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) eitE/ћ
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 = (kћ)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:
¡ò Pnl(r)Pn¡¯l¡¯(r)dr = dnn¡¯dll¡¯
¡¤ 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
Are the bound-state and continuum state eigenfunctions, Eqs. (6) and (7), orthogonal? Try y1s and yks, k = 1 a.u.
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 ©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 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 ¢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(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).