The simplest atom is the hydrogen atom.  It consists of a proton of mass mp = 1.7´10-27 kg and charge qe = 1.6´10-19 C and an electron of mass me = 9.1´10-31kg and charge -qe.  The dominant part of the interaction between the two particles is the electrostatic interaction.  The electrostatic potential energy is

U(r) = -(1/(4pe0))(qe2/r) = -e2/r

in SI units.

If we confine ourselves to studying the relative motion of the two particles, and if we neglect any external forces, and if we treat the particles as spinless particles, then the Hamiltonian of the system is

.

The reduced mass m of the system is nearly the same as the electron mass me, and the center of mass of the system is nearly in the same place as the proton.  We therefore often call the relative particle "the electron" and the center of mass "the proton".

H0 is the Hamiltonian of a fictitious particle moving in a central potential.  The eigenfunctions of H0 are of the form

ynlm(r) = Rnl(r)Ylm(q,f) = (unl(r)/r)Ylm(q,f).

The energy levels of the hydrogen atom are En = -(1/n2)EI, n = 1, 2, 3, .... ;  n is called the principal quantum number.  Given n, l can take on n possible values  l = 0, 1, 2, ..., (n - 1); n characterizes an electron shell, which contains n subshells characterized by l.  Each subshell contains 2l + 1 distinct states.  The energy levels are degenerate.  We often denote the energy eigenstates of the hydrogen atom by {|n, l ,m; s, ms; i, mi>}.  Here s and i denote the electron and proton spin respectively.  Since H0 does not operate on s and i, we often simply denote the eigenstates by {|n, l, m>}.  Because H0 has degenerate eigenstate, that choice of eigenbasis is not unique.  Different angular momentum coupling schemes will yield different eigenbases, for example {|n, l, s, j, mj; i, mi> }.

Simple spectra (one-electron atoms):

The eigenvalue equation for the Hamiltonian H0 of a particle with potential energy -Ze2/r is

.

Common eigenfunctions of H0, L2, and Lz are of the form ynlm(r) = Rnl(r)Ylm(q,f).

H0ynlm = Eynlm,    L2ynlm = l(l+1)h2ynlm,    Lzynlm = mhynlm.

If E > 0, then a solution exists for any value of E and l.

If E < 0, then solutions exist only for discrete values of E, En = - m(Ze)2/(2n2h2).

Definitions:

 n = 1 ground state ionization potential n = 2 resonance level resonance potential

For hydrogen EI = 13.595eV, ER =10.196eV.

Definition of shells:

 K L M N O 1s 2s 3s 4s 5s 2p 3p 4p 5p 3d 4d 5d 4f 5f 5g

To find corrections to the energy eigenvalues and eigenfunctions due to the spin of the electron and the nucleus, we use perturbation theory.

Wso, the spin-orbit term:

The electron is a spin ½ particle and has magnetic moment Me = g(mB/h)S, where g = -2 and mB  is the Bohr magneton.  The energy of this magnetic moment in a magnetic field B is W = -Me×B.   In the rest frame of the electron the nucleus is orbiting the electron.  This moving charge produces a magnetic field at the position of the electron.  This magnetic field will be proportional to the orbital angular momentum L of the electron.  The correction term -Me×B is therefore proportional to S×L, and it is called the spin-orbit coupling term Wso.  Wso is responsible for the fine structure of the one-electron atom.  If <Wso> is small compared to <H0>,  then we can use perturbation theory to find corrections to the energy eigenvalues and eigenfunctions.

The correction term for the spin-orbit interaction is Wso = -Me×B = (a/h2) S×L.  (Here B is the magnetic field due to the relative orbital motion of electron and nucleus.)  The total angular momentum of the electron is given by J = L + S.  Squaring J we obtain J2 = L2 + S2 + 2S×L, or

S×L = (1/2)(J2 - L2 - S2).

Wso is a scalar operator.  It does not operate on the proton spin variables and commutes with J.  Its eigenvalues are therefore independent of the magnetic quantum number mj.  Wso also commutes with L2 and matrix elements between states with different values of l are zero.

The eigenfunctions of H0 may be chosen as {|n, l, s, j, mj ; i, mi>}.  These basis functions are eigenfunctions of Wso.  In that basis the corrections to the energy level due to the spin-orbit interaction are given by

<Wso>nlj = (a/2)(j(j+1) - l(l+1) - s(s+1)).

Example: Let l = 1, s = 1/2.  Then the possible values of j are j = 3/2 and j = 1/2.  The spin orbit interaction splits this 2s+1L = 2P state into two 2s+1Lj levels, 2P1/2 and  2P3/2.

<Wso>n,1,1/2 = -2(a/2),  <Wso>n,1,3/2 = (a/2).

Whf, the hyperfine structure term:

The nucleus may also have spin I and a magnetic moment Mn = gI(mN/h)I, where gI is the g-factor of the nucleus and mN  is the nuclear magneton  mN = (me/mp)mB.  In the magnetic field produced by the orbiting electron, the energy of this magnetic moment is Whf = -Mn×B.  Whf is responsible for the hyperfine structure of the atom, and <Whf> is a factor of ~me/mp smaller than <Wso>.

The correction term for the hyperfine structure is Whf = -Mp×B = (b/h2) I×J.  (Here B is the magnetic field due to the due to the relative orbital motion of electron and nucleus and the electron spin.)  The total angular momentum of the atom is given by F = J + I.  Squaring F we obtain F2 = J2 + I2 + 2I×J, or

I×J = (1/2)(F2 - J2 - I2).

When calculating the matrix elements of Whf, we choose {|n, l, s, j, i, f, mf>} as the eigenbasis of H0.  In that basis the corrections to the energy level due to the hyperfine interaction are given by

<Whf>nljf = (b/2)(f(f+1) - j(j+1) - i(i+1)).

Example: Let l = 1, s = 1/2, j = 3/2, i = 3/2.  Then the possible values of f are f = 3, 2, 1, and 0.  (When adding two angular momenta J1 and J2 to form J = J1 + J2,  the possible values of j are j1 + j2,  j1 + j2 - 1, ..., |j1 - j2|.)    The hyperfine interaction splits the 2s+1L = 2P3/2 level into 4 sublevels with f = 3, 2, 1, and 0.

<Whf>f=0 = -(15/2)(b/2),  <Wso>f=1 = -(11/2)(b/2),

<Whf>f=2 = -(3/2)(b/2),  <Wso>f=3 = (9/2)(b/2).

The atom in a weak external magnetic field (Zeeman effect):

A weak external magnetic B field in the z-direction yields an equidistant splitting of each of the hyperfine sublevels into 2F + 1 components, according to the magnetic quantum number mf.

DEn,l,j,f,mf = gF mB B mf.

Here gF = gJ [f(f+1) + j(j+1) - i(i+1)]/[2f(f +1)] - gI(mN/mB)[f(f+1) - j(j+1) + i(i+1)]/[2f(f +1)] ,

and gJ is the Landé g - factor, gJ = 1+[j(j+1) + s(s+1) - l(l+1)]/[2j(j +1)].

The second term in gF is much smaller than the first term, so

DEn,l,j,f,mf » (1+[j(j+1) + s(s+1) - l(l+1)]/[2j(j +1)] ) [f(f+1) + j(j+1) - i(i+1)]/[2f(f +1)] mB B mf.

To calculate transition probabilities when an is atom interacting with a monochromatic plane electromagnetic wave we use time-dependent perturbation theory and treat the electromagnetic field classically.

Assume that at t = -¥ a system is in an eigenstate |fi> of the Hamiltonian H0.  At t = t1 the system is perturbed and the Hamiltonian becomes H = H0 + W(t).  The probability of finding the system in the eigenstate |ff> of the Hamiltonian H0 at t = t2 is given by

to first order in the perturbation W.

For the interaction of a plane wave with an atom the term first order in W is WDE(t) = -p×E,  the energy of an electric dipole in an electric field.

p = qr,   E = E0coswt k,   WDE(t)  = -qE0z coswt.

Therefore <ff|WDE|fi> proportional to <ff|z|fi>

The matrix elements of WDE are proportional to the matrix elements of z, because E is in the z direction.  If

,

then

.

The integrant is a product of three spherical harmonics and the integral can be given in terms of Clebsch-Gordan coefficients.

The integral is zero unless mf = mi and lf = li ± 1.  If we choose another direction for the polarization of E, i.e. ,  then we find  mf = mi ± 1 and lf = li ± 1

The dipole transition selection rules therefore are

Dl = ± 1, Dm = 0, ± 1.

These selection rules result as a consequence of the properties of the spherical harmonics.

An electromagnetic field is most likely to induce a transition between an initial and a final state if these selection rules are satisfied.  If these selection rules are not satisfied a transition is less likely and is said to be forbidden.

When deriving the dipole transition selection rules Dl = ±1, Dm = 0, ±1, we assumed that the Hamiltonian H0 = p2/(2m) +V(r)  is perturbed by WDE(t).  We neglect the spin orbit interaction.  If H0 contains a spin-orbit coupling term f(r)L·S, then the eigenstates of H0 are {|l, s j, mj>} and not {|l, s; m, ms>}.  The dipole selection rules then become Dj = 0, ±1, (except ji = jf = 0), Dl = ±1, Dmj = 0, ±1.  These selection rules follow from the Wigner-Eckart theorem.

If H0 also includes the hyperfine interaction, then the dipole selection rules then become Df = 0, ±1, Dj = 0, ±1,  Dl = ±1, Dmf = 0, ±1.

Dipole transitions are single photon transitions.  An atom absorbs or emits a photon when perturbed by an electromagnetic wave.   This is referred to as absorption or stimulated emission.  An atom can also emit a photon via spontaneous emission.  Atoms in excited states randomly emit single photons in all directions according to statistical rules via spontaneous emission

Consider the simple case of a two-level system with lower level 1 and upper level 2.  Let N1 be the number density of atoms in level 1 and N2 be the number density in level 2.  Let u(f12) be the energy density per unit frequency interval of the light at frequency f12 = (E2 - E1)/h.

 The rate of spontaneous emission is independent of u(f12).  It is proportional to N2. Rspon. emiss. = A21N2. The rate of stimulated emission depends on u(f12).  It is proportional to u(f12)N2. Rstim. emiss. = B21u(f12)N2. The rate absorption depends on u(f12).  It is proportional to u(f12)N1. Rabsorb. = B12u(f12)N1.

The proportional constants A21, B21, and B12 are called the Einstein coefficients.

Simple quantum mechanics predicts B21 = B12 and lets us calculate the value of B21 = B12 using time-dependent perturbation theory.  But as long as we treat the electromagnetic field classically, we cannot calculate the probability for spontaneous emission of a photon this way.  To calculate A21 we also need to quantize the radiation field.

We can however avoid this problem by making statistical arguments.  In a cavity in thermal equilibrium the probabilities that states 1 and 2 are occupied are proportional to the Boltzmann factors exp(-E1/(kT)) and exp(-E2/(kT)), respectively, and in equilibrium the probability of up transitions must exactly balance the probability of down transitions.  We therefore need

(A21+B21u(f12))exp(-E2/(kT)) = B12u(f12)exp(-E1/(kT)) = B21u(f12)exp(-E1/(kT)).

In a cavity in thermal equilibrium u(f12) is given by Planks law

u(f12) = (8phf123/c3)(exp(hf12/kT -1)-1,

and we have

A21 = B218phf123/c3.

A calculation of the absorption or stimulated transition probability therefore also yields the spontaneous transition probability.