2.4 Numerical Integration of the Schrödinger Equation
73
which shows that the DAF can be used to find the kth derivative of the wave function
at any point in space given that we know the wave function (and not its derivatives) on
an equally spaced grid (the distance between the points is h in the above expression.
Evaluating the expression at the same grid points we have
ψ
(k)
DAF (x i ) =
j
D
(k)
DAF (x i − x j )ψ DAF (x j )hdx
(2.130)
or in matrix notation
ψ
(k)
DAF = D
(k)
DAF ψ DAF
(2.131)
where ψ is now a column vector and D is a square matrix. Dropping the DAF subscript
for clarity gives
ψ
(k)
= D
(k)
ψ.
(2.132)
Now the Schrödinger equation is an eigenvalue equation
(D
(2)
− U)f − λf = 0
(2.133)
where
λ = −h
2 k
2
.
(2.134)
Each eigenvector or column of the above eigenvalue problem is a solution to the
secular equation and is associated eigenvalues. The negative eigenvalue corresponds
to the bound states of the system.
Note: This DAF procedure will only work if there are significant regions where the
wave function is essentially zero! It will not work for the hard sphere model potential
or the Coulomb potential. For the radial wave function the DAF method only works
for bound states with a sufficiently strong repulsive wall at the origin to force the wave
function to be essentially zero well before r = 0. The DAF matrix is highly banded
as a result of the Gaussian factor but is not tridiagonal like the finite difference
or Numerov methods. One cannot use numerical propagation. However, there are
extensions of the DAF presented here to accurately include periodic functions.
2.4.3 Approximating the Wave Function
The Finite Expansion
Let us approximate the wave function as a finite sum of N known analytical basis
functions
(r) =
N
i=1
a i f i (r)
(2.135)
73
which shows that the DAF can be used to find the kth derivative of the wave function
at any point in space given that we know the wave function (and not its derivatives) on
an equally spaced grid (the distance between the points is h in the above expression.
Evaluating the expression at the same grid points we have
ψ
(k)
DAF (x i ) =
j
D
(k)
DAF (x i − x j )ψ DAF (x j )hdx
(2.130)
or in matrix notation
ψ
(k)
DAF = D
(k)
DAF ψ DAF
(2.131)
where ψ is now a column vector and D is a square matrix. Dropping the DAF subscript
for clarity gives
ψ
(k)
= D
(k)
ψ.
(2.132)
Now the Schrödinger equation is an eigenvalue equation
(D
(2)
− U)f − λf = 0
(2.133)
where
λ = −h
2 k
2
.
(2.134)
Each eigenvector or column of the above eigenvalue problem is a solution to the
secular equation and is associated eigenvalues. The negative eigenvalue corresponds
to the bound states of the system.
Note: This DAF procedure will only work if there are significant regions where the
wave function is essentially zero! It will not work for the hard sphere model potential
or the Coulomb potential. For the radial wave function the DAF method only works
for bound states with a sufficiently strong repulsive wall at the origin to force the wave
function to be essentially zero well before r = 0. The DAF matrix is highly banded
as a result of the Gaussian factor but is not tridiagonal like the finite difference
or Numerov methods. One cannot use numerical propagation. However, there are
extensions of the DAF presented here to accurately include periodic functions.
2.4.3 Approximating the Wave Function
The Finite Expansion
Let us approximate the wave function as a finite sum of N known analytical basis
functions
(r) =
N
i=1
a i f i (r)
(2.135)