68
2 The Quantum Approach to the Two-Body Problem
Gaussian quadratures There are many different Gaussian quadratures (i.e., Gauss–
Hermite, Gauss–Laguerre, Gauss–Legendre, Gauss–Mehler to name a few). These
quadratures are highly accurate since the quadrature weights and abscissas (quadrature points) are determined to make some integrals exact as follows:
X nm ≡
b
a
xψ n (x)ψ m (x)W (x) =
N
i=1
ψ n (x i )x i ψ m (x i )w i
(2.106)
The N abscissas x i and N weights w i are chosen to make these integrals exact. It should
be noted that w i = W (x i ). The weighting function W (x) is a positive definite function
like e
−x in a Gauss–Laguerre quadrature or e
−x
2 for a Gauss–Hermite quadrature.
The X matrix is symmetric and hence its eigenvalues x i are real. In fact the abscissa
or quadrature points x i are just the eigenvalues of the matrix X. It is interesting to
note that the mth eigenvector of X is
√
w i ψ m (x i ) and thus one can easily evaluate the
quadrature weights w i . This method requires that we know the analytical functions
ψ n (x) and that we can analytically evaluate the matrix elements in Eq. 2.106. Then
one can approximately evaluate similar integrals of the form
I =
b
a
f (x)W (x)dx ≈
N
i=1
w i f (x i )
(2.107)
In this case, the abscissas are not equally spaced and none of quadrature points and
abscissas are the same if one goes from an N-term quadrature to a M- term quadrature.
In fact, the abscissas in a N term Gauss–Hermite quadratures are the zero of H N+1 (x).
Likewise, the abscissas for a Gauss–Legendre quadrature are the zeros of Legendre
polynomials, etc. These methods, therefore, do not allow the reuse of previously
calculated values that is instead easy to implement when using constant step size
methods by doubling the number of quadrature points.
Accordingly, we will discuss here, in more detail, only the following fixed stepsize
approaches: the Laplacian operator, the wave function, and the potential.
2.4.2 Approximation to the Laplacian
Finite Difference
Let us first numerically approximate the Laplacian using finite differences to the
differential operator. In the case of the second derivative, we have
ψ
(x) =
d
2
ψ(x)
dx 2
ψ(x i+1 ) − 2ψ(x i ) + ψ(x i−1 )
h 2
(2.108)
2 The Quantum Approach to the Two-Body Problem
Gaussian quadratures There are many different Gaussian quadratures (i.e., Gauss–
Hermite, Gauss–Laguerre, Gauss–Legendre, Gauss–Mehler to name a few). These
quadratures are highly accurate since the quadrature weights and abscissas (quadrature points) are determined to make some integrals exact as follows:
X nm ≡
b
a
xψ n (x)ψ m (x)W (x) =
N
i=1
ψ n (x i )x i ψ m (x i )w i
(2.106)
The N abscissas x i and N weights w i are chosen to make these integrals exact. It should
be noted that w i = W (x i ). The weighting function W (x) is a positive definite function
like e
−x in a Gauss–Laguerre quadrature or e
−x
2 for a Gauss–Hermite quadrature.
The X matrix is symmetric and hence its eigenvalues x i are real. In fact the abscissa
or quadrature points x i are just the eigenvalues of the matrix X. It is interesting to
note that the mth eigenvector of X is
√
w i ψ m (x i ) and thus one can easily evaluate the
quadrature weights w i . This method requires that we know the analytical functions
ψ n (x) and that we can analytically evaluate the matrix elements in Eq. 2.106. Then
one can approximately evaluate similar integrals of the form
I =
b
a
f (x)W (x)dx ≈
N
i=1
w i f (x i )
(2.107)
In this case, the abscissas are not equally spaced and none of quadrature points and
abscissas are the same if one goes from an N-term quadrature to a M- term quadrature.
In fact, the abscissas in a N term Gauss–Hermite quadratures are the zero of H N+1 (x).
Likewise, the abscissas for a Gauss–Legendre quadrature are the zeros of Legendre
polynomials, etc. These methods, therefore, do not allow the reuse of previously
calculated values that is instead easy to implement when using constant step size
methods by doubling the number of quadrature points.
Accordingly, we will discuss here, in more detail, only the following fixed stepsize
approaches: the Laplacian operator, the wave function, and the potential.
2.4.2 Approximation to the Laplacian
Finite Difference
Let us first numerically approximate the Laplacian using finite differences to the
differential operator. In the case of the second derivative, we have
ψ
(x) =
d
2
ψ(x)
dx 2
ψ(x i+1 ) − 2ψ(x i ) + ψ(x i−1 )
h 2
(2.108)