In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, 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
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
for a particle encountering a rectangular
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
energy barrier. It is usually assumed, as here, that a
free particle
In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. I ...
impinges on the barrier from the left.
Although classically a particle behaving as a
point mass
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization (science philosophy), idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension (metaph ...
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
In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic field that does not propagate as an electromagnetic wave but whose energy is spatially concentrated in the vicinity of the source (oscillati ...
. The likelihood that the particle will pass through the barrier is given by the
transmission coefficient
The transmission coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A transmission coefficient describes the amplitude, intensity, or total power of a transmitte ...
, whereas the likelihood that it is reflected is given by the
reflection coefficient
In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected wa ...
.
Schrödinger's wave-equation allows these coefficients to be calculated.
Calculation
The time-independent Schrödinger equation for the wave function
reads
where
is the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
,
is the (reduced)
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
,
is the
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
,
the energy of the particle and