Lecture Note for Sept. 17, 2002 (Tue.)

 

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) eitE

 

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

 

       y (r) = ynlm (r,J,j) = Rnl (r) Ylm (J,j)

 

     For computation, it is more convenient to use

 

              Pnl (r) = rRnl (r)

 

     Otherwise, Rns(r) do no vanish at r = 0, while others do

 

     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 Z2

 

     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)/r2

 

     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 = p/ћ =  continuum electron wave vector, plays the role of the principal quantum number n in the discrete spectrum

 

              E = - Z2/2n2

                          

              n2 = - Z2/2E  (E > 0 for continuum states)

 

              n = -iZ/(2E)1/2

 

              E = p2/2m = ()2/2m = k2/2 (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

 

yklm (r,J,j) = Rkl (r) Ylm (J,j)

 

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:

 

              Pnl(r)Pnl(r)dr = dnndll

 

     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 y1s and yks, 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 = k2/2 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 p

 

     The Dirac Hamiltonian contains the rest mass (= mc2) term

 

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

 

       Example: E(1s) = mc2 - 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 (Pnk) and the small component (Qnk) are switched

 

       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 ¥ (same as a 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(2s1/2), H(2p1/2) and H(2p3/2) are degenerate

 

     Relativistic energies for the hydrogen atom depend on n and j.  The Dirac energy for H(2s1/2) and H(2p1/2) are degenerate, but the energy for H(2p3/2) is higher

 

     In reality, the Lamb shift removes the degeneracy between H(2s1/2) and H(2p1/2).  With the Lamb shift, H(2p1/2) is lower than H(2s1/2)

 

     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 U91+(2s1/2) and U91+(2p1/2), U91+(2p3/2).