Why more quantum mechanics: from ideal models to real atoms

Quantum Mechanics I, PHYS-372, built the theory from its foundations and ended with a triumph: the hydrogen atom solved exactly, with energies \(E_n = -13.6\ \text{eV}/n^2\), degeneracy \(n^2\), and radial functions that reproduce the Balmer, Lyman and Paschen series. Those exact solutions are the skeleton of the subject, but they describe idealized systems: a single spinless electron, a point nucleus of infinite mass, no relativity, no magnetic effects and no other particles. Real atoms, molecules and solids break every one of those assumptions. This course is about how quantum mechanics copes with the real world, where exact solutions are rare and approximation methods, symmetry and the rules for many particles do most of the work.

The plan follows the second half of a standard textbook course. Lessons 2 and 3 complete the theory of angular momentum by adding the one ingredient PHYS-372 left out, spin, and then learning to combine angular momenta. Lessons 4 and 5 treat systems of identical particles, where a single symmetry requirement produces the Pauli exclusion principle, the periodic table and the behavior of electrons in metals. Lessons 6 to 9 develop the approximation methods that physicists use every day: time-independent perturbation theory, applied to the fine structure and Zeeman effect of hydrogen; the variational principle, applied to helium and the simplest molecule; and the WKB approximation, applied to tunneling and alpha decay. Lessons 10 and 11 turn to processes rather than states: transitions driven by time-dependent perturbations, the golden rule that governs every spectral line, and scattering. The capstone, lesson 12, brings spin, entanglement and two-level systems together in magnetic resonance and quantum bits.

How ideal are the ideal models?

A good way to see what remains to be done is to compare the PHYS-372 predictions with experiment. The hydrogen spectrum is the cleanest case. The Bohr formula with an infinitely heavy proton gives a ground state of \(-13.6057\) eV. Letting the proton move, by replacing the electron mass with the reduced mass \(\mu = m_e m_p/(m_e + m_p)\), shifts every level by about one part in 1,836 and brings the Lyman alpha line, the transition from \(n = 2\) to \(n = 1\), to 121.568 nm, within a thousandth of a nanometer of the measured 121.567 nm. That agreement is so good that it is easy to miss what it hides. With a spectrometer of high resolution, each hydrogen line splits into close components. The \(n = 2\) level, fourfold degenerate in the PHYS-372 solution (eightfold with spin), is actually three levels separated by tens of micro-electronvolts. These splittings, the fine structure, are smaller than the gross structure by a factor of order \(\alpha^2 \approx 5 \times 10^{-5}\), where \(\alpha = e^2/4\pi\varepsilon_0\hbar c \approx 1/137\) is the fine-structure constant. Explaining them requires relativity and electron spin, and the tool for calculating them is perturbation theory (lessons 6 and 7).

Helium is a harsher test. With two electrons, the Schrödinger equation contains the repulsion \(e^2/4\pi\varepsilon_0|\mathbf r_1 - \mathbf r_2|\) between them, and no one has ever solved it exactly. If we simply drop the repulsion, each electron sits in a hydrogen-like orbit around a nucleus of charge \(Z = 2\), and the ground-state energy is \(2 \times (-Z^2 \times 13.6\ \text{eV}) = -108.8\) eV. The measured value, the energy needed to remove both electrons, is \(-79.0\) eV. The ideal model is wrong by almost 40 parts in 100. Two approximation methods rescue the situation: first-order perturbation theory, treating the repulsion as a correction, gives \(-74.8\) eV; the variational method, letting each electron see a partly screened nucleus, gives \(-77.5\) eV. Modern numerical variational calculations, descended from Egil Hylleraas's work in 1929, agree with experiment to better than one part in a billion. The program below reproduces the simple estimates.

import math
R_inf = 13.605693   # Rydberg energy in eV (infinite nuclear mass)
me_mp = 1 / 1836.15267
alpha = 1 / 137.035999
mu = 1 / (1 + me_mp)                      # reduced-mass factor for hydrogen
print("Ideal model versus a real atom, hydrogen and helium")
print(f"hydrogen ground state, infinite nuclear mass : {-R_inf:9.4f} eV")
print(f"hydrogen ground state, reduced mass          : {-R_inf * mu:9.4f} eV")
lyman = R_inf * mu * (1 - 1 / 4)
print(f"Lyman alpha photon energy (n=2 to 1)         : {lyman:9.4f} eV")
print(f"  wavelength                                 : {1239.84198 / lyman:9.3f} nm")
fs = R_inf * alpha ** 2 / 16                # n = 2 fine-structure splitting 2p3/2 - 2p1/2
print(f"n = 2 fine-structure splitting (preview)     : {fs * 1e6:9.2f} micro-eV")
print(f"  ratio to the Lyman alpha energy            : {fs / lyman:9.2e}")
print()
Z = 2
E_noint = -2 * Z ** 2 * R_inf
E_first = E_noint + (5 / 4) * Z * R_inf
Zeff = Z - 5 / 16
E_var = -2 * Zeff ** 2 * R_inf
E_meas = -79.005
for name, E in [("no electron repulsion", E_noint), ("plus first-order repulsion", E_first),
                ("variational, Z_eff = 27/16", E_var), ("measured", E_meas)]:
    print(f"helium ground state, {name:27s}: {E:8.2f} eV  error {100 * (E - E_meas) / E_meas:+6.1f} %"
          if name != "measured" else f"helium ground state, {name:27s}: {E:8.2f} eV")

Ideal model versus a real atom, hydrogen and helium
hydrogen ground state, infinite nuclear mass :  -13.6057 eV
hydrogen ground state, reduced mass          :  -13.5983 eV
Lyman alpha photon energy (n=2 to 1)         :   10.1987 eV
  wavelength                                 :   121.568 nm
n = 2 fine-structure splitting (preview)     :     45.28 micro-eV
  ratio to the Lyman alpha energy            :  4.44e-06

helium ground state, no electron repulsion      :  -108.85 eV  error  +37.8 %
helium ground state, plus first-order repulsion :   -74.83 eV  error   -5.3 %
helium ground state, variational, Z_eff = 27/16 :   -77.49 eV  error   -1.9 %
helium ground state, measured                   :   -79.00 eV

The numbers frame the course. The reduced-mass correction to hydrogen is exact and simple; the fine structure is a small correction of relative size \(10^{-5}\) to \(10^{-6}\), perfectly suited to perturbation theory; and the electron–electron repulsion in helium is not small at all, yet clever approximation still gets within two parts in 100 using pencil and paper. Learning which method suits which problem is a large part of what this course teaches.

What the ideal hydrogen atom leaves out

It helps to list the physics omitted from the PHYS-372 hydrogen atom, roughly in decreasing order of size, because each item appears later in the course:

Each of these is tiny compared with the Coulomb energy, which is why the PHYS-372 solution works so well, and each has been measured with extraordinary precision. The hydrogen \(1s\)–\(2s\) transition frequency is known to about one part in \(10^{15}\), making hydrogen one of the most stringent tests of physical theory anywhere.

Many particles and the role of symmetry

The second great limitation of PHYS-372 is that it treated one particle at a time. Most of physics and all of chemistry involve many electrons. The state of \(N\) particles is a wave function of \(3N\) coordinates (plus spins), and solving for it exactly is hopeless except in special cases. Two ideas make progress possible. The first is approximation: treat each electron as moving in the average field of the others, as in the Hartree method of 1928 and the Hartree–Fock method of 1930, or use variational trial functions. The second is a deep symmetry requirement that has no classical counterpart. Electrons are identical, not merely similar: there is no experiment, even in principle, that can tell which electron is which. Quantum mechanics expresses this by requiring that the many-electron wave function change sign whenever two electrons are exchanged. That one requirement, explored in lesson 4, gives the Pauli exclusion principle, the shell structure of atoms, the stability of matter, the chemical bond, and the enormous pressure inside white dwarf stars.

Particles of integer spin, such as photons and helium-4 atoms, obey the opposite rule: their wave function is unchanged by exchange. These bosons like to occupy the same state, which is the principle behind lasers, superfluid helium and Bose–Einstein condensates, first made in 1995 from rubidium and sodium atoms cooled to millionths of a kelvin. The distinction between fermions and bosons, connected to spin by the spin–statistics theorem, runs through lessons 4 and 5.

Approximation methods: a preview

Almost every real calculation in quantum mechanics uses one of a small set of approximation methods, and choosing the right one depends on what is small in the problem.

Two methods work in very different ways. Perturbation theory starts from an exactly solved problem and adds corrections in powers of a small parameter; it is systematic and can be carried to higher orders, but it fails if the perturbation is not small or if levels are degenerate and are not handled with care. The variational method starts from a guess for the wave function and adjusts it to minimize the energy; it always gives an upper bound to the true ground-state energy, works even when nothing is small, and is the basis of most of computational chemistry. The WKB method is different again: it treats the wave function as locally a plane wave whose wavelength changes slowly, connecting quantum mechanics to the classical picture of a particle moving through a potential.

Processes, not just states

PHYS-372 asked mainly about stationary states and their energies. Much of experimental physics asks about rates instead: how fast an excited atom emits a photon, how likely a neutron is to scatter from a nucleus, how quickly a radioactive nucleus decays. Lesson 10 develops time-dependent perturbation theory, in which a weak perturbation drives transitions between stationary states, and derives Fermi's golden rule, the formula for a transition rate into a continuum of final states. The golden rule, together with the selection rules that follow from angular momentum, explains which spectral lines appear and how bright they are. Lesson 11 develops scattering theory, which is how physicists probe structure they cannot see directly, from Rutherford's discovery of the nucleus in 1911 to the experiments at TRIUMF in Vancouver and at Fermilab near Chicago today.

Two-level systems and the quantum frontier

The simplest quantum system of all has only two states, like a spin one-half in a magnetic field. It is also among the most important. Every magnetic resonance imaging scanner works by driving the spins of hydrogen nuclei in the body between two states with radio waves, and listening to them relax back. Atomic clocks, including the cesium standard that defines the second at 9,192,631,770 cycles, are two-level systems interrogated with exquisite precision. And quantum computers store information in qubits, two-level systems whose superpositions and entanglement give quantum algorithms their power. The 2025 Nobel Prize in Physics went to John Clarke, Michel Devoret and John Martinis for showing, in the 1980s, that a superconducting electrical circuit large enough to see could behave as a single quantum system with quantized energy levels and tunneling, the experiments behind today's superconducting qubits. The capstone of this course works through the physics of these two-level systems using the tools developed in every earlier lesson.

Units and scales

Atomic physics has natural scales that make estimates quick. Lengths are measured in Bohr radii, \(a_0 = 4\pi\varepsilon_0\hbar^2/m_e e^2 = 0.0529\) nm; energies in rydbergs, \(13.6\) eV, or hartrees, \(27.2\) eV; and the fine-structure constant \(\alpha\) sets the ratio of each successive layer of corrections. The gross structure of hydrogen is of order \(\alpha^2 m_e c^2\), the fine structure of order \(\alpha^4 m_e c^2\), the Lamb shift of order \(\alpha^5 m_e c^2\) with logarithms, and the hyperfine structure is smaller still by the ratio of the electron mass to the proton mass. Keeping these orders in mind makes it possible to see at a glance which corrections matter for a given experiment.

These are order-of-magnitude scales, before the numerical factors that the detailed calculations supply; the actual \(n = 2\) fine-structure splitting is \(4.5 \times 10^{-5}\) eV and the ground-state hyperfine splitting is \(5.9 \times 10^{-6}\) eV. Even so, the hierarchy explains why each layer can be treated as a perturbation of the one above it.