1000 Solved Problems in Modern Physics

3.3 Solutions 197

Fig. 3.19Class I and Class II wave functions

3.49 The analysis for the reflection and transmission of stream of particles from
the square well potential is similar to that for a barrier (Problem 3.30) except
that the potentialVbmust be replaced by−V 0 and in the region 2,k 2 must be
replaced byik 2. Thus, from Eq. (9) of Problem 3.30, we get

τ=

4 k 1 k 2 exp(−ik 1 a)

(k 2 +k 1 )^2 exp(−ik 2 a)−(k 2 −k 1 )^2 exp(ik 2 a)

The fraction of transmitted particles whenk 2 a=nπis determined by the

imaginary exponential terms in the denominator.

e+inπ=cosnπ±isin(nπ)=cosnπ

=1; (n= 0 , 2 , 4 ·)

=−1; (n= 1 , 3 , 5 ...)

Fig. 3.20Transmission
coefficientTas a function of
the ratioE/Vofor attractive
square well potential

Thereforeτ∗= 1

A typical graph forTas a function ofE/V 0 is shown in Fig. 3.20.

In general we get the transmission coefficient

T=

∣

∣

∣

∣

∣

1 +

V 02 sin^2 k 2 a

4 E(E+V 0 )

∣

∣

∣

∣

∣

− 1

The transmission coefficient goes to zero atE=0 because of the 1/Eterm

in the denominator. ForE/V 0 1, narrow transmission bands occur when-