Atomic StructureInitial questions: How can we identify atoms or molecules from their spectra? Putanother way, given a set of lines, how can we be confident that we can figure out thecomposition?In the next several lectures we’re going to consider radiative processes related to transi-tions in atoms and molecules. In this lecture we’ll take a basic look at the structure of atomsthemselves. To do this right we need to use the equations of quantum mechanics, whichwould mean the Dirac equation or, for a good approximation, the Schr¨odinger equation withthe Pauli exclusion principle. We will, in fact, use the Schr¨odinger equation. However, beforegoing into those details it’s helpful to see how one can get some insight with semiclassicalapproaches.First, let’s think of an atom purely classically. Imagine that we treat a hydrogen atomas an electron moving in circles around a proton. The electron is therefore accelerated, soit radiates. The total energy at a given instant is the (negative) potential energy plus thekinetic energy, and for a circular orbit the total energy is negative (it’s half the potentialenergy, by the virial theorem). Therefore, loss of energy means that the electron moves closerto the proton, so the acceleration is greater and the energy loss is greater. Classically, thisprocess would run away and within a tiny fraction of a second the atom would collapse.As a semi-classical try to deal with this, Bohr suggested a quantization rule that theangular momentum of the electron had to be an integer multiple of ~. If we assume thisbut otherwise keep our classical “solar system” picture, then for the ground state we haveV = −e2/r, K =12mev2= e2/2r (from the virial theorem), and L = mevr = ~. The totalenergy is −e2/2r, which for this angular momentum is E = −mee4/2~2= −13.6 eV. Thishappens to be exactly right, and if you put in 2~, 3~, and so on you get the right energyspacing (E(n~) ∝ 1/n2).But why should the angular momentum be quantized? Let’s take a different approach:from the uncertainty principle we know that if an electron is confined to a small volume thenit has a large momentum. In particular, let’s say that p = ~/∆x, or if the electron is withina distance r of the proton then p = ~/r. Then, independent of quantization effects, we’dlike to know the ground state of the atom, which is where the total energy is minimized.The total energy is p2/2me− e2/r, which is ~2/(2mer2) − e2/r. This reaches a minimum atr = ~2/(mee2), so that again we get the exact answer E = −mee4/2~2.This is too good to be true. In fact, we’re just lucky to get the right factors in thesecases, although getting the dependences right is not an accident. It is, however, helpful tohave this general picture before moving on to the equations.The fully quantum mechanical way to understand the structure of atoms, if they’renonrelativistic, is to use the Schr¨odinger equation. As you know, this takes the formHΨ = i~∂Ψ/∂t (1)where H is the Hamiltonian operator and Ψ is the wavefunction. In classical physics theHamiltonian is the total energy, i.e., the sum of the kinetic and potential energies. In quan-tum mechanics, it is the sum of the operators for the kinetic and potential energies. Con-veniently, the kinetic energy depends only on momenta (or derivatives of position), whereasthe potential energy depends only on positions. One can select a representation in whichone wants to write the operators; in the coordinate representation p = −i~∇, so if we usethe nonrelativistic expression EK= p2/2m then the Schr¨odinger equation becomesµ−~22me∇2+ V¶Ψ = i~∂Ψ/∂t . (2)If the solution is time-independent, then Ψ = ψ(r) exp(−iEt/~), so we get the time-independent equationµ−~22me∇2+ V¶ψ = Eψ . (3)Before specializing to electric fields, let’s think about how this would be generalized torelativistic energies. I want to say up front that I don’t expect you to grasp all of this fully (Icertainly don’t understand all the implications!), and in no way do I intend to test you on it. Ido, however, want to show you a little of the thinking that has gone into relativistic quantummechanics, so that we can have a better perspective on the nonrelativistic approximationswe’ll be using.Imagine that we have no potentials to worry about. Then E2= p2c2+ m2c4for somegeneral particle of mass m, so if we square the time-dependent version of the Schr¨odingerequation and rearrange we get·µ∇2−1c2∂2∂t2¶−³mc~´2¸Ψ = 0 . (4)This is called the Klein-Gordon equation, because it was originally written down by Schr¨odinger.Looks fine, right? The difficulty, as discussed in Shu (beginning of chapter 25) is that forthis equation (unlike the Schr¨odinger equation) one can’t interpret Ψ as a wave functionsuch that |Ψ|2is the probability density. That’s why Schr¨odinger rejected this equation andit got named after other people. It does turn out that the Klein-Gordon equation can beused instead as a field equation for scalar (spin 0) particles.The problem here is the introduction of a second time derivative. Dirac looked for ageneralization of the Schr¨odinger equation that kept the linearity in time but had symmetrybetween time and space (as required by special relativity). Thus, the Hamiltonian operator(without a potential) would beH = a · Pc + bmc2, (5)where a and b are constants. This can be equated to H = (P2c2+ m2c4)1/2; squaring andsolving, we get the requirementsaxax= ayay= azaz= bb = 1axay+ ayax= ayaz+ azay= azax+ axaz= 0axb + bax= ayb + bay= azb + baz= 0 .(6)We write it in this way instead of, say, writing axb+bax= 2axb because in fact these equationscan’t be solved if you use ordinary numbers for ax, ay, az, and b. Instead, you have to usematrices. These 4x4 matrices are given in many places (e.g., Shu, page 268). As a result,the wave function needs to have four components (as Shu emphasizes, this does not make ita four-vector; it’s just some internal space for the particles treated in this way). If we sayΨ → (Ψ1, Ψ2, Ψ3, Ψ4) and solve the matrix equation that way, then the probability densitybecomes ρ = |Ψ1|2+ |Ψ2|2+ |Ψ3|2+ |Ψ4|2. Dirac’s equation turns out to represent electronsbeautifully, taking into account their spin, being relativistic, and the whole shebang. We’vediverted here because it’s useful to see the full correct way of doing things every now andthen, and Rybicki and Lightman don’t cover