PHY4604–Introduction to Quantum Mechanics
Fall 2004
Problem Set 6
Oct. 11, 2004
Due: Oct. 18, 2004
Reading: Griffiths Ch. 2
According to classical mechanics, a particle approaching a “potential barrier” (e.g.
a local maximum Vmax in the potential V (x)) will be reflected back if its energy E is
less than Vmax but will move past Vmax if E is greater than Vmax . (One consequence
is that a particle inside a “square well”, like a box where V > E outside, is trapped
and has no probability of being found outside.) In quantum mechanics, on the other
hand, such a particle has a finite probability of penetrating the barrier, whatever
its energy, and a finite probability of being reflected from it. It can “leak out” of
potential wells.
The probability of transmission (reflection) can be expressed in terms of the transmission coefficient T (reflection coefficient R ), defined as the ratio of the probability
flux of the transmitted (reflected) wave to the probability flux of the incident wave.
Recall a general probability current is written
ih̄ ∗
j=−
[ψ ∇ψ − ((∇ψ ∗ )ψ)]
(1)
2m
and the transmission and reflection coefficients are defined:
T =
|jT |
,
|jI |
R=
|jR |
|jI |
(2)
where T, R and I represent the transmitted, reflected, and incident fluxes, respectively. Note number conservation means |jT | + |jR | = |jI | and therefore T + R = 1.
1
1. dψ/dx Matching Condition. At boundaries ψ must be continuous. What
happens to the 1st derivative dψ/dx depends on the value of the potential at
the boundary. Integrate the Schrödinger equation once from −² to ², then let
² → 0 to show
Ã
dψ
∆
dx
!
=
Z ²
2m
lim
dx V (x)ψ(x),
h̄2 ²→0 −²
(3)
where ∆(dψ/dx) is the (possible) discontinuity in dψ/dx at x = 0. Show that
for a smooth function V (x), ∆(dψ/dx) = 0, but for a δ-function potential
V = −αδ(x), we have ∆(dψ/dx) 6= 0, i.e. the wave function ψ has a kink.
Consider the potential V (x) = −αδ(x) further in problems 2 and 3:
2. Delta function bound states.
Look for solutions to the Schrödinger equation with E < 0 (they will involve
ekx and e−kx ). Write down the general solutions on both sides of the δ-function,
then match the coefficients by using the continuity of the wave function at x = 0
and the condition derived in problem 1. Prove there is only one bound state
and find its energy and wave function.
3. Delta function scattering states. Do the same thing for E > 0 (here the
solutions will involve eikx and e−ikx ). Find the coefficients A, B, and C assuming
particles are incident on the δ-function from the left as shown. Show, using the
definitions given in the introduction, that
T =
|B|2
,
|A|2
R=
|C|2
|A|2
and express these finally in terms of m, α, E, and h̄.
2
(4)