**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

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

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (43)

Related people (5)

Related concepts (8)

Related courses (2)

Related lectures (22)

Finite potential well

The finite potential well (also known as the finite square well) is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a "box", but one which has finite potential "walls". Unlike the infinite potential well, there is a probability associated with the particle being found outside the box. The quantum mechanical interpretation is unlike the classical interpretation, where if the total energy of the particle is less than the potential energy barrier of the walls it cannot be found outside the box.

Probability current

In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability. Specifically, if one thinks of probability as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. It is a real vector that changes with space and time. Probability currents are analogous to mass currents in hydrodynamics and electric currents in electromagnetism. As in those fields, the probability current (i.e.

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 transmitted wave relative to an incident wave. Different fields of application have different definitions for the term.

PHYS-207(c): General physics : quanta

Le cours traite les ondes électromagnétiques (optique géométrique et optique physique) et donne
une introduction à la physique quantique.

PHYS-425: Quantum physics III

To introduce several advanced topics in quantum physics, including
semiclassical approximation, path integral, scattering theory, and
relativistic quantum mechanics

Quantum Physics I

Covers the basics of quantum physics, including wave packet dynamics and probability density functions.

Brownian Motion and Chemical ReactionsMOOC: Advanced statistical physics

Explores Brownian motion's role in chemical reactions and particle escape mechanisms over potential barriers.

Proton Barrier Penetration: Transmission Coefficient Calculation

Covers the calculation of the transmission coefficient for proton barrier penetration and its application in determining average passage time.

Jean-Michel Sallese, Adil Koukab

A time-resolved multi-gate ion sensitive field effect transducer, including a silicon layer, a P-doped region in the silicon layer and a first electrode in electric connection with the P doped region, a N-doped region in the silicon layer and a second elec ...

2023Daniel Kuhn, Mengmeng Li, Tobias Sutter

We propose a policy gradient algorithm for robust infinite-horizon Markov Decision Processes (MDPs) with non-rectangular uncertainty sets, thereby addressing an open challenge in the robust MDP literature. Indeed, uncertainty sets that display statistical ...

2023Christian Enz, Assim Boukhayma, Antonino Caizzone

A Time-of-flight optical device and a 3D optical detector comprising a CMOS integrated circuit with an array of photosensitive pixels that are, at least in part, interconnected to form macro-pixels (180). Each macro-pixel (180) groups a plurality of indivi ...

2020