All About Atoms
The last chapter showed that confining an electron quantizes its energy, and that the hydrogen atom is nature's neatest trap. Now we open up the atom completely. Five properties so ordinary we rarely notice them — atoms are stable, they combine, they are built by systematic rules, they emit and absorb light at sharp wavelengths, and they carry angular momentum and magnetism — all flow from quantum physics. To explain them we need a fuller label for each electron: not one quantum number but five, including the electron's own intrinsic spin. We meet the experiments that forced these ideas (Stern–Gerlach, Einstein–de Haas, magnetic resonance), the single rule that makes chemistry possible — the Pauli exclusion principle — and watch it build the periodic table shell by shell. We then read the atom's deepest secrets in its X-ray spectrum, the tool Moseley used to order the elements by nuclear charge, and finish with one of quantum physics' great gifts to technology: the laser.
- Five quantum numbers fix each atomic electron: principal \(n\), orbital \(\ell\), orbital magnetic \(m_\ell\), spin \(s\), and spin magnetic \(m_s\).
- Orbital angular momentum is quantized, \(L = \sqrt{\ell(\ell+1)}\,\hbar\) with measurable component \(L_z = m_\ell\,\hbar\); the electron's intrinsic spin gives \(S = \sqrt{s(s+1)}\,\hbar\) with \(s = \tfrac12\) and \(S_z = m_s\,\hbar\).
- Each angular momentum carries a magnetic moment scaled by the Bohr magneton \(\mu_B = \dfrac{eh}{4\pi m} = 9.27\times10^{-24}\,\mathrm{J/T}\); the Stern–Gerlach experiment proves these orientations are quantized.
- Magnetic resonance spin-flips a proton when \(hf = 2\mu_z B\) — the physics behind NMR and MRI.
- The Pauli exclusion principle (no two electrons share all quantum numbers) fills shells (\(2n^2\) states) and subshells (\(2(2\ell+1)\) states), building the periodic table.
- X rays give a cutoff \(\lambda_{\min} = \dfrac{hc}{K_0}\) and characteristic \(K\)-lines whose Moseley plot \(\sqrt{f} \propto (Z-1)\) orders the elements by atomic number; lasers run on stimulated emission and population inversion.
What Is Physics?
A primary goal of physics is to understand the atom. A century ago researchers struggled merely to prove that atoms exist; today we photograph them with scanning tunneling microscopes, drag them across surfaces to build structures like the quantum corral of the previous chapter, and hold a single ion in a trap indefinitely to study it in perfect isolation. With confinement and quantization established, we now ask the next questions: how is an atom's full set of electrons arranged, what gives an atom angular momentum and magnetism, and why do the elements repeat their properties so faithfully down each column of the periodic table?
Some Properties of Atoms
Several everyday facts about atoms are so basic we hardly notice them, yet each is a quantum result. Atoms are stable — the matter around us has persisted unchanged for billions of years. Atoms combine into molecules and stack into rigid solids, so a floor of atoms holds you up even though each atom is mostly empty space. And the elements are built systematically: plotting the energy needed to remove an atom's most loosely bound electron (its ionization energy) against atomic number traces a sawtooth that repeats period after period, peaking at the inert noble gases and dropping to the reactive alkali metals.
The numbers of elements in the six complete periods of the table are \(2,\ 8,\ 8,\ 18,\ 18,\ 32\) — and, remarkably, quantum physics predicts exactly these counts. Atoms also emit and absorb light only in sharp lines, because they exist only in discrete quantum states and a photon must bridge a gap exactly.
Because each atom's magnetic moment is coupled to its angular momentum, lining up the moments also lines up the angular momenta. With no external torque, total angular momentum must stay zero, so the cylinder as a whole spins the other way to compensate. The experiment proved that an atom's magnetic moment and angular momentum are coupled, and dramatically showed that quantum angular momenta can produce visible, everyday rotation.
Electron Spin
Whether trapped in an atom or moving freely, every electron carries an intrinsic spin angular momentum \(\vec{S}\) — "intrinsic" meaning it is a basic property like mass or charge, not the result of anything literally spinning. Its magnitude is set by a spin quantum number \(s\), always \(\tfrac12\) for an electron, and its component along any axis by a spin magnetic quantum number \(m_s = \pm\tfrac12\). Postulated by Uhlenbeck and Goudsmit from atomic spectra and later grounded by Dirac's relativistic theory, spin completes the description of an electron.
All states with the same \(n\) form a shell; counting the allowed \(\ell\) and \(m_\ell\) values and doubling for the two spins shows a shell holds \(2n^2\) states. All states with the same \(n\) and \(\ell\) form a subshell of \(2(2\ell+1)\) states, all at (essentially) the same energy.
The allowed values of the quantum numbers are tightly linked, as summarized below.
| Quantum number | Symbol | Allowed values |
|---|---|---|
| Principal | \(n\) | \(1, 2, 3, \dots\) |
| Orbital | \(\ell\) | \(0, 1, 2, \dots, (n-1)\) |
| Orbital magnetic | \(m_\ell\) | \(0, \pm1, \pm2, \dots, \pm\ell\) |
| Spin | \(s\) | \(\tfrac12\) |
| Spin magnetic | \(m_s\) | \(\pm\tfrac12\) |
Angular Momenta and Magnetic Dipole Moments
Every quantum state has an orbital angular momentum with a quantized magnitude, and only its component along a chosen axis — never the full vector — can be measured.
The Stern–Gerlach Experiment
In 1922 Stern and Gerlach sent a beam of neutral silver atoms through a strongly non-uniform magnetic field. A neutral atom feels no \(q\vec{v}\times\vec{B}\) force, but its magnetic dipole does feel a force in a field gradient. Starting from the dipole energy \(U = -\vec{\mu}\cdot\vec{B} = -\mu_z B\), the deflecting force along the gradient is the key result.
For silver, every electron's moment cancels except the spin of one valence electron whose orbital moment is zero, so \(\mu_z = \pm\mu_B\) and the deflecting force takes only the two values \(F_z = \pm\mu_B(dB/dz)\). (The faint spots first became visible when Stern, a cigar smoker, breathed sulfur onto the plate and turned the silver into black silver sulfide.) Modern versions with cesium beams show the same clean two-peak split.
Magnetic Resonance
A proton also has spin and a spin magnetic moment, but because it is positive, its moment points along its spin. In a field \(\vec{B}\) the proton has two allowed orientations — a lower-energy spin-up state aligned with the field and a higher-energy spin-down state against it. The gap between them is set by the moment and the field.
Sweeping the external field while watching the RF energy absorbed produces a nuclear magnetic resonance (NMR) spectrum whose peaks form a signature unique to each molecule. Ethanol, for instance, shows three groups of lines for its \(\mathrm{CH_3}\), \(\mathrm{CH_2}\), and \(\mathrm{OH}\) protons. The same physics, imaged spatially, is the basis of the MRI scanner.
The Pauli Exclusion Principle
So far each trap has held a single electron. Once two or more electrons share a trap, a new rule governs them — one that applies to any particle with half-integer spin (electrons, protons, neutrons, all with \(s = \tfrac12\)). Wolfgang Pauli stated it in 1925.
In an atom this means no two electrons can share the same four values \((n, \ell, m_\ell, m_s)\); they must differ in at least one. Without this rule every electron would collapse into the lowest level, there would be no chemistry, and neither molecules nor we could exist. The exclusion principle is the reason matter has structure at all.
Multiple Electrons in Rectangular Traps
Before tackling real atoms, return to the rectangular traps of Chapter 39 and add electrons one at a time, obeying Pauli. A one-dimensional trap needs the single quantum number \(n\) plus \(m_s\), a rectangular corral needs \((n_x, n_y, m_s)\), and a box needs \((n_x, n_y, n_z, m_s)\). Each spatial level can therefore hold two electrons — one spin up, one spin down — after which it is full and the next electron must climb to the next level.
Seven electrons fill them as \(2 + 4 + 1\): two in the lowest level, four in the degenerate pair, one in the \((2,2)\) level. The system's ground-state energy is therefore \(2(2) + 4(5) + 1(8) = 32\) in units of \(h^2/8mL^2\). This bookkeeping — levels, degeneracy, two-per-state, sum the energies — is exactly how we will populate real atoms.
Building the Periodic Table
In a multielectron atom each electron moves in the field of the nucleus and all the other electrons, so the energy depends mostly on \(n\) and a little on \(\ell\). Subshells are labelled by a letter for \(\ell\), and the Pauli principle caps how many electrons each can hold.
| \(\ell\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Letter | s | p | d | f |
| Capacity \(2(2\ell+1)\) | 2 | 6 | 10 | 14 |
Filling subshells in order of energy reproduces the chemistry of the elements. Neon (10 electrons) closes its \(1s\), \(2s\), and \(2p\) subshells; closed subshells have zero net angular momentum and a spherically symmetric cloud, so neon is chemically inert, like all noble gases. Sodium (11) adds one electron in a \(3s\) state outside an inert neon-like core, and that single loosely bound valence electron makes sodium a reactive alkali metal. Chlorine (17) is one electron short of filling its \(3p\) subshell, leaving a "hole" eager to be filled — which is why sodium and chlorine bond so readily into NaCl. Iron (26) has the configuration \(1s^2 2s^2 2p^6 3s^2 3p^6 3d^6 4s^2\): notice the \(4s\) subshell fills before the \(3d\) is complete, because that arrangement gives the lower total energy — a reminder that subshells are not always filled in the naive numerical order.
X Rays and the Ordering of the Elements
Fire kilovolt electrons at a solid target and it emits X rays of two kinds. As an incident electron is deflected by target atoms it sheds energy in photons, producing a broad continuous spectrum. The most energetic possible photon comes from an electron that loses all its kinetic energy \(K_0\) in a single collision, setting a sharp short-wavelength cutoff.
Riding on the continuous spectrum are sharp characteristic peaks. An energetic electron knocks a deep electron out of the \(K\) shell (\(n=1\)); an electron from a higher shell drops in to fill the hole, emitting a characteristic photon. A drop from the \(L\) shell (\(n=2\)) gives the \(K_\alpha\) line, from the \(M\) shell (\(n=3\)) the \(K_\beta\) line.
This proved that the true basis for the periodic table is the atomic number \(Z\) (the number of protons), not atomic mass as had been assumed. The deep \(K\) electrons sit close to the nucleus and so are sensitive probes of its charge, which is why characteristic X-ray spectra are far more regular than the optical spectra of the heavily screened outer electrons. Moseley's plot let him fill gaps in the table and settle disputed claims of new elements.
Lasers and Laser Light
The laser (light amplification by stimulated emission of radiation) is one of quantum physics' great technological gifts. Its light is intensely monochromatic, coherent (wave trains kilometers long), directional (a beam can mark a spot only kilometers wide on the Moon), and sharply focusable to enormous intensity. All four traits follow from how the light is made.
An atom can interact with light three ways. In absorption it takes up a photon and climbs to a higher state, \(hf = E_x - E_0\). In spontaneous emission it falls of its own accord, emitting a photon in a random direction and phase (ordinary light). In stimulated emission a passing photon of the right energy triggers the drop, and the emitted photon is identical to the trigger — same energy, phase, polarization, and direction. Stimulated emission is what makes laser light coherent.
Long-lived metastable states make the inversion last. In the helium–neon laser, an electric discharge pumps helium atoms to a metastable level whose energy nearly matches a neon level; collisions transfer the energy to neon, building an inversion there. A first spontaneous photon then triggers a cascade of identical stimulated photons that bounce between two mirrors (one slightly leaky) to build a coherent beam at \(632.8\,\mathrm{nm}\). Among countless uses: barcode and disc reading, surgery, surveying, welding, fiber-optic communication, and holography.
Putting It to Work
Problem. An electron is in a state with orbital quantum number \(\ell = 3\). Find (a) the magnitude \(L\) of its orbital angular momentum, (b) the number of allowed values of \(m_\ell\), and (c) the smallest semi-classical angle between \(\vec{L}\) and the \(z\) axis.
Solution. Use \(L = \sqrt{\ell(\ell+1)}\,\hbar\); the allowed \(m_\ell\) run from \(-\ell\) to \(+\ell\); the smallest angle uses the largest \(L_z = \ell\hbar\) in \(\cos\theta = L_z/L\).
The angular momentum can take seven orientations, and even the most "upright" one tilts \(30^\circ\) off the axis — it can never align perfectly, because \(L_z = \ell\hbar\) is always less than \(L = \sqrt{\ell(\ell+1)}\,\hbar\). That gap is the fingerprint of space quantization.
Problem. Silver atoms pass through a Stern–Gerlach magnet of magnitude \(B = 0.50\,\mathrm{T}\). (a) What is the energy difference between the two spin orientations of the valence electron? (b) What photon frequency would drive a transition between them?
Solution. The two orientations have \(\mu_z = \pm\mu_B\), so the energy gap is \(\Delta E = 2\mu_B B\), and \(f = \Delta E/h\).
The split is minute — tens of microelectron-volts — and the matching photon is a microwave. Tiny as it is, that energy difference is what fans the single silver beam into two distinct spots, and the same kind of splitting underlies magnetic resonance.
Problem. A proton sits in a field \(B = 0.78\,\mathrm{T}\) with spin-magnetic-moment component \(\mu_z = 1.41\times10^{-26}\,\mathrm{J/T}\). What frequency of radiation will spin-flip it?
Solution. Apply the magnetic-resonance condition \(hf = 2\mu_z B\).
About \(33\,\mathrm{MHz}\) — squarely in the radio band, which is why NMR and MRI machines use RF coils rather than light. Because the local field differs slightly from proton to proton, the precise resonance frequency reveals each proton's chemical surroundings.
Problem. Seven non-interacting electrons are confined to a square corral (\(L_x = L_y = L\)). What is the ground-state energy of the system, as a multiple of \(h^2/8mL^2\)?
Solution. Fill the one-electron levels two at a time (Pauli), lowest first. Levels (in units of \(h^2/8mL^2\)): \((1,1)=2\); degenerate \((2,1),(1,2)=5\); \((2,2)=8\).
Two electrons fill the lowest level, four fill the doubly degenerate level (two states, two spins each), and the seventh sits alone in the next level. The total, \(32\,h^2/8mL^2\), is the lowest energy Pauli allows — the model atom's "ground state."
Problem. A cobalt target (\(Z = 27\)) shows a \(K_\alpha\) line at \(178.9\,\mathrm{pm}\), plus a fainter impurity line at \(143.5\,\mathrm{pm}\). Identify the impurity.
Solution. By Moseley, \(\sqrt{f} = \sqrt{c/\lambda} \propto (Z-1)\), so dividing the two relations eliminates the constant \(C\).
An atomic number of 30 is zinc. Note the shorter \(K_\alpha\) wavelength belongs to the higher-\(Z\) element, since a larger nuclear charge means a bigger \(K\)-to-\(L\) energy jump. This is exactly how Moseley assigned elements to their places by nuclear charge.
Problem. A helium–neon laser emits \(P = 2.3\,\mathrm{mW}\) at \(\lambda = 632.8\,\mathrm{nm}\). At what rate does it emit photons?
Solution. Each photon carries \(E = hc/\lambda\), so the rate is the power divided by that energy, \(R = P\lambda/hc\).
Even a modest classroom laser pours out roughly \(7\times10^{15}\) photons each second, every one identical in wavelength, phase, and direction — the hallmark of stimulated emission, and the reason the beam stays so tight and pure over long distances.
Chapter Summary
Stable, combine, built systematically; emit/absorb light via \(hf = E_{\text{high}} - E_{\text{low}}\); carry coupled angular momentum and magnetism.
Five numbers \((n,\ell,m_\ell,s,m_s)\) fix a state; a shell holds \(2n^2\), a subshell \(2(2\ell+1)\).
\(L = \sqrt{\ell(\ell+1)}\,\hbar\), \(L_z = m_\ell\hbar\); spin \(S = \sqrt{s(s+1)}\,\hbar\), \(S_z = m_s\hbar\).
Scaled by \(\mu_B = eh/4\pi m\); Stern–Gerlach shows orientation is quantized (space quantization).
Spin-flip a proton with \(hf = 2\mu_z B\); NMR/MRI read molecular signatures.
No two electrons share all quantum numbers; closed subshells (s²p⁶…) explain noble gases, alkali metals, halogens.
Cutoff \(\lambda_{\min} = hc/K_0\); characteristic \(K\)-lines; Moseley plot \(\sqrt{f}\propto(Z-1)\) orders elements by \(Z\).
Stimulated emission gives identical photons; needs a population inversion and metastable states (He–Ne at 632.8 nm).
Problems
Take \(h = 6.63\times10^{-34}\,\mathrm{J\cdot s} = 4.14\times10^{-15}\,\mathrm{eV\cdot s}\), \(\hbar = h/2\pi = 1.055\times10^{-34}\,\mathrm{J\cdot s}\), the Bohr magneton \(\mu_B = 9.27\times10^{-24}\,\mathrm{J/T}\), \(hc = 1240\,\mathrm{eV\cdot nm}\), and Boltzmann's constant \(k = 8.62\times10^{-5}\,\mathrm{eV/K}\). Use \(L = \sqrt{\ell(\ell+1)}\,\hbar\), \(L_z = m_\ell\hbar\), and a shell capacity of \(2n^2\).
- How many (a) subshells and (b) electron states are in the \(n = 2\) shell? How many (c) subshells and (d) electron states are in the \(n = 4\) shell?
- For an electron with \(\ell = 2\), list all allowed values of \(m_\ell\), and find the magnitude \(L\) of the orbital angular momentum and its largest projection \(L_z\).
- An electron is in a state with \(\ell = 5\). Find the magnitude of \(\vec{L}\) and the minimum possible semi-classical angle between \(\vec{L}\) and the \(z\) axis.
- Verify by direct counting that a shell with principal quantum number \(n\) contains exactly \(2n^2\) electron states.
- Which of these proposed subshells cannot exist: \(2p\), \(4f\), \(3d\), \(1p\)? Explain using the rule relating \(\ell\) to \(n\).
- What is the magnitude of the spin angular momentum \(S\) of an electron, and what are its two allowed components \(S_z\) along a chosen axis?
- In a Stern–Gerlach experiment the field has magnitude \(B = 0.50\,\mathrm{T}\) at the silver atoms. Find (a) the energy difference between the two moment orientations and (b) the frequency of radiation that would induce a transition between them.
- A silver atom (\(M = 1.8\times10^{-25}\,\mathrm{kg}\), speed \(750\,\mathrm{m/s}\)) crosses a Stern–Gerlach region of length \(w = 3.5\,\mathrm{cm}\) in a gradient \(dB/dz = 1.4\,\mathrm{T/mm}\). Using \(d = \tfrac12\,[\mu_B(dB/dz)/M]\,(w/v)^2\), find the deflection \(d\).
- In an NMR experiment the RF source is at \(34\,\mathrm{MHz}\) and resonance occurs at \(B = 0.78\,\mathrm{T}\). Taking \(hf = 2\mu_z B\), find the proton's moment component \(\mu_z\).
- State the Pauli exclusion principle, and use it to explain why a single spatial state in a trap can hold at most two electrons.
- Seven non-interacting electrons are confined to a one-dimensional infinite well (one-electron energies \(\propto n^2\)). What multiple of \(h^2/8mL^2\) is the ground-state energy of the system? Do not neglect spin.
- A cubical box (\(L_x = L_y = L_z = L\)) holds eight non-interacting electrons. What multiple of \(h^2/8mL^2\) gives the system's ground-state energy?
- Write the ground-state electron configuration of (a) neon (\(Z=10\)) and (b) chlorine (\(Z=17\)), and explain in one sentence each why neon is inert and chlorine is reactive.
- Through what minimum potential difference must an electron be accelerated to produce X rays with a cutoff wavelength of \(0.100\,\mathrm{nm}\)?
- The \(K_\alpha\) line of iron has wavelength \(193\,\mathrm{pm}\). What is the energy difference between the two iron states responsible for the transition?
- A target's \(K_\alpha\) line lies at \(154\,\mathrm{pm}\) (copper, \(Z = 29\)). Using the Moseley relation, estimate the \(K_\alpha\) wavelength of a nickel target (\(Z = 28\)).
- A gas laser emits at \(\lambda = 550\,\mathrm{nm}\) between the ground state and an excited state. At room temperature (\(kT = 0.0259\,\mathrm{eV}\)) and with no pumping, find the thermal population ratio \(N_x/N_0\), and comment on why pumping is essential.
- A pulsed laser emits \(0.150\,\mathrm{J}\) per pulse at \(694.4\,\mathrm{nm}\). How many photons are emitted in each pulse?