Simple Nature - Light and Matter

22 Microscopic circuits are etched on the surface of a silicon

chip. The equivalent of a wire in such an integrated circuit is called

a “trace.” We consider the case where the trace is narrow enough to

make quantum effects relevant, and we treat an electron inside the

trace using the two-dimensional Schr ̈odinger equation. We describe

the trace as an infinite strip running parallel to thexaxis, extending

fromy= 0 toy=b. The potential is

U=

{

0, 0 < y < b

+∞, y≤0 ory≥b

For convenience of notation, leta=π/b. Consider the following

wavefunctions:

Ψ 1 =ei(−kx−ωt)sinay

Ψ 2 =e−iωtsinkx eay

Ψ 3 =e−iωtekxsinay

Ψ 4 =e−iωt(sin 2axsinay+ sinaxsin 2ay)

The symbolsωandkstand for real constants. Identify the wave-

functions that have the following properties. Exactly one of the

wavefunctions has each property. Explain all answers.

(a) cannot be a solution of the Schr ̈odinger equation for this poten-

tial

(b) is a traveling wave solution

(c) is a solution that could represent the case where 0< y < bis

classically forbidden

(d) is not separable

Key to symbols:

√easy typical challenging difficult very difficult

An answer check is available at http://www.lightandmatter.com.

1014 Chapter 14 Additional Topics in Quantum Physics