| rectangular barrier | |
|---|---|
| Name | Rectangular barrier |
| Caption | Schematic potential profile for a one-dimensional rectangular barrier |
| Field | Quantum mechanics |
| Introduced | 1920s |
| Related | Schrödinger equation, Tunnel diode, Scanning tunneling microscope |
rectangular barrier
A rectangular barrier is an idealized one-dimensional potential energy profile consisting of a region of constant, elevated potential energy bounded by two abrupt steps. In Quantum mechanics it serves as a paradigmatic scattering and tunneling model that illustrates non-classical transmission and reflection of particles described by the Schrödinger equation. The rectangular barrier underpins understanding of phenomena in solid-state physics, nuclear physics, and nanoscale devices.
The canonical rectangular barrier potential V(x) is defined piecewise: V(x)=V0 for x1
Solving the time-independent Schrödinger equation for the barrier yields piecewise plane-wave or evanescent solutions. For E>V0 the solutions inside the barrier are oscillatory with wavevector k2 = sqrt(2m(E−V0))/ħ, while for E
From the matched solutions one derives the transmission coefficient T(E) and reflection coefficient R(E)=1−T(E) for flux-conserving one-dimensional scattering. For E
When the barrier is paired with adjacent potential wells or leads, resonant transmission (Breit–Wigner–type resonances) can occur for energies matching quasi-bound states inside the barrier region. Fabry–Pérot–like interference between reflections at the two interfaces produces sharp resonances for thin barriers and narrow resonant widths determined by coupling to the leads. In nuclear physics, analogous potential barriers explain alpha decay rates via Gamow's tunneling model, linking to work by George Gamow and the concept of barrier penetration in radioactive decay. Resonances are mathematically connected to complex-energy poles of the scattering matrix (S-matrix) studied in scattering theory.
Time-dependent analysis uses wavepackets to study transit times, dwell times, and wavepacket reshaping upon interaction with the barrier. Numerical propagation of Gaussian or other minimal-uncertainty packets reveals phenomena such as partial reflection of low-momentum components, forward reshaping that can lead to apparent superluminal group delays (Hartman effect), and temporal broadening due to dispersion. These time-domain behaviors relate to operational definitions of tunneling time explored in experimental programs at institutions like National Institute of Standards and Technology and theoretical analyses by authors such as Mark Büttiker.
The rectangular barrier model justifies qualitative and quantitative features of many technologies: tunneling currents in metal–insulator–metal junctions, field emission from sharp tips, Josephson junction behavior in superconductivity when modeled with insulating barriers, and quantum well infrared photodetectors. Experimental realizations include semiconductor heterostructures fabricated by molecular beam epitaxy at facilities like Bell Labs and IBM Research, where sharp potential steps approximate rectangular profiles. The model also informs nanofabrication of quantum dots and resonant tunneling diodes used in high-frequency electronics.
Variants of the rectangular barrier include finite-depth wells (inverted barriers), multiple barrier arrays forming superlattices (as in Esaki and Tsu superlattice theory), and higher-dimensional barriers where transverse modes and angular momentum channels appear. Periodic arrays yield band structures described by Bloch's theorem and Kronig–Penney style models; disordered barrier ensembles connect to Anderson localization studied by Philip Anderson. Extensions also employ scattering matrix formalisms and transfer-matrix methods widely used in computational condensed matter and photonics.
Category:Quantum mechanics models Category:Quantum tunneling