Homework Assignment 05: One-Dimensional Systems
(Due: 03/03/2016)
Problem 1 (20%) A one-dimensional Hamiltanian has a ground-state eigenfunction of
A
,
ψ0(x) =
cosh(x / a)
with the constant a > 0, and V(x) → 0 as
|x |
 1. The Hyperbolic cosine function is defined
a
e x +e −x
.
2
(a) [ 5%] Determine A .
(b) [15%] Find the exact eigenvalue E0 and the exact potential V (x) .
as cosh(x) =
Problem 2 (40%+30%) A particle of mass m is trapped inside of an infinite potential well of
width a :
⎧
⎪ 0
(| x |< a / 2)
V(x) = ⎪
⎨
⎪
∞ (| x |≥ a / 2)
⎪
⎪
⎩
(a) [40%] If an additional repulsive narrow potential is applied that can be modeled as λδ(x)
with λ > 0 , determine the eigenenergies and eigenfunctions.
[Hints: This is a symmetric potential, and we can consider the symmetric and anit-symmetric
solutions separately. Then if degeneracy exists between these two types of eigenfunctions, the
general solution is linear combination of symmetric and anti-symmetric eigenfunctions with the
same eigenvalues. If an equation has no analytical solution, just denote your solutions to this
equation by some symbols, such as kn , and then you can write down the eigenfunctions and
eigenenergies in terms of kn . ]
(b) [Bonus 30%] If an additional attractive potential −λδ(x) is applied, find the condition for
existence of a state with negative energy.
Problem 3 (40%) Consider a series of δ -functions given by
λ 2 n=+∞
V(x) =
∑ δ(x −nd)
2m n=−∞
as an approximate to a periodic potential. Here λ > 0 .
(a) [30%] Show that
λ
cos(kd) = cos(αd) +
sin(αd)
2α
where α = 2mE /  , E the eigenenergy, and k a wave number.
(b) [10%] Schematically draw F(θ) = cos(θ) +
existence of forbidden energy bands.
λ
sin(θ) , where θ = αd . Then argue the
2α