Simple Nature - Light and Matter

a/An incident wave is partially

reflected and partially transmitted

at a step in the potentialU. The

complex wavefunctions are rep-

resented using a complex plane

perpendicular to the direction of

propagation, so that they look

like corkscrews. The incident and

reflected wavefunctions actually

superposed, but are drawn as

separate entities and offset for

purposes of visualization.

obvious that we should be able to get away with recycling that result,

both because our quantum-mechanical wavefunctions are complex

and because the Schr ̈odinger equation is dispersive, so we can no

longer assume, as we did there, that a wave packet simply glides

along rigidly (example 4, p. 968).

To sidestep the problem of dispersion, we will carry out our anal-

ysis using an infinitely long wave-train with a definite wavelength.

Let the incident wave have unit amplitude and travel to the right,

ΨI=ei(kx−ωt) (x <0),

as in example 3, p. 967. Recall that the wavenumberkis basically

just momentum,p=~k.

For the reflected and transmitted parts of the wave, we take

ΨR=Rei(−kx−ωt) (x <0),

and

ΨT=Tei(k

′x−ωt)

(x >0),

where the reflected and transmitted amplitudes RandT are un-
known, and our goal is to find them. The different sign inside the
exponential for ΨRcorresponds to the opposite direction of motion
at the same speedv, while in the expression for ΨTwe have motion
to the right, but with a different momentump′=~k′as required by
conservation of energy.
Demanding continuity of Ψ gives

1 +R=T.

The derivatives are∂ΨI/∂x=ikΨI,∂ΨR/∂x=−ikΨR, and∂ΨT/∂x=
ik′ΨT, and evaluating these atx= 0,t= 0 givesik,−ikR, andik′T.
If the derivative is to be the same forx→ 0 −and forx→ 0 +, we
need to haveik−Rik=iTk′, or

1 −R=

k′

k

T,

But these two equations are exactly the same as the ones found on

p. 381 for a classical, nondispersive wave, the only difference being

the replacement ofv/v′withk′/k. To keep the writing simple, let

α= k′/k. With this replacement, the solutions are the same as

before,R= (1−α)/(1 +α) andT = 2/(1 +α). For a particle of

energyE, we can find the momentum ratioαusing conservation

of energy, α =

√

(E−U 2 )/(E−U 1 ). There is partial reflection

not just in the case of a sudden rise in the potential, but also at

a sudden drop (U 2 < U 1 ), which is surprising and seems to violate

the correspondence principle, but actually does not, as discussed in

example 18 on p. 907.

Section 14.5 Methods for solving the Schrodinger equation ̈ 971