19.3 Homework
- A general one dimensional scattering problem could be characterized by an (arbitrary) poten-
tialV(x) which is localized by the requirement thatV(x) = 0 for|x|> a. Assume that the
wave-function is
ψ(x) =
{
Aeikx+Be−ikx x <−a
Ceikx+De−ikx x > a
Relating the “outgoing” waves to the “incoming” waves by the matrixequation
(
C
B
)
=
(
S 11 S 12
S 21 S 22
)(
A
D
)
show that
|S 11 |^2 +|S 21 |^2 = 1
|S 12 |^2 +|S 22 |^2 = 1
S 11 S∗ 12 +S 21 S 22 ∗ = 0
Use this to show that theSmatrix is unitary.
- Calculate theSmatrix for the potential
V(x) =
{
V 0 |x|< a
0 |x|> a
and show that the above conditions are satisfied.
- The odd bound state solution to the potential well problem bearsmany similarities to the
zero angular momentum solution to the 3D spherical potential well. Assume the range of the
potential is 2. 3 × 10 −^13 cm, the binding energy is -2.9 MeV, and the mass of the particle is
940 MeV. Find the depth of the potential in MeV. (The equation to solve is transcendental.)
- Find the three lowest energy wave-functions for the harmonic oscillator.
- Assume the potential for particle bound inside a nucleus is given by
V(x) =
{
−V 0 x < R
̄h^2 ℓ(ℓ+1)
2 mx^2 x > R
and that the particle has massmand energye >0. Estimate the lifetime of the particle inside
this potential.
