Simple Nature - Light and Matter

energy and momentum is E = p 2 /2m, so the group velocity isdE/ dp = p/m = v, exactly what it should be. It is only the phasevelocity that differs by a factor of two from what we would haveexpected, but the phase velocity is not the physically importantthing.13.3.3 Bound statesElectrons are at their most interesting when they’re in atoms,that is, when they are bound within a small region of space. Wecan understand a great deal about atoms and molecules based onsimple arguments about such bound states, without going into anyof the realistic details of atom. The simplest model of a bound stateis known as the particle in a box: like a ball on a pool table, theelectron feels zero force while in the interior, but when it reachesan edge it encounters a wall that pushes back inward on it witha large force. In particle language, we would describe the electronas bouncing off of the wall, but this incorrectly assumes that theelectron has a certain path through space. It is more correct todescribe the electron as a wave that undergoes 100% reflection atthe boundaries of the box.Like a generation of physics students before me, I rolled myeyes when initially introduced to the unrealistic idea of putting aparticle in a box. It seemed completely impractical, an artificialtextbook invention. Today, however, it has become routine to studyelectrons in rectangular boxes in actual laboratory experiments. The“box” is actually just an empty cavity within a solid piece of silicon,amounting in volume to a few hundred atoms. The methods forcreating these electron-in-a-box setups (known as “quantum dots”)were a by-product of the development of technologies for fabricatingcomputer chips.For simplicity let’s imagine a one-dimensional electron in a box,i.e., we assume that the electron is only free to move along a line.The resulting standing wave patterns, of which the first three areshown in the figure, are just like some of the patterns we encounteredwith sound waves in musical instruments. The wave patternsmust be zero at the ends of the box, because we are assuming thewalls are impenetrable, and there should therefore be zero probabilityof finding the electron outside the box. Each wave pattern islabeled according to n, the number of peaks and valleys it has. Inquantum physics, these wave patterns are referred to as “states” ofthe particle-in-the-box system.The following seemingly innocuous observations about the particlein the box lead us directly to the solutions to some of the mostvexing failures of classical physics:The particle’s energy is quantized (can only have certain values).Each wavelength corresponds to a certain momentum, and a givenf / Three possible standingwavepatterns for a particle in abox.Section 13.3 Matter As a Wave 859