantum tunneling. The transmission is exponentially suppressed with the barrier width, which can be understood from the functional form of the wave function: Outside of the barrier it oscillates with wave vector whereas within the barrier it is exponentially damped over a distance If the barrier is much wider than this decay length, the left and right part are virtually independent and tunneling as a consequence is suppressed.

E > V0

where

and 1.8 in the above figure). Note that the probabilities and amplitudes as written are for any energy (above/below) the barrier height.

E = V0

is

This expression can be obtained by calculating the transmission coefficient from the constants stated above. For this purpose the ratio

:

is defined as follows:

.

Now, when calculating the limit as x approaches 1 (using L'Hôpital's rule),

approaches 1 can be obtained:

in the evaluated value for the limit, the above expression for T is successfully reproduced.

Remarks and applications

The calculation presented above may at first seem unrealistic and hardly useful. However it has proved to be a suitable model for a variety of real-life systems. One such example are interfaces between two conducting materials. In the bulk of the materials, the motion of the electrons is quasi-free and can be described by the kinetic term in the above Hamiltonian with an effective mass. Often the surfaces of such materials are covered with oxide layers or are not ideal for other reasons. This thin, non-conducting layer may then be modeled by a barrier potential as above. Electrons may then tunnel from one material to the other giving rise to a current.

The operation of a scanning tunneling microscope (STM) relies on this tunneling effect. In that case, the barrier is due to the gap between the tip of the STM and the underlying object. Since the tunnel current depends exponentially on the barrier width, this device is extremely sensitive to height variations on the examined sample.

The above model is one-dimensional, while space is three-dimensional. One should solve the Schrödinger equation in three dimensions. On the other hand, many systems only change along one coordinate direction and are translationally invariant along the others; they are separable. The Schrödinger equation may then be reduced to the case considered here by an ansatz.

For another, related model of a barrier, see Delta potential barrier (QM) constant.

See also

• Morse/Long-range potential

• Step potential

• Finite potential well

References