Lecture Note for

II. Hydrogen Atom: Nonrelativistic and Relativistic
Solutions

¡¤ MCDF2002 code uses relativistic notations

¡¤ Nonrelativistic structure theory uses the LS
coupling

¡¤ Relativistic structure theory uses the jj
coupling

¡¤ Real atomic
properties are described better by intermediate coupling than pure LS or
pure jj coupling

¡¤ Low-lying levels of light atoms (Z < 40)
are better described by the LS coupling than the jj coupling

¡¤ Energy levels of heavier atoms are closer to
the jj coupling than the LS coupling

¡¤ Relativistic structure and collision
theories have been developed in parallel to nonrelativistic theories

¡¤ The LS coupling will be used to describe
theories in this course, and modifications necessary to use the jj coupling
will be pointed out as needed.

II A. Nonrelativistic,
bound-state wave functions of H:

¡¤ Atomic
structure theory in most cases works with time-independent
solutions, i.e., steady-state solutions

*Y** *(*r**,t*) = *y** *(** r**) e

Radial and
angular parts are separable (central field approx.)

*y** *(** r**) =

¡¤ For computation, it is
more convenient to use

*P _{nl }*(

¡¤ Otherwise, *R _{ns}*(

¡¤ Nonrelativistic wave functions are the
solutions of the Schrödinger equation

¡¤ See Eqs. (1)—(3) of the lecture note. Atomic unit is used for the total energy

¡¤ Total energies for the hydrogenic ions
(known as the isoelectronic sequence) scales as *Z*^{2}

¡¤ The total energies for the hydrogenic ions
depend on *n* but not on *l*, although the differential equation for
the radial function contains *l*-dependent term, *l*(*l*+1)/*r*^{2}

¡¤ This energy degeneracy on *n* and *l*
is true only for pure Coulomb field, such as the
hydrogenic isoelectronic sequence

¡¤ For the discrete spectrum, allowed values of
*l* are 0, 1, 2,¡¦,n-1.

¡¤ For the continuous spectrum, ** k**
=

*E* = - *Z*^{2}/2*n*^{2}

*n*^{2}
= - *Z*^{2}/2*E*
(*E* > 0 for continuum states)

*n*
= -*iZ*/(2*E*)^{1/2}

*E* = *p*^{2}/2*m* = (*kћ*)^{2}/2*m*
= *k*^{2}/2 (in a.u.)

*n* = -*iZ*/*k*
(in a.u.)

*k* = -*iZ*/*n*
(in a.u.)

Momentum
= wave vector in a.u.

Ö(kinetic energy
in rydberg) = momentum in a.u.

¡¤ Unlike the discrete spectrum, allowed values
of *l* are infinite, i.e., *l* = 0, 1, 2,¡¦,¥ are allowed for any *k*

¡¤ Hence, the stationary
continuum orbital

*y** _{klm }*(

requires an infinite
expansion in *l*

¡¤ In practice, the upper limit of *l* for
a given *k* depends on the continuum energy and the desired accuracy: Higher the
energy and accuracy, more *l* are needed

¡¤ Orthonormality requirements are similar in
the wave functions for the discrete and continuous spectra. Since the spherical harmonics are
orthonormal by themselves, radial functions must be made orthonormal if *l*
and *m* are the same:

¡ò *P _{nl}*(

¡¤ The normalization of a continuum radial
function is defined by Eq. (8) of the lecture note. *T* and *T¡¯* in Eq. (8) can either be the wave vector *k* or the
continuum energy *E*

¡¤ Continuum radial functions with different
wave vectors *k* or energies *E* are orthogonal, i.e., the integral
of their product vanishes

¡¤ Question

Are the
bound-state and continuum state eigenfunctions, Eqs. (6) and (7),
orthogonal? Try *y*_{1s} and *y*_{k}_{s}, *k* = 1
a.u.

¡¤ Challenge

What is the
continuum solution for *k* = 0?

¡¤ The total energy *E* is also the
kinetic energy *w* of the continuum electron only for a one-electron atom.

¡¤ In a many-electron system, *w* = *k*^{2}/2
©ö *E*.

¡¤ Unlike wave functions for the bound states,
continuum wave functions (= Coulomb functions) have many different forms:

Coulomb
function for attractive potential (electron-nucleus interaction)

Coulomb
function for repulsive potential (electron-electron interaction; proton-proton
interaction)

Coulomb
function for incoming wave

Coulomb
function for outgoing wave

Coulomb
function for stationary wave

Analytic
expressions for all 5 cases

(See the QM
textbook by Landau and Lifshitz)

¡¤ In addition, radial functions may be
normalized on the momentum scale (*k*) or the energy scale (*w*). So, there are many different forms of
Coulomb functions for a given continuum energy!

Continuum
radial functions are defined per momentum interval or energy interval

¡¤ Must use the correct
form of the continuum wave functions.
Otherwise, may get nonsense results

¡¤ Coulomb functions are defined in terms of
the confluent hypergeometric functions (CHFs). In general, CHFs are infinite
series. Only when the first parameter
*a* in *F*(*a,b*;*z*) is 0 or a negative integer, CHFs
become a polynomial

¡¤ For cross sections, energy normalized radial
functions are used more often

II B. Bound
Solutions of H: Relativistic

¡¤ Dirac Equation

Unlike the
Schrödinger equation, the Dirac equation is a first-order, 4-component coupled
differential equation. It contains
two radial functions, the large-component and
small-component radial functions.

¡¤ Each radial function has spin-up and
spin-down angular components, making the total wave function to have 4
components

¡¤ The Dirac Hamiltonian is linear in ** p**,
while the Schrödinger Hamiltonian is quadratic in

¡¤ The Dirac Hamiltonian contains the rest mass
(= *mc*^{2}) term

¡¤ The Dirac Hamiltonian has positive and
negative energy bands. The bound
levels of H belong to the positive energy band:

Example: *E*(1s)
= *mc*^{2} - 13.6 eV

¡¤ The negative energy spectrum is continuous,
i.e., no bound levels in the negative energy spectrum

¡¤ The Dirac wave functions are identified by
the principal quantum number *n*, the Dirac quantum number *k*, and the projection *m*
of the angular momentum *j*

¡¤ The Dirac quantum number *k* combines *j* and *l*
but *k* is not an
angular momentum variable

(See Table
II)

¡¤ The large component reduces to
nonrelativistic radial functions, while the small component vanishes in the
nonrelativistic limit

¡¤ Eq. (20) of the lecture note follows the
phase convention used by M. E. Rose [*Relativistic Electron Theory* (John
Wiley & Sons, New York, 1961)]

¡¤ There are different phase conventions:

Sometimes,
the positions of the large component (*P _{n}*

Sometimes,
the phase factor *i* is placed in front of the large component, instead of
the small component

¡¤ The fine-structure constant *a* appears explicitly in relativistic
wave functions and energy eigenvalues

¡¤ The speed of light *c* in atomic unit
is 1/*a* » 137

¡¤ In the limit c ¢ç ¥ (same as *a** *¢ç 0), all relativistic
formulas should reduce to corresponding nonrelativistic formulas

¡¤ Work out Ex. 1 and 2 to familiarize with the
transformation between relativistic and nonrelativistic formulas

¡¤ Nonrelativistic energies for H(2s_{1/2}),
H(2p_{1/2}) and H(2p_{3/2}) are degenerate

¡¤ Relativistic energies for the hydrogen atom
depend on *n* and *j*.
The Dirac energy for H(2s_{1/2}) and H(2p_{1/2}) are
degenerate, but the energy for H(2p_{3/2}) is higher

¡¤ In reality, the Lamb shift removes the
degeneracy between H(2s_{1/2}) and H(2p_{1/2}). With the Lamb shift, H(2p_{1/2})
is lower than H(2s_{1/2})

¡¤ Although the Dirac Hamiltonian introduced
spin and relativity to the Schr`dinger
Hamiltonian, it did not include the interaction of the bound electron with its
own electromagnetic field.

¡¤ This is known as the quantum electrodynamic
(QED) corrections. There is no Hamiltonian to represent QED corrections

¡¤ The most important QED corrections are the
self-energy correction, and the vacuum polarization correction

¡¤ These corrections apply to every level. The self-energy correction is positive
and large, while the vacuum polarization correction is negative (most of the
time) and small.

¡¤ An observed wavelength is determined by the
difference between two energy levels, including the difference in the QED
corrections

¡¤ In addition to Ex. 3, calculate the
transition energy between U^{91+}(2s_{1/2}) and U^{91+}(2p_{1/2}),
U^{91+}(2p_{3/2}).