Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

1. Solution of the radial Schrödinger equation, Eq.(3.5), with the Lennard-
Jones potential (3.9) for scattering states (i.e. with positive energy). One
can simply use Numerov’s method with a uniform grid and outwards
integration only (there is no danger of numerical instabilities, since the
solution is oscillating). One has however to be careful and to avoid the
divergence (as ∼ r −12 ) for r → 0. A simple way to avoid running into
trouble is to start the integration not from r min = 0 but from a small but
nonzero value. A suitable value might be r min ∼ 0.5σ, where the wave
function is very small but not too close to zero. The first two points can be
calculated in the following way, by assuming the asymptotic (vanishing)
form for r → 0:
χ ′′ (r) ≃ 2mɛ
¯h 2 σ 12
r 12 χ(r) =⇒ χ(r) ≃ exp ⎛
⎝−
√
2mɛσ 12
25¯h 2
r−5 ⎞
⎠ (3.15)
(note that this procedure is not very accurate because it introduces and
error of lower order, i.e. worse, than that of the Numerov’s algorithm.
In fact by assuming such form for the first two steps of the recursion,
we use a solution that is neither analytically exact, nor consistent with
Numerov’s algorithm).
The choice of the units in this code is (once again) different from that of
the previous codes. It is convenient to choose units in which ¯h 2 /2m is a
number of the order of 1. Two possible choices are meV/Å 2 , or meV/σ 2 .
This code uses the former choice. Note that m here is not the electron
mass! it is the reduced mass of the H-Kr system. As a first approximation,
m is here the mass of H.
2. Calculation of the phase shifts δ l (E). Phase shifts can be calculated by
comparing the calculated wave functions with the asymptotic solution at
two different values r 1 and r 2 , both larger than the distance r max beyond
which the potential can be considered to be negligible. Let us write
χ l (r 1 ) = Akr 1 [j l (kr 1 ) cos δ l − n l (kr 1 ) sin δ l ] (3.16)
χ l (r 2 ) = Akr 2 [j l (kr 2 ) cos δ l − n l (kr 2 ) sin δ l ] , (3.17)
from which, by dividing the two relations, we obtain an auxiliary quantity
K
K ≡ r 2χ l (r 1 )
r 1 χ l (r 2 ) = j l(kr 1 ) − n l (kr 1 ) tan δ l
(3.18)
j l (kr 2 ) − n l (kr 2 ) tan δ l
from which we can deduce the phase shift:
tan δ l = Kj l(kr 2 ) − j l (kr 1 )
Kn l (kr 2 ) − n l (kr 1 ) . (3.19)
The choice of r 1 and r 2 is not very critical. A good choice for r 1 is about
at r max . Since the LJ potential decays quite quickly, a good choice is
r 1 = r max = 5σ. For r 2 , the choice is r 2 = r 1 + λ/2, where λ = 1/k is the
wave length of the scattered wave function.
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