QUANTUM PHYSICS 1
PROBLEM SET 3
due October 22, 2008
A. A. Scattering from square step
A particle of mass mand kinetic energy Emoves from the left and scatters from a square well potential:
V(x) = 0,for x < 0
V0for x > 0,(1)
with V0>0. Find the transmission and reflection coefficients for both when the energy Eis larger than V0and
smaller than V0. Verify that the sum of the probabilities for the particle to go through the potential and to bounce
back equals one.
B. B. All you wanted to know about Dirac’s δ-function and were afraid to ask
We can define the δ-function buy its behavior inside integrals:
Zb
a
f(x)δ(x−y)≡f(y),for a < y < b, (2)
for any well-behaved function f(x) , usually called the test-function. Show that
a) R∞
−∞ dx(x3−1)δ(x−1) = 0
b) δ(cx) = 1
|c|δ(x) ( Hint: Insert both sides of the equation in the definition of δabove and change variables.)
c) dθ(x)
dx =δ(x) where θ(x) is the step function
θ(x) = 1 if x > 0;
0 if x < 0.(3)
(and θ(0) = 1/2 if it ever matters).
d) What is the Fourier transform of δ(x)
F(k) = Z∞
−∞
dx
√2πe−ikxδ(x) =? (4)
Use Plancherel’s theorem (see text) to show that
δ(x) = Z∞
−∞
dk
2πeikx,(5)
which is a relation we used in class after a hand waving “proof”.
e) Show that
Z∞
−∞
dxf(x)δ′(x) = −f′(0).(6)
Feel free to assume that f(x)→as fast as necessary as x→ ±∞.
f) Another way of defining the δ-function is through the relation
δ(x)≡lim
α→∞ rα
πe−αx2.(7)
Show that the result of d) is the same using this new definition. Feel free to exchange the order of limits and
integrations and assume that the test functions are as well behaved as necessary.
