Atomic Physics §

From the Bohr model to the Schrödinger equation §

In 1913, Niels Bohr proposed that the electron in hydrogen occupies only certain allowed circular orbits with quantized angular momentum $L = n\hbar$. This ad hoc quantization reproduced the observed $1/n^2$ energy spacing and explained the hydrogen spectrum. The Bohr model was a crucial stepping stone: it identified the correct energy formula and the role of angular momentum quantization, but it could not explain line intensities, fine structure, or multi-electron atoms.

The full quantum-mechanical treatment replaces orbits with wavefunctions. Solving the Schrödinger equation for the Coulomb potential $V(r) = -e^2/(4\pi\varepsilon_0 r)$ yields the same energy eigenvalues:

The quantum states of hydrogen are labeled by four quantum numbers:

• Principal $n = 1, 2, 3, \ldots$ — sets the energy.
• Orbital angular momentum $\ell = 0, 1, \ldots, n-1$ — labeled s, p, d, f, …
• Magnetic $m_\ell = -\ell, \ldots, +\ell$ — orientation in a magnetic field.
• Spin $s = 1/2$ for the electron.

The Bohr radius $a_0 = 4\pi\varepsilon_0 \hbar^2/(m_e e^2) \approx 0.529$ Å sets the size scale of the atom.

Energy-level diagram §

The figure below shows the first four energy levels of hydrogen and the principal transitions:

Worked example: Lyman-alpha §

The Lyman-alpha line corresponds to the transition $n=2 \to n=1$:

Wavelength: $\lambda = hc/E = 1240/10.2 \approx 121.6$ nm (ultraviolet).

Authoritative sources

• Bohr (1913), Phil. Mag. 26, 1
• Griffiths, Introduction to Quantum Mechanics, Ch. 4

Why the levels split §

In the basic Schrödinger treatment, all states with the same $n$ but different $\ell$ are degenerate (e.g., 2s and 2p have the same energy). In reality, three relativistic effects lift this degeneracy:

1. Relativistic kinetic energy correction: the electron speed in inner orbits approaches $c$, so the kinetic energy $p^2/2m$ must be replaced by the relativistic expression. This shifts all levels slightly.

2. Spin-orbit coupling: in the electron’s rest frame, the orbiting proton constitutes a current loop, producing a magnetic field $\mathbf{B}_{\text{so}}$. The electron’s spin magnetic moment $\boldsymbol{\mu}_s = -g_s \mu_B \mathbf{S}/\hbar$ interacts with this field:

The factor of $1/2$ is the Thomas precession correction.

3. Darwin term: a contact interaction affecting only $s$-states ($\ell=0$).

The total angular momentum $\mathbf{J} = \mathbf{L} + \mathbf{S}$ is the good quantum number. For a given $\ell$, the spin couples to give $j = \ell \pm 1/2$ (except $\ell=0$ where only $j=1/2$).

The fine-structure formula §

The combined correction to the energy is:

For $n=2$: the $2p_{3/2}$ level is slightly higher than the $2p_{1/2}$ level, and the $2s_{1/2}$ level is degenerate with $2p_{1/2}$ (to this order). The $2p_{3/2} \to 2s_{1/2}$ transition produces the famous 2.1 μm “fine-structure line” of hydrogen.

Diagram: fine-structure splitting of n=2 §

Authoritative sources

• Foot, Atomic Physics, Ch. 2
• Berestetskii, Lifshitz & Pitaevskii, Quantum Electrodynamics, §43

Why multi-electron atoms are harder §

In hydrogen, the electron sees a pure $-e^2/r$ Coulomb potential and the problem is exactly solvable. In a multi-electron atom with nuclear charge $Ze$, each electron feels:

1. The attractive Coulomb potential of the nucleus: $-Ze^2/(4\pi\varepsilon_0 r)$.
2. The repulsive Coulomb potential of all other electrons: $+e^2/(4\pi\varepsilon_0 r_{ij})$.

The electron–electron repulsion terms make the Schrödinger equation unsolvable in closed form. The key physical insight is screening (or shielding): inner electrons partially cancel the nuclear charge, so an outer electron sees an effective nuclear charge $Z_{\text{eff}} < Z$. This is why the energy of an orbital depends on both $n$ and $\ell$: orbitals with smaller $\ell$ (more penetrating, e.g. s orbitals) spend more time near the nucleus and feel a larger $Z_{\text{eff}}$, lowering their energy relative to higher-$\ell$ orbitals with the same $n$.

The filling rules §

• Pauli exclusion principle: no two electrons in an atom can share the same set of four quantum numbers $(n, \ell, m_\ell, m_s)$. Each spatial orbital holds at most two electrons (one spin-up, one spin-down).
• Aufbau principle (Madelung rule): orbitals fill in order of increasing $n + \ell$; ties are broken by lower $n$ first. This is an empirical rule with known exceptions (e.g., Cr: $[\text{Ar}]\,3d^5 4s^1$ rather than $3d^4 4s^2$; Cu: $[\text{Ar}]\,3d^{10} 4s^1$ rather than $3d^9 4s^2$), where exchange energy and near-degeneracy shift the ground state.
• Hund’s rule: degenerate orbitals (same $n, \ell$) are singly occupied with parallel spins before any pairing occurs.

Orbital-filling diagram §

The diagram below shows the filling order for the first 36 elements (up to krypton), with the $n+\ell$ rule:

The periodic table §

The periodic table organizes elements by atomic number $Z$. Elements in the same column (group) have the same valence electron configuration, explaining chemical periodicity. The nucleus provides the positive charge $+Ze$ that binds the electrons; its internal structure (protons, neutrons, strong force) is the subject of nuclear physics.

Example. Carbon ($Z = 6$): configuration $1s^2\, 2s^2\, 2p^2$. The two 2p electrons occupy separate orbitals with parallel spins (Hund’s rule), giving carbon its tetravalent chemistry.

Authoritative sources

• Foot, Atomic Physics, Ch. 3
• Cowan, The Theory of Atomic Structure and Spectra (1980)

Connection to the hydrogen energy levels §

The Rydberg formula is a direct restatement of the hydrogen energy-level formula. Starting from $E_n = -13.6/n^2$ eV, the photon energy for a transition $n_i \to n_f$ is:

Converting to wavelength $\lambda = c/\nu = hc/(h\nu)$:

So the Rydberg constant is not an independent empirical parameter — it is fixed by the same physics that gives $E_n = -13.6/n^2$.

Emission, absorption, and selection rules §

• Emission: an electron drops from a higher to a lower level, emitting a photon of energy $h\nu = E_{\text{high}} - E_{\text{low}}$.
• Absorption: an electron absorbs a photon of exactly the right energy to jump to a higher level.
• Selection rules: electric-dipole transitions require $\Delta \ell = \pm 1$ and $\Delta m_\ell = 0, \pm 1$. Transitions violating these rules are “forbidden” (they can still occur via weaker multipole mechanisms, but with much lower probability).

Spectral-line diagram §

The figure below shows the hydrogen emission spectrum (visible Balmer series) and the corresponding absorption spectrum:

Multi-electron spectral lines §

In multi-electron atoms, the energy levels depend on both $n$ and $\ell$ (due to screening), so the spectrum is richer. The sodium D-line at 589 nm arises from the $3p \to 3s$ transition. The fine structure splits this into two closely spaced lines at 589.0 and 589.6 nm (the $3p_{1/2}$ and $3p_{3/2}$ upper levels).

Authoritative sources

• Foot, Atomic Physics, Ch. 4
• Rydberg, Zur Theorie der Spektrallinien (1890)

Mechanism §

In the absence of a field, states with the same $n, \ell$ but different $m_\ell$ are degenerate. An external magnetic field $\mathbf{B} = B\hat{z}$ couples to the orbital and spin magnetic moments:

where $\mu_B = e\hbar/(2m_e) \approx 5.79 \times 10^{-5}$ eV/T is the Bohr magneton and $g_s \approx 2$.

Each sub-level shifts by $\Delta E = \mu_B (m_\ell + g_s m_s) B$. Since the selection rule $\Delta m_\ell = 0, \pm 1$ still applies, a single spectral line splits into multiple components.

Normal vs. anomalous Zeeman effect §

• Normal Zeeman effect: occurs when spin is effectively zero (or when the total spin of the upper and lower states is the same). A line splits into three components (a triplet): $\Delta m = 0$ (π line, unshifted) and $\Delta m = \pm 1$ (σ lines, shifted by $\pm \mu_B B$).

• Anomalous Zeeman effect: occurs when spin-orbit coupling is significant (the common case). The level splits into $2J+1$ sub-levels labeled by $m_J$, and the pattern is more complex. In the Paschen-Back regime (very strong field), spin-orbit coupling is broken and the splitting reverts to a pattern based on $m_\ell$ and $m_s$ separately.

Diagram: normal Zeeman splitting §

The Stark effect (brief note) §

An external electric field similarly splits or shifts energy levels (the Stark effect). For hydrogen, the linear Stark effect mixes states of the same $n$ but different parity (e.g., 2s and 2p), lifting the $m_\ell$ degeneracy. The Stark effect is generally smaller than the Zeeman effect for typical laboratory fields but is important in astrophysics (pressure broadening in stellar atmospheres).

Authoritative sources

• Foot, Atomic Physics, Ch. 5
• Zeeman, Sur la modification que subit la polarisation de la lumière émise par les rayons d’un champ magnétique (1897)

The LCAO method §

The Linear Combination of Atomic Orbitals (LCAO) method approximates molecular orbitals as superpositions of atomic orbitals on the constituent atoms. For a diatomic molecule AB:

where $\phi_A$ and $\phi_B$ are atomic orbitals on atoms A and B, and $S = \langle \phi_A | \phi_B \rangle$ is the overlap integral. The $+$ combination is the bonding orbital (lower energy, electron density concentrated between the nuclei); the $-$ combination is the antibonding orbital (higher energy, node between the nuclei).

Molecular orbital diagram for H₂ §

Types of bonds §

• Covalent bond: two atoms share a pair of electrons in a bonding molecular orbital. The bonding MO is lower in energy than the atomic orbitals; the antibonding MO is higher. H₂ has a bond energy ≈ 4.5 eV and bond length ≈ 0.74 Å.
• Ionic bond: one atom transfers an electron to another; the resulting ions are held by Coulomb attraction. Example: NaCl. The bond energy is the difference between the ionization energy of the donor, the electron affinity of the acceptor, and the Madelung (lattice) energy.
• Metallic bond: valence electrons are delocalized over the entire lattice, forming a “sea” of electrons that binds the positive ion cores. Described by band theory: the discrete atomic levels broaden into energy bands.

Van der Waals interactions §

The Lennard-Jones potential models weak van der Waals (dispersion) interactions between neutral atoms or molecules:

The $r^{-12}$ term models Pauli repulsion at short range; the $r^{-6}$ term is the London dispersion attraction (induced dipole–induced dipole).

Authoritative sources

• Pauling, The Nature of the Chemical Bond (1960)
• Lennard-Jones, On the determination of molecular fields (1924)

An electron in a hydrogen atom makes a transition from $n = 3$ to $n = 1$. Compute the wavelength of the emitted photon. (Use $E_n = -13.6/n^2$ eV and $hc = 1240$ eV·nm.)