19 In section 14.5.1, p. 970, we found the solution to the
Schr ̈odinger equation for a particle arriving at a potential barrier, in
the case where the far side of the barrier is classically allowed. We
now consider the case in which the barrier is not just high enough to
make the region beyond it classically forbidden — we let the height
of the barrier be infinite. Let the potential beU(x) ={
0, x < 0
+∞, x >0,and let the incident wave beΨI=ei(kx−ωt) (x <0).Determine the form of the complete solution to the Schr ̈odinger
equation on the left side of the barrier, including both the incident
wave and the reflected wave.
√20 As discussed on p. 986, suppose that for a particle in a box,
we have
OE = 1 ,
OE = 4 ,
Ψ =c +c′ , and
|c|=|c′|.Show that〈Ψ|OEΨ〉= 2.5. .Solution, p. 105321 Consider the wavefunctionsΨ 1 =ex+it and
Ψ 2 =et+ix.To keep the writing simple, we use a system of units (not SI) such
that these expressions make sense, and in which~= 1.
(a) Show by direct substitution in the time-dependent Schr ̈odinger
equation (withU= constant) that one of these is a solution and the
other is not.
(b) Make an independent argument, requiring no calculations, to the
effect that the invalid one violates one of the fundamental principles
1-5 of quantum mechanics listed in section 14.6.5, p. 990.
Problems 1013