Potentialbarriere

Potentialbarriere (German for "potential barrier") denotes a region in a physical system where the potential energy exceeds that in surrounding regions, creating a barrier for the motion of particles. In classical mechanics, a particle with total energy E below the barrier height V cannot cross the barrier and is reflected; if E > V, the particle can surmount it with reduced kinetic energy. In quantum mechanics, even when E < V, there is a finite probability for the particle to penetrate and traverse the barrier, a phenomenon known as quantum tunneling.

One-dimensional models use the Schrödinger equation -ħ^2/(2m) d^2ψ/dx^2 + V(x) ψ = E ψ. For a finite rectangular barrier of height V0 and width a, matching wavefunctions at the interfaces yields a transmission coefficient T(E) that is exponentially small for E < V0, T ~ exp(-2κa) with κ = sqrt{2m(V0 − E)}/ħ. For E > V0, T increases and exhibits oscillations and, in double-barrier structures, resonances due to constructive interference.

Common barrier shapes include rectangular, triangular, and delta-function barriers. Practical relevance appears in nuclear alpha decay via tunneling through the Coulomb barrier, in semiconductor devices such as tunnel diodes and resonant tunneling diodes, and in scanning tunneling microscopy. In superconducting electronics, Josephson junctions rely on quantum tunneling of Cooper pairs through thin insulating barriers.

Analytical and numerical methods include exact solutions for simple barriers, the WKB approximation for high and wide barriers, and transfer-matrix techniques for layered structures. In mesoscopic transport, the transmission coefficient T relates to conductance via the Landauer formula.