Chapter 4 Linear Differential Operators

142 CHAPTER 4. LINEAR DIFFERENTIAL OPERATORS
of states is that of a box of length R. We see that changing the potential
does not create or destroy eigenstates, it just moves them around.
The spike is not exactly a delta function because of level repulsion between
nearly degenerate eigenstates. The interloper elbows the nearby levels out of
the way, and all the neighbours have to make do with a bit less room. The
stronger the coupling between the states on either side of the delta-shell, the
stronger is the inter-level repulsion, and the broader the resonance spike.
Normalization factor
We now evaluate R
so as to find the the normalized wavefunctions
0
dr|ψk| 2 = N −2
k , (4.142)
χk = Nkψk. (4.143)
Let ψk(r) be a solution of
Hψ = − d2
+ V (r) ψ = k
dr2 2 ψ (4.144)
satisfying the boundary condition ψk(0) = 0, but not necessarily the boundary
condition at r = R. Such a solution exists for any k. We scale ψk by
requiring that ψk(r) = sin(kr + η) for r > R0. We now use Lagrange’s
identity to write
(k 2 − k ′2
R
) dr ψk ψk ′ =
0
R
0
dr {(Hψk)ψk ′ − ψk(Hψk ′)}
= [ψkψ ′ k ′ − ψ′ kψk ′]R
0
= sin(kR + η)k ′ cos(k ′ R + η)
−k cos(kR + η) sin(k ′ R + η). (4.145)
Here, we have used ψk,k ′(0) = 0, so the integrated out part vanishes at the
lower limit, and have used the explicit form of ψk,k ′ at the upper limit.
Now differentiate with respect to k, and then set k = k ′ . We find
R
2k dr(ψk)
0
2 = − 1
2 sin
2(kR + η) + k R + ∂η
. (4.146)
∂k