Illustrated Physics
physics, drawn
Concept

WKB approximation — where it appears

A way of solving wave equations in a slowly varying medium by letting the wave keep its local wavelength and adjust its amplitude as it goes. It fails where the wavelength changes appreciably within one wavelength: at turning points, and in the badlands where quantum reflection happens.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A current that grows thirty decades with the field. The current density of electrons tunnelling out of a cold metal against the field at its surface, on a logarithmic scale, from the Fowler–Nordheim law, for work functions of 2.7 eV (lanthanum hexaboride), 4.5 eV (tungsten) and 5.6 eV (a surface barely willing to let electrons go). For tungsten the current is 0.009 A/m² at 2 GV/m, 5·10⁵ at 4 GV/m and 6·10⁹ at 8 GV/m — rising by more than twenty orders of magnitude between 1 and 4 GV/m. The work function sits in the exponent as a three-halves power, so lowering it from 4.5 to 2.7 eV raises the current at 3 GV/m by a factor of 2·10⁵. A current density of a million amperes per square metre, an ordinary working value for a field emitter, needs about 4.2 GV/m on tungsten.

The electrons a field pulls from cold metal

A metal holds its electrons behind a step a few electronvolts high, and to get them out the usual way is to heat the metal white-hot until some climb over it. Put a strong enough field outside instead and the step tilts into a triangle a couple of nanometres wide, and electrons leave a cold metal by tunnelling through it. The current grows thirty decades as the field rises tenfold, a sharp needle at a thousand volts reaches the field that a flat plate would need a million volts for, and the electrons that come out all have nearly the same energy — which is why the brightest, sharpest electron microscopes are fed from a single etched tungsten point.

quantum · Tunnelling An attractive surface that reflects slow atoms. The probability that an atom approaching a surface is reflected by the surface's attraction alone, before reaching the surface, against its wavenumber in units of the length β₄ = √(2mC₄)/ħ set by the attraction, from the Schrödinger equation solved with everything reaching the surface absorbed. Fast atoms fall in and stick. Slow ones are turned back by a potential that only ever pulls them forward, with a reflection that approaches one as R ≈ 1 − 4.0 kβ₄. For atoms near a solid β₄ is a fraction of a micrometre, and kβ₄ = 0.1 is reached at speeds of millimetres per second.

The cliff a slow atom bounces off

A ball rolling towards the edge of a cliff goes over it, and so does every classical particle that meets a drop in potential energy. A quantum particle may bounce back. The faster its potential energy falls, the more of its wave is reflected, and as the particle slows the reflection rises towards certainty, however gently the drop is shaped. Slow atoms approaching a surface that attracts them, and would capture them if they arrived, are turned back by that attraction alone, in a region where nothing pushes them away.

quantum · Matter waves

Named alongside it

The objects these essays reach for when they reach for this one.

BarrierCasimir polder forceElectric fieldElectron microscopeFermi levelField emissionMatter waveQuantum reflectionReflection coefficientSchrodinger equationTunnellingWork function

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