Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics
Подождите немного. Документ загружается.
Computational Techniques References 151
sometimes taking a step “uphill”. That is, given a set of
coordinates q and a desired distribution function f(q),
a trial step is taken from the ith configuration q
i
to the
next, depending on whether the ratio f(q
i
+1)/ f(q
i
) is
greater or less than one. If the ratio is greater than one,
the step is accepted, but if it is less than one, the step is
accepted with a probability given by the ratio.
8.4.3 Monte Carlo Integration
The basic idea of Monte Carlo integration is that if a large
number of points is generated uniformly randomly in
some n-dimensional space, the number falling inside
a given region is proportional to the volume, or definite
integral, of the function defining that region. Though this
idea is as true in one dimension as it is in n, unless there
is a large number (“large” could be as little as three) of
dimensions or the boundaries are quite complicated, the
numerical quadrature schemes described previously are
more accurate and efficient. However, since the Monte
Carlo approach is based on just sampling the function at
representative points rather than evaluating the function
at a large number of finely spaced quadrature points, its
advantage for very large problems is apparent.
For simplicity, consider the Monte Carlo method for
integrating a function of only one variable; the gener -
alization to n dimensions being straightforward. If we
generate N random points uniformly on (a, b),thenin
the limit of large N the integral is
b
a
f(x) dx ≈
1
N
f(x)
±
f
2
(x)
−
f(x)
2
N
,
(8.111)
where
f(x)
≡
1
N
N
i=1
f(x
i
) (8.112)
is the arithmetic mean. The probable error given is
appropriately a statistical one rather than a rigorous
error bound and is the one standard error limit. From
this one can see that the error decreases as only N
1/2
,
more slowly than the rate of decrease for the quadrature
schemes based on interpolation. Also, the accuracy is
greater for relatively smooth functions, since the Monte
Carlo generation of points is unlikely to sample nar-
rowly peaked features of the integrand well. To estimate
the integral of a multidimensional function with compli-
cated boundaries, simply find an enclosing volume and
generate points uniformly randomly within it. Keeping
the enclosing volume as close as possible to the vol-
ume of interest miminizes the number of points which
fall outside, and therefore increases the efficiency of the
procedure.
The Monte Carlo integral is related to techniques
for generating random numbers according to prescribed
distributions described in Sect. 8.4.2. If we consider
a normalized distribution w(x), known as the weight
function, then with the change of variables defined by
y(x) =
x
a
w(x
)dx
, (8.113)
the Monte Carlo estimate of the integral becomes
b
a
f(x) dx ≈
1
N
f
[
x (y)
]
w
[
x (y)
]
,
(8.114)
assuming that the transformation is invertible. Choosing
w(x) to behave approximately as f(x) allows a more
efficient generation of points within the boundaries of
the integrand. This occurs since the uniform distribution
of points y results in values of x distributed according to
w and therefore “close” to f . This procedure, generally
termed the reduction of variance of the Monte Carlo
integration, improves the efficiency of the procedure to
the extent that the transformed function f/w can be
made smooth, and that the sampled region is as small as
possible but still contains the volume to be estimated.
References
8.1 J. Stoer, R. Bulirsch: Introduction to Numerical Anal-
ysis (Springer, Berlin, Heidelberg 1980)
8.2 R. L. Burden, J. D. Faires, A. C. Reynolds: Numerical
Analysis (Prindle, Boston 1981)
8.3 W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vet-
terling: Numerical Recipes, the Art of Scientific
Computing (Cambridge Univ. Press, Cambridge
1992)
8.4 M. Abramowitz, I. A. Stegun (Eds.): Handbook
of Mathematical Functions, Applied Mathematics
Series, Vol. 55 (National Bureau of Standards, Wash-
ington, Dover, New York 1968)
Part A 8
152 Part A Mathematical Methods
8.5 D. E. Knuth: The Art of Computer Programming,Vol.2
(Addison-Wesley, Reading 1981)
8.6 G. Arfken: Mathematical Methods for Physicists (Aca-
demic Press, Orlando 1985)
8.7 H. Jeffreys, B. S. Jeffreys: Methods of Mathematical
Physics (Cambridge Univ. Press, Cambridge 1966)
8.8 S. E. Koonin, D. C. Meredith: Computational Physics
(Addison-Wesley, Redwood City 1990)
8.9 P. M. Morse, H. Feshbach: Methods of Theoretical
Physics (McGraw-Hill, New York 1953)
8.10 R. Courant, D. Hilbert: Methods of Mathematical
Physics (Interscience, New York 1953)
8.11 P. J. Huber: Robust Statistics (Wiley, New York 1981)
8.12 H. D. Young: Statistical Treatment of Experimental
Data (McGraw-Hill, New York 1962)
8.13 P. R. Bevington: Data Reduction and Error Analy-
sis for the Physical Sciences (McGraw-Hill, New York
1969)
8.14 D. C. Champeney: Fourier Transforms and Their Phys-
ical Applications (Academic, New York 1973)
8.15 R. W. Hamming: Numerical Methods for Scientists
and Engineers (McGraw-Hill, New York 1973)
8.16 D. F. Elliott, K. R. Rao: Fast Transforms: Algorithms,
Analyses, Applications (Academic, New York 1982)
8.17 D. Zwillinger: CRC Standard Mathematical Tables and
Formulae, 31st edn. (Chapman & Hall/CRC, New York
2002)
8.18 J. Lambert: Numerical Methods for Ordinary Differ-
ential Equations (Wiley, New York 1991)
8.19 L. F. Shampine: Numerical Solution of Ordinary Dif-
ferential Equations (Chapman Hall, New York 1994)
8.20 J. Butcher: The Numerical Analysis of Ordinary Differ-
ential Equations: Runge–Kutta and General Linear
Methods (Wiley, New York 1987)
8.21 G. Hall, J. Watt: Modern Numerical Methods for Ordi-
nary Differential Equations (Clarendon, Oxford 1976)
8.22 I. Gladwell, R. Wait (Eds.): A Survey of Numerical
Methods for Partial Differential Equations (Claren-
don, Oxford 1979)
8.23 K. Rektorys: Variational Methods in Mathematics,
Science, and Engineering, 2nd edn. (Reidel, Boston
1980)
8.24 D. Cook, D. S. Malkus, M. E. Plesha: Concepts and Ap-
plications of Finite Element Analysis, 3rd edn. (Wiley,
New York 1989)
8.25 G. Nurnberger: Approximation by Spline Functions
(Springer, Berlin, Heidelberg 1989)
8.26 C. DeBoor: Practical Guide to Splines (Springer, New
York 1978)
8.27 C. J. Joachain: Quantum Collision Theory (Elsevier,
New York 1983)
8.28 L. S. Rodberg, R. M. Thaler: Introduction to the Quan-
tum Theory of Scattering (Academic, New York 1967)
8.29 M. L. Goldberger, K. M. Watson: Collision Theory (Wi-
ley, New York 1964)
8.30 P. G. Ciarlet: Introduction to Numerical Linear Al-
gebra and Optimisation (Cambridge Univ. Press,
Cambridge 1989)
8.31 G. Golub, C. Van Loan: Matrix Computations,
2nd edn. (Johns Hopkins Univ. Press, Baltimore
1989)
8.32 W. Hackbusch: Iterative Solution of Large Sparse
Systems of Equations (Springer, New York 1994)
8.33 A. George, J. Liu: Computer Solution of Large Sparse
Positive Definite Systems (Prentice-Hall, Englewood
Cliffs 1981)
8.34 M. H. Kalos, P. A. Whitlock: The Basics of Monte Carlo
Methods (Wiley, New York 1986)
Part A 8
153
Hydrogenic Wa
9. Hydrogenic Wave Functions
This chapter summarizes the solutions of the
one-electron nonrelativistic Schrödinger equation,
and the one-electron relativistic Dirac equation,
for the Coulomb potential. The standard notations
and conventions used in the mathematics
literature for special functions have been chosen
in preference to the notations customarily
used in the physics literature whenever there
is a conflict. This has been done to facilitate
the use of standard reference works such as
Abramowitz and Stegun [9.1], the Bateman
project [9.2, 3], Gradshteyn and Ryzhik [9.4],
Jahnke and Emde [9.5], Luke [9.6, 7], Magnus,
Oberhettinger,andSoni [9.8], Olver [9.9],
Szego [9.10], and the new NIST Digital Library
of Mathematical Functions project, which is
preparing a hardcover update [9.11]ofAbramowitz
and Stegun [9.1] and an online digital library
of mathematical functions [9.12]. The section
on special functions contains many of the
formulas which are needed to check the results
quoted in the previous sections, together
with a number of other useful formulas. It
9.1 Schrödinger Equation .......................... 153
9.1.1 Spherical Coordinates ................ 153
9.1.2 Parabolic Coordinates ................ 154
9.1.3 Momentum Space ..................... 156
9.2 Dirac Equation .................................... 157
9.3 The Coulomb Green’s Function.............. 159
9.3.1 The Green’s Function
for the Schrödinger Equation...... 159
9.3.2 The Green’s Function
for the Dirac Equation................ 161
9.4 Special Functions................................. 162
9.4.1 Confluent Hypergeometric
Functions ................................. 162
9.4.2 Laguerre Polynomials ................ 166
9.4.3 Gegenbauer Polynomials............ 169
9.4.4 Legendre Functions ................... 169
References .................................................. 170
includes a brief introduction to asymptotic
methods.
References to the numerical evaluation of
special functions are given.
9.1 Schrödinger Equation
The nonrelativistic Schrödinger equation for a hydro-
genic ion of nuclear charge Z in atomic units is
−
1
2
∇
2
−
Z
r
ψ
(
r
)
= Eψ
(
r
)
.
(9.1)
9.1.1 Spherical Coordinates
The separable solutions of (9.1) in spherical coordinates
are
ψ
(
r
)
= Y
m
(
θ, φ
)
R
(
r
)
,
(9.2)
where Y
m
(
θ, φ
)
is a spherical harmonic as defined by
Edmonds [9.13]andR
(
r
)
is a solution of the radial
equation
−
1
2
d
2
dr
2
+
2
r
d
dr
−
(
+1
)
r
2
−
Z
r
R
(
r
)
= ER
(
r
)
.
(9.3)
The general solution to (9.3)is
R
(
r
)
=r
exp
(
ikr
)
[
A
1
F
1
(
a;c;z
)
+ BU
(
a, c, z
)
]
,
(9.4)
where
1
F
1
and U are the regular and irregular solu-
tions of the confluent hypergeometric equation defined
Part A 9
154 Part A Mathematical Methods
in (9.130)and(9.131)below,and
k =
√
2E , (9.5)
a = +1 −ik
−1
Z , (9.6)
c = 2 +2 , (9.7)
z =−2ikr . (9.8)
A and B are arbitrary constants. The solution given in
(9.4) has an r
−−1
singularity at r = 0 unless B = 0or
a is a non-positive integer. The leading term for small
r is proportional to r
when B = 0 and/or a is a non-
positive integer. The large r behavior of the solution for
(9.4) follows from (9.134), (9.135), and (9.164)below.
Bound state solutions, with energy
E =−
1
2
Z
2
n
−2
(9.9)
are obtained when a =−n + +1wheren > is the
principal quantum number. The properly normalized
bound state solutions are
R
n,
(
r
)
=
2Z
n
2
Z(n− −1)!
(n+)!
2Zr
n
×exp
(
−Zr/n
)
L
(2+1)
n−−1
(
2Zr/n
)
,
(9.10)
where L
(2+1)
n−−1
is the generalized Laguerre polynomial
defined in (9.187). The relation in (9.188)showsthat
1
F
1
and U are linearly dependent in this case, so that
(9.4) is no longer the general solution of (9.3). A lin-
early independent solution for this case can be obtained
by replacing the L
(2+1)
n−−1
(
2Zr/n
)
in (9.10) by the second
(irregular) solution M
(2+1)
n−−1
(
2Zr/n
)
of the Laguerre
equation [see (9.194), (9.196), and (9.197)]. The first
three R
n,
are
R
1,0
(
r
)
= 2Z
3/2
exp
(
−Zr
)
, (9.11)
R
2,0
(
r
)
=
1
2
Z
3/2
(
2 −Zr
)
exp
−
1
2
Zr
,
(9.12)
R
2,1
(
r
)
=
1
2
Z
3/2
1
3
1/2
Zr exp
−
1
2
Zr
.
(9.13)
Additional explicit expressions, together with graphs
of some of them, can be found in Pauling and Wil-
son [9.14].
The R
n,
can be expanded in powers of 1/n [9.15]
R
n,
(
r
)
=−
2Z
2
(
n +
)
!
(
n − −1
)
!n
2+4
1/2
× r
−1/2
∞
k=0
g
()
k
(
8Zr
)
1/2
n
−2k
,
(9.14)
where the functions g
()
k
(
z
)
are finite linear combina-
tions of Bessel functions:
g
()
k
(
z
)
= z
3k
k
m=0
a
()
k,m
J
2+2m+k+1
(
z
)
.
(9.15)
The coefficients a
()
k,m
in (9.15) are calculated recursively
from
a
()
k,m
=
(
2 +2m +k +1
)
32
(
2k+m
)(
2 +m +2k+1
)
×
1
(
2 +2m +k −1
)
×
(
2 +2m +k −1
)
a
()
k−1,m
+ 32
(
k−m +1
)(
2 +m −k
)
a
()
k,m−1
,
(9.16)
starting with the initial condition
a
()
0,0
= 1 . (9.17)
The expansion (9.14) converges uniformly in r for r in
any bounded region of the complex r plane. However, it
converges fast enough so that a few terms give a good
description of R
n,
only if r is small. The square root
in (9.14) has not been expanded in inverse powers of
n because it has a branch point at 1/n = 1/ which
would reduce the radius of convergence of the expansion
to 1/. In some cases, large n expansions of matrix
elements can be obtained by inserting (9.14)forR
n,
and integrating term by term; examples can be found
in Drake and Hill [9.15]. An asymptotic expansion in
powers of 1/n, which is valid from r equaltoanarbitrary
fixed positive number through the turning point at r =
2n
2
/Z out to r =∞, can be assembled from (9.133),
(9.166)–(9.181), and (9.188)below.
The R
n,
are not a complete set because the con-
tinuum has been left out. The Sturmian functions ρ
k,
,
given by
ρ
k,
(
β;r
)
=
β
3
k !
Γ
(
k+2 +3
)
(
βr
)
e
−βr/2
× L
(2+2)
k
(
βr
)
,
(9.18)
Part A 9.1
Hydrogenic Wave Functions 9.1 Schrödinger Equation 155
do form a complete orthonormal set. The positive con-
stant β, which is independent of k and , sets the length
scale for the basis set (9.18).
9.1.2 Parabolic Coordinates
The Schrödinger equation (9.1) is separable in para-
bolic coordinates ξ, η, φ, which are related to spherical
coordinates r, θ, φ via
ξ =r +z = r
[
1 +cos
(
θ
)
]
,
(9.19)
η =r −z = r
[
1 −cos
(
θ
)
]
, (9.20)
φ = φ. (9.21)
This separability in a second coordinate system is related
to the existence of a “hidden” O(4) symmetry, which is
also responsible for the degeneracy of the bound states
[9.16, 17]. The solutions in parabolic coordinates are
particularly convenient for derivations of the Stark ef-
fect and the Rutherford scattering cross section. The
separable solutions of (9.1) in parabolic coordinates are
ψ
(
r
)
= exp
(
imφ
)
N
(
η
)
Ξ
(
ξ
)
,
(9.22)
where
N
(
η
)
= η
|m|/2
exp
1
2
ik
1
η
[
A
1
F
1
(
a
1
;c;−ik
1
η
)
+ BU
(
a
1
, c, −ik
1
η
)
]
, (9.23)
Ξ
(
ξ
)
= ξ
|m|/2
exp
1
2
ik
2
ξ
[
C
1
F
1
(
a
2
;c;−ik
2
ξ
)
+ DU
(
a
2
, c, −ik
2
ξ
)
]
, (9.24)
with
1
F
1
and U defined in (9.130), (9.131)below,and
k
1
=±k
2
=±
√
2E , (9.25)
a
1
=
1
2
(
|m|+1
)
−ik
−1
1
µ, (9.26)
a
2
=
1
2
(
|m|+1
)
−ik
−1
2
(
Z −µ
)
,
(9.27)
c =|m|+1 . (9.28)
A, B, C,andD are arbitrary constants; µ is the
separation constant. An important special case is the
well-known Coulomb function
ψ
C
(
r
)
= Γ
1 −ik
−1
Z
exp
1
2
πk
−1
Z +ik· r
×
1
F
1
ik
−1
Z;1;i
(
kr −k· r
)
,
(9.29)
which is obtained by orienting the z-axis in the k direc-
tion and taking m = 0, −k
1
= k
2
=|k|, µ = Z +
1
2
i|k|.
ψ
C
is normalized to unit incoming flux [see (9.34)be-
low]. In applications, Z is often replaced by −Z
1
Z
2
,so
that the Coulomb potential in (9.1) becomes +Z
1
Z
2
/r.
Equation (9.232), the addition theorem for the spheri-
cal harmonics ([9.13] p. 63 Eq. 4.6.6), and the λ =c = 1
special case of (9.163) below can be used to expand ψ
C
into an infinite sum of solutions of the form (9.2):
ψ
C
(
r
)
= 4π
∞
=0
m=−
Γ
+1 −ik
−1
Z
(
2 +1
)
!
×
(
−2ik
)
e
πk
−1
Z/2
Y
∗
m
(
θ
k
,φ
k
)
Y
m
(
θ, φ
)
× r
e
ikr
1
F
1
+1 −ik
−1
Z;2 +2;−2ikr
,
(9.30)
where k, θ
k
,andφ
k
are the spherical coordinates of k.
ψ
C
can be split into an incoming plane wave and an
outgoing spherical wave with the aid of (9.134)below:
ψ
C
(
r
)
= ψ
in
(
r
)
+ψ
out
(
r
)
,
(9.31)
where
ψ
in
(
r
)
= exp
ik· r −
1
2
πk
−1
Z
× U
ik
−1
Z;1;i
(
kr −k· r
)
,
(9.32)
ψ
out
(
r
)
=−
Γ
1 −ik
−1
Z
Γ
ik
−1
Z
exp
ikr −
1
2
πk
−1
Z
× U
1 −ik
−1
Z;1;−i
(
kr −k· r
)
.
(9.33)
The functions ψ
in
and ψ
out
can be expanded for kr −k· r
large with the aid of (9.164). The result is
ψ
in
(
r
)
∼exp
ik· r −ik
−1
Z ln
(
kr −k· r
)
×
∞
n=0
(
−i
)
n
n!
Γ
ik
−1
Z +n
Γ
ik
−1
Z
2
×
(
kr −k· r
)
−n
, (9.34)
ψ
out
(
r
)
∼−
iΓ
1 −ik
−1
Z
Γ
ik
−1
Z
(
kr −k· r
)
×exp
ikr −ik
−1
Z ln
(
kr −k· r
)
×
∞
n=0
i
n
n!
Γ
1 −ik
−1
Z +n
Γ
1 −ik
−1
Z
2
×
(
kr −k· r
)
−n
. (9.35)
Part A 9.1
156 Part A Mathematical Methods
Because (9.1) is an elliptic partial differential equa-
tion, its solutions must be analytic functions of the
Cartesian coordinates (except at r = 0, where the so-
lutions have cusps). The n = 0 special case of (9.138)
shows that ψ
in
and ψ
out
are logarithmically singular at
k· r = kr. Thus ψ
in
and ψ
out
are not solutions to (9.1)at
k· r = kr. The logarithmic singularity cancels when ψ
in
and ψ
out
are added to form ψ
C
,whichis a solution to
(9.1).
Bound state solutions, with energy
E =−
1
2
Z
2
(
n
1
+n
2
+|m|+1
)
−2
, (9.36)
are obtained when a
1
=−n
1
and a
2
=−n
2
where n
1
and
n
2
are non-negative integers. The properly normalized
bound state solutions, which can be put into one–one cor-
respondence with the bound state solutions in spherical
coordinates, are
ψ
n
1
,n
2
,m
(η,ξ,φ)
=
β
2|m|+4
n
1
!n
2
!
2πZ
(
n
1
+|m|
)
!
(
n
2
+|m|
)
!
×exp
imφ −
1
2
β
(
η +ξ
)
×
(
ηξ
)
|m|/2
L
(|m|)
n
1
(
βη
)
L
(|m|)
n
2
(
βξ
)
,
(9.37)
where
β = Z
(
n
1
+n
2
+|m|+1
)
−1
. (9.38)
9.1.3 Momentum Space
The nonrelativistic Schrödinger equation (9.1) becomes
the integral equation
1
2
p
2
φ
(
p
)
−
Z
2π
2
φ
p
(
p− p
)
2
d
3
p
= Eφ
(
p
)
(9.39)
in momentum space. Its solutions are related to the so-
lutions in coordinate space via the Fourier transforms
ψ
(
r
)
=
(
2π
)
−3/2
exp
(
i p· r
)
φ
(
p
)
d
3
p , (9.40)
φ
(
p
)
=
(
2π
)
−3/2
exp
(
−ip· r
)
ψ
(
r
)
d
3
r .
(9.41)
AtrickofFock’s [9.16, 18] can be used to expose the
“hidden” O(4) symmetry of hydrogen and construct the
bound state solutions to (9.39). Let p, θ
p
, φ
p
and p
,
θ
p
, φ
p
be the spherical coordinates of p and p
. Change
variables from p, p
to χ, χ
via p =
√
−2E tan
(
χ/2
)
and p
=
√
−2E tan
χ
/2
. This brings (9.39)tothe
form
2π
2
Z
−1
√
−2E
sec
4
(
χ/2
)
φ
(
p
)
=
sec
4
χ
/2
φ
p
sin
2
χ
dχ
sin
θ
p
dθ
p
dφ
p
2 −2
[
cos
(
χ
)
cos
(
χ
)
+sin
(
χ
)
sin
(
χ
)
cos
(
γ
)
]
,
(9.42)
where γ
is the angle between p and p
. Equation (9.42)
is solved by introducing spherical coordinates and spher-
ical harmonics in four dimensions via a natural extension
of the procedure used in three dimensions. Going to polar
coordinates on x and y yields the cylindrical coordinates
r
2
, φ, z; the further step of going to polar coordinates on
r
2
and z yields spherical coordinates r
3
, θ, φ.Ifthereis
a fourth coordinate w, spherical coordinates in four di-
mensions are obtained via the additional step of going
to polar coordinates on r
3
and w. The result is
x =r
4
sin
(
χ
)
sin
(
θ
)
cos
(
φ
)
, (9.43)
y = r
4
sin
(
χ
)
sin
(
θ
)
sin
(
φ
)
, (9.44)
z =r
4
sin
(
χ
)
cos
(
θ
)
, (9.45)
w =r
4
cos
(
χ
)
. (9.46)
The volume element, which is easily obtained via the
same series of transformations, is
dV =r
3
4
dr
4
dΩ
4
, (9.47)
dΩ
4
= sin
2
(
χ
)
dχ sin
(
θ
)
dθ dφ.
(9.48)
The four-dimensional spherical harmonics [9.2,Vol.2,
Chap. XI] are
Y
n,,m
(
χ, θ, φ
)
= 2
+1
!
n
(
n − −1
)
!
2π
(
n +
)
!
sin
(
χ
)
× C
+1
n−−1
[
cos
(
χ
)
]
Y
m
(
θ, φ
)
,
(9.49)
where C
+1
n−−1
is a Gegenbauer polynomial and n ≥ +
1 is an integer. They have the orthonormality property
Y
∗
n,,m
(
χ, θ, φ
)
Y
n
,
,m
(
χ, θ, φ
)
dΩ
4
= δ
n,n
δ
,
δ
m,m
. (9.50)
Part A 9.1
Hydrogenic Wave Functions 9.2 Dirac Equation 157
Equations (9.229)and(9.230) with λ = 1, equation
(9.231), and the addition theorem for the three dimen-
sional spherical harmonics Y
m
can be used to show
that
1 −2
cos
(
χ
)
cos
χ
+ sin
(
χ
)
sin
χ
cos
γ
t +t
2
−1
=
∞
n=1
n−1
=0
m=−
2π
2
n
t
n−1
Y
n,,m
(
χ, θ, φ
)
× Y
∗
n,,m
χ
,θ
,φ
(9.51)
holds for |t|< 1, where γ
is the angle between p and p
.
Multiply both sides of (9.51)byY
n,,m
χ
,θ
,φ
dΩ
4
(where dΩ
4
is dΩ
4
with χ, θ, φ replaced by χ
, θ
, φ
)
and use the orthogonality relation (9.50). The result can
be rearranged to the form
2π
2
n
−1
t
n−1
Y
n,,m
(
χ, θ, φ
)
=
Y
n,,m
χ
,θ
,φ
sin
2
χ
dχ
sin
(
θ
)
dθ dφ
1−2
[
cos
(
χ
)
cos
(
χ
)
+sin
(
χ
)
sin
(
χ
)
cos
(
γ
)
]
t+t
2
.
(9.52)
Analytic continuation can be used to show that (9.52)
is valid for all complex t despite the fact that (9.51)is
restricted to |t| < 1. Comparing the t = 1 case of (9.52)
with (9.42)showsthatE =−
1
2
Z
2
n
−2
in agreement with
(9.9), and that
φ
(
p
)
=
normalizing
factor
cos
4
(
χ/2
)
× Y
n,,m
(
χ, θ, φ
)
.
(9.53)
Transforming from χ back to p brings these to the form
φ
(
p
)
= Y
m
θ
p
,φ
p
F
n,
(
p
)
,
(9.54)
where the properly normalized radial functions are
F
n,
(
p
)
= 2
2+2
n
2
!
2(n− −1)!
πZ
3
(n+)!
np
Z
×
Z
2+4
n
2
p
2
+Z
2
+2
× C
+1
n−−1
n
2
p
2
−Z
2
n
2
p
2
+Z
2
.
(9.55)
The first three F
n,
are
F
1,0
(
p
)
= 4
2
πZ
3
Z
4
p
2
+Z
2
2
, (9.56)
F
2,0
(
p
)
=
32
√
πZ
3
Z
4
4p
2
−Z
2
4p
2
+Z
2
3
, (9.57)
F
2,1
(
p
)
=
128
√
3πZ
3
Z
5
p
4p
2
+Z
2
3
. (9.58)
The F
n,
satisfy the integral equation
1
2
p
2
F
n,
(
p
)
−
Z
πp
∞
0
Q
p
2
+ p
2
2pp
F
n,
p
p
dp
= EF
n,
(
p
)
,
(9.59)
which can be obtained by inserting (9.54)in(9.39). Here
Q
is the Legendre function of the second kind, which
is defined in (9.233)below.
9.2 Dirac Equation
The relativistic Dirac equation for a hydrogenic ion of
nuclear charge Z can be reduced to dimensionless form
by using the Compton wavelength
/
(
mc
)
for the length
scale and the rest mass energy mc
2
for the energy scale.
The result is
−iα · ∇+β −
Zα
r
ψ
(
r
)
= Eψ
(
r
)
,
(9.60)
where α = e
2
/
(
c
)
is the fine structure constant, and α,
β are the usual Dirac matrices:
α =
0 σ
σ 0
,β=
I 0
0 −I
.
(9.61)
Here σ is a vector whose components are the two by two
Pauli matrices, and I is the two by two identity matrix
Part A 9.2
158 Part A Mathematical Methods
given by
σ
x
=
01
10
,σ
y
=
0 −i
i0
,
σ
z
=
10
0 −1
, I =
10
01
.
(9.62)
The solutions to (9.60) in spherical coordinates have the
form
ψ
(
r
)
=
G
(
r
)
χ
m
κ
(
θ, φ
)
iF
(
r
)
χ
m
−κ
(
θ, φ
)
,
(9.63)
where, for positive energy states, G
(
r
)
is the radial part
of the large component and iF
(
r
)
is the radial part of
the small component. For negative energy states, G
(
r
)
is the radial part of the small component and iF
(
r
)
is
the radial part of the large component. χ is the two
component spinor
χ
m
κ
=
−
κ
|κ|
κ +
1
2
−m
2κ +1
1/2
Y
|κ+
1
2
|−
1
2
,m−
1
2
κ +
1
2
+m
2κ +1
1/2
Y
|κ+
1
2
|−
1
2
,m+
1
2
.
(9.64)
The relativistic quantum number κ is related to the total
angular momentum quantum number j by
κ =±
j +
1
2
.
(9.65)
Because j takes on the values
1
2
,
3
2
,
5
2
, ... , κ is restricted
to the values ±1, ±2, ±3, ... . The spinor χ
m
κ
obeys
the useful relations
σ ·
ˆ
r χ
m
κ
=−χ
m
−κ
, (9.66)
σ · Lχ
m
κ
=−
(
κ +1
)
χ
m
κ
, (9.67)
where
ˆ
r =r/r and L = r × p with p =−i∇. Equations
(9.66), (9.67), and the identity
σ · p =
σ ·
ˆ
r
ˆ
r · p +
iσ · L
r
(9.68)
can be used to derive the radial equations, which are
d
dr
+
1 +κ
r
G
(
r
)
−
1 +E +
Zα
r
F
(
r
)
= 0 ,
(9.69)
d
dr
+
1 −κ
r
F
(
r
)
−
1 −E −
Zα
r
G
(
r
)
= 0 .
(9.70)
Equations (9.158), (9.159), (9.161), and (9.162)below
can be used to show that the general solution to (9.69)
and (9.70)is
G
(
r
)
=r
γ
exp
(
−λr
)(
1 +E
)
1/2
{
A
[
f
2
(
r
)
+ f
1
(
r
)
]
+B
[
f
4
(
r
)
+ f
3
(
r
)
]
}
,
(9.71)
F
(
r
)
=r
γ
exp
(
−λr
)(
1 −E
)
1/2
{
A
[
f
2
(
r
)
− f
1
(
r
)
]
+B
[
f
4
(
r
)
− f
3
(
r
)
]
}
,
(9.72)
where
f
1
(
r
)
=
Zαλ
−1
−κ
1
F
1
(
a;c;2λr
)
,
(9.73)
f
2
(
r
)
= a
1
F
1
(
a+1;c;2λr
)
,
(9.74)
f
3
(
r
)
=U
(
a, c, 2λr
)
,
(9.75)
f
4
(
r
)
=
Zαλ
−1
+κ
U
(
a+1, c, 2λr
)
, (9.76)
λ =
(
1 +E
)
1/2
(
1 −E
)
1/2
, (9.77)
γ =−1 +
κ
2
−Z
2
α
2
1/2
, (9.78)
a = 1 +γ −λ
−1
EZα, (9.79)
c = 3 +2γ. (9.80)
A and B are arbitrary constants. Because γ is in general
not an integer, the solutions have a branch point at r =0,
and become infinite at r = 0whenκ =±1, which makes
γ negative. The solutions for E < −1andE > +1are
in the continuum, which implies that one of the factors
(
1 +E
)
1/2
,
(
1 −E
)
1/2
is real with the other imaginary.
Square integrable solutions, with energy
E
n,κ
=
Z
|Z|
1 +
Z
2
α
2
(
n +1 +γ
)
2
−1/2
, (9.81)
are obtained when a =−n where n is a non-negative
integer. The properly normalized square integrable solu-
tions are
G
n,κ
(
r
)
= C
n,κ
(
2λr
)
γ
exp
(
−λr
)
1 +E
n,κ
1/2
×
g
(2)
n,κ
(
r
)
+g
(1)
n,κ
(
r
)
,
(9.82)
F
n,κ
(
r
)
= C
n,κ
(
2λr
)
γ
exp
(
−λr
)
1 −E
n,κ
1/2
×
g
(2)
n,κ
(
r
)
−g
(1)
n,κ
(
r
)
,
(9.83)
g
(1)
n,κ
(
r
)
=
Zαλ
−1
−κ
1/2
L
(2+2γ)
n
(
2λr
)
,
(9.84)
Part A 9.2
Hydrogenic Wave Functions 9.3 The Coulomb Green’s Function 159
g
(2)
n,κ
(
r
)
=−
(
n +2 +2γ
)
Zαλ
−1
−κ
−1/2
× L
(2+2γ)
n−1
(
2λr
)
,
(9.85)
C
n,κ
=
2λ
4
n!
ZαΓ
(
n +3 +2γ
)
.
(9.86)
When n = 0, |Zαλ
−1
|=|κ|, and the value of κ whose
sign is the same as the sign of Zαλ
−1
is not permitted.
Also, L
(2+2γ)
−1
(
2λr
)
is counted as zero, so that g
(2)
0,κ
(
r
)
=
0.
The eigenvalues and eigenfunctions for the first four
states for Z > 0 will now be written out explicitly in
terms of the variable ρ = Zαr.Forthe1S
1/2
ground
state, with n =0, j =
1
2
, κ =−1, the formulae are
E
0,−1
=
1 −Z
2
α
2
, (9.87)
G
0,−1
(
r
)
=
4Z
3
α
3
1 +E
0,−1
Γ
1 +2E
0,−1
(
2ρ
)
E
0,−1
−1
e
−ρ
,
(9.88)
F
0,−1
(
r
)
=−
4Z
3
α
3
1 −E
0,−1
Γ
1 +2E
0,−1
×
(
2ρ
)
E
0,−1
−1
e
−ρ
. (9.89)
Theformulaeforthe2S
1/2
excited state, with n = 1,
j =
1
2
, κ =−1, and for the 2 P
1/2
excited state, with
n = 1, j =
1
2
, κ = 1, can be written together. They are
E
1,κ
=
1
2
+
1
2
1 −Z
2
α
2
1/2
, (9.90)
G
1,κ
(
r
)
=
Z
3
α
3
2E
1,κ
−κ
1 +E
1,κ
2E
2
1,κ
Γ
4E
2
1,κ
+1
× ρ
2E
2
1,κ
−2
1
e
−ρ
1
/2
×
2E
1,κ
−κ −1
2E
1,κ
+κ
−ρ
1
,
(9.91)
F
1,κ
(
r
)
=−
Z
3
α
3
2E
1,κ
−κ
1 −E
1,κ
2E
2
1,κ
Γ
4E
2
1,κ
+1
× ρ
2E
2
1,κ
−2
1
e
−ρ
1
/2
×
2E
1,κ
−κ +1
2E
1,κ
+κ
−ρ
1
,
(9.92)
where ρ
1
= ρ/E
1,κ
.Forthe2P
3/2
excited state, with
n = 0, j =
1
2
, κ =−2, the formulae are
E
0,−2
=
1 −
1
4
Z
2
α
2
, (9.93)
G
0,−2
(
r
)
=
Z
3
α
3
1 +E
0,−2
2Γ
1 +4E
0,−2
ρ
2E
0,−2
−1
e
−ρ/2
,
(9.94)
F
0,−2
(
r
)
=−
Z
3
α
3
1 −E
0,−2
2Γ
1 +4E
0,−2
ρ
2E
0,−2
−1
e
−ρ/2
.
(9.95)
9.3 The Coulomb Green’s Function
The abstract Green’s operator for a Hamiltonian H is
the inverse G
(
E
)
=
(
H −E
)
−1
. It is used to write the
solution to
(
H −E
)
|ξ=|η in the form |ξ=G|η.It
has the spectral representation
G
(
E
)
=
j
1
E
j
−E
|e
j
e
j
| . (9.96)
The sum over j in (9.96) runs over all of the spectrum
of H, including the continuum. For the bound state part
of the spectrum, the numbers E
j
and vectors |e
j
are the
eigenvalues and eigenvectors of H. For the continuous
spectrum, |e
j
e
j
|is a projection valued measure [9.19].
The representation (9.96)showsthatG
(
E
)
has first
order poles at the eigenvalues. The reduced Green’s op-
erator (also known as the generalized Green’s operator),
which is the ordinary Green’s operator with the singu-
lar terms subtracted out, remains finite when E is at an
eigenvalue. It can be calulated from
G
(red)
(
E
k
)
= lim
E→E
k
∂
∂E
[
(
E − E
k
)
G
(
E
)
]
.
(9.97)
The coordinate and momentum space representatives of
the abstract Green’s operator are the Green’s functions.
The nonrelativistic Coulomb Green’s function has been
discussed by Hostler and Schwinger [9.20, 21]. A uni-
fied treatment of the Coulomb Green’s functions for
the Schrödinger and Dirac equations has been given by
Swainson and Drake [9.22]. Reduced Green’s functions
are discussed in the third of the Swainson–Drake papers,
and in the paper of Hill and Huxtable [9.23].
Part A 9.3
160 Part A Mathematical Methods
9.3.1 The Green’s Function for the
Schrödinger Equation
The Green’s function G
(S)
for the Schrödinger equation
(9.98) is a solution of
−
1
2
∇
2
−
Z
r
−E
G
(S)
r, r
; E
= δ
r −r
.
(9.98)
An explicit closed form expression for G
(S)
is
G
(S)
r, r
; E
=
Γ
(
1 −ν
)
2π|r−r
|
×
W
ν,
1
2
(
z
2
)
∂
∂z
1
M
ν,
1
2
(
z
1
)
− M
ν,
1
2
(
z
1
)
∂
∂z
2
W
ν,
1
2
(
z
2
)
,
(9.99)
where M
ν,1/2
and W
ν,1/2
are the Whittaker functions
defined in (9.132)and(9.133)below,and
ν = Z
(
−2E
)
−1/2
, (9.100)
z
1
=
(
−2E
)
1/2
r +r
−|r −r
|
, (9.101)
z
2
=
(
−2E
)
1/2
r +r
+|r −r
|
. (9.102)
Thebranchonwhich
(
−2E
)
1/2
is positive should be
taken when E < 0. When E > 0, the branch which cor-
responds to incoming (or outgoing) waves at infinity can
be selected with the aid of the asymptotic approximation
G
(S)
r, r
; E
≈
Γ
(
1 −ν
)
2π|r−r
|
z
ν
2
exp
−
1
2
z
2
,
(9.103)
which holds when z
2
z
1
. This approximation is ob-
tained by using (9.130), (9.132), (9.133), and (9.164)in
(9.99). A number of useful expansions for G
(S)
can be
obtained from the integral representation
G
(S)
r, r
; E
=
2Z
ν
∞
0
coth
1
2
ρ
2ν
sinh
(
ρ
)
× I
0
!
ν
−1
Z sinh
(
ρ
)
2rr
1/2
[
1 +cos
(
Θ
)
]
"
×exp
−ν
−1
Z
r +r
cosh
(
ρ
)
dρ,
(9.104)
where Θ is the angle between r and r
. These expansions,
and other integral representations, can be found in [9.20–
22]. The partial wave expansion of G
(S)
is
G
(S)
r, r
; E
=
,m
g
(S)
r, r
;ν
Y
m
(
θ, φ
)
Y
∗
m
θ
,φ
.
(9.105)
The radial Green’s function g
(S)
is a solution of the radial
equation
−
1
2
d
2
dr
2
+
2
r
d
dr
−
(
+1
)
r
2
−
Z
r
−E
g
(S)
r, r
;ν
=
δ
r −r
rr
. (9.106)
The standard method for calculating the Green’s func-
tion of a second order ordinary differential equation
([9.24] pp. 354–355) yields
g
(S)
r, r
;ν
=
(
2Z
)
2+2
Γ
(
+1 −ν
)
(
2 +1
)
!ν
2+1
exp
−ν
−1
Z
r +r
×
rr
1
F
1
+1 −ν;2 +2;2ν
−1
Zr
<
[3pt]
× U
+1 −ν, 2 +2, 2ν
−1
Zr
>
,
(9.107)
where r
<
is the smaller of the pair r, r
and r
>
is the
larger of the pair r, r
.Matrixelementsofg
(S)
can be
calculated with the aid of the formula for the double
Laplace transform, which is
∞
0
dr
∞
0
dr
rr
+1
exp
−λr−λ
r
g
(S)
r, r
;ν
=
2
(
2 +1
)
!
−ν +1
ν
2Z
2+3
4Z
2
(
νλ +Z
)(
νλ
+Z
)
2+2
×
2
F
1
(
2 +2,−ν +1; −ν +2;1 −ζ
)
,
(9.108)
where
ζ =
2νZ
λ +λ
(
νλ +Z
)(
νλ
+Z
)
,
(9.109)
Matrix elements with respect to Slater orbitals can be
calculated from (9.108) by taking derivatives with re-
spect to λ and/or λ
to bring down powers of r and r
.
Matrix elements with respect to Laguerre polynomials
can be calculated by using (9.108) to evaluate integrals
Part A 9.3