12.07.05 · quantum / perturbation

Stark and Zeeman effects in LL3 framing

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Anchor (Master): Landau & Lifshitz, Quantum Mechanics, Vol. 3 (Pergamon, 1977), Ch. VI §§76-77 and Ch. V §37; Schrödinger 1926 *Annalen der Physik* 80; Landé 1923 *Zeits. f. Physik* 15

Intuition Beginner

An atom left alone emits light at sharp, specific colours: each colour is a jump between two energy levels. Place that atom in a strong electric or magnetic field and the sharp lines break apart into several closely spaced lines. The field has reached inside the atom and nudged the energy levels, splitting what used to be a single level into a small fan of nearby levels. The splitting in an electric field is the Stark effect; the splitting in a magnetic field is the Zeeman effect.

Why should a field split a level at all? An atom is a tiny arrangement of charges that can be oriented in different ways relative to the field. Each orientation costs a slightly different amount of energy, the way a compass needle costs different energy pointing along a magnet versus against it. States that had the same energy when the atom was free now spread out by orientation, so one level becomes several. Turn the field off and the fan collapses back to a single line.

The size of the split tells you what is being measured. In a magnetic field the spacing grows in simple proportion to the field strength, and counting the lines reveals how the atom's internal rotation and its electron's built-in spin add together. In an electric field hydrogen behaves unusually: its levels shift in direct proportion to the field, a behaviour no other atom shares, and that peculiarity is a fingerprint of hydrogen's special symmetry. Most other atoms shift only in proportion to the field squared.

These splittings were among the first hard numbers that the new quantum theory had to reproduce, and getting them right was a make-or-break test. The patterns are not decoration: they encode the angular momentum of the state, the strength of the field, and the way spin and orbital motion combine.

Visual Beginner

Picture a single horizontal line marking an atomic energy level. As you slowly turn up an external field, read left to right, the one line splits into a small ladder of evenly spaced lines that spread apart further as the field grows. The number of rungs and their spacing is what the experiment measures.

For a magnetic field the simplest case gives three rungs, evenly spaced, and the middle rung sits exactly where the original line was. For hydrogen in an electric field the rungs spread out symmetrically above and below the original line. The picture to keep is a fan that opens wider as the field gets stronger.

Worked example Beginner

Take the lowest excited level of hydrogen, the one labelled $n = 2$ , and place the atom in an electric field of strength $E$ . This level is fourfold degenerate when the atom is free: four different states share the same energy. The electric field splits this single level into three distinct energies.

Step 1. The shift of each state turns out to be a multiple of one basic amount, $3 e a_{0} E$ , where $e$ is the electron charge and $a_{0}$ is the Bohr radius, the natural size of the hydrogen atom. This combination $e a_{0} E$ is the typical energy an electron-sized dipole picks up in the field.

Step 2. Of the four states, one is pushed up by $3 e a_{0} E$ , one is pushed down by $3 e a_{0} E$ , and the remaining two are not shifted at all; they stay at the original energy.

Step 3. So the four states land at three energies: $+ 3 e a_{0} E$ , $0$ , and $- 3 e a_{0} E$ relative to the unperturbed level. The middle energy is shared by two states.

Step 4. Put in numbers. With a field of $E = 1 \times 1 0^{7}$ volts per metre, the basic amount $3 e a_{0} E$ works out to roughly $1.6 \times 1 0^{- 23}$ joules, about $0.1$ milli-electron-volts. Small compared with the level energy itself, but easily large enough to see as a split spectral line.

What this tells us: the field does not move the level as a whole — it spreads it symmetrically, up by as much as down, with a gap that grows in step with the field. That straight-line growth with field strength is the signature of hydrogen, and it traces back to the four original states not all having the same shape.

Check your understanding Beginner

Formal definition Intermediate+

Let $\hat{H}_{0}$ be an atomic Hamiltonian with eigenstates $∣ n ℓ m ⟩$ and a static external field switched on as a perturbation. The two cases of interest add the coupling of the atom to a uniform field.

For a uniform electric field $E = E \hat{z}$ , the Stark perturbation is the dipole coupling $$ \hat W_{\mathrm{Stark}} = e E \hat z = e E, r\cos\theta, $$ with $e > 0$ the elementary charge and $\overset{z}{^}$ the position operator along the field. For a uniform magnetic field $B = B \hat{z}$ , the Zeeman perturbation couples the field to the total magnetic moment, $$ \hat W_{\mathrm{Zeeman}} = -\boldsymbol\mu\cdot\mathbf B = \frac{\mu_B}{\hbar} B,(\hat L_z + g_s \hat S_z), \qquad \mu_B = \frac{e\hbar}{2 m_e c}, $$ where $μ_{B}$ is the Bohr magneton, $\hat{L}_{z}$ and $\hat{S}_{z}$ are the orbital and spin angular-momentum components along the field, and $g_{s} \approx 2$ is the electron spin $g$ -factor. The orbital moment is $μ_{L} = - (μ_{B} /ℏ) L$ and the spin moment is $μ_{S} = - g_{s} (μ_{B} /ℏ) S$ ; the factor-of-two discrepancy between them is the origin of the anomalous Zeeman effect.

The defining distinction is whether the unperturbed level is degenerate with respect to parity. A level is non-degenerate in parity when all states sharing its energy have the same parity; then $⟨ ψ ∣ \overset{z}{^} ∣ ψ ⟩ = 0$ by parity and the leading Stark shift is second order, the quadratic Stark effect $Δ E^{(2)} = - \frac{1}{2} α_{pol} E^{2}$ , with $α_{pol}$ the static dipole polarizability. A level is parity-degenerate when states of opposite parity coincide in energy, as happens in hydrogen because the Coulomb degeneracy joins all $ℓ = 0, 1, \dots, n - 1$ at fixed $n$ ; then $\hat{W}_{Stark}$ has nonzero matrix elements within the degenerate subspace and degenerate perturbation theory gives a linear Stark effect $Δ E^{(1)} \propto E$ .

For the magnetic case, the weak-field (anomalous Zeeman) regime holds when $\hat{W}_{Zeeman}$ is small compared with the fine-structure spin-orbit splitting; the good quantum numbers are ${n, J, m_{J}}$ and $Δ E = g_{J} μ_{B} B m_{J}$ . The strong-field (Paschen-Back) regime holds when $\hat{W}_{Zeeman}$ dominates the spin-orbit term; the good quantum numbers revert to ${m_{ℓ}, m_{s}}$ and $Δ E = μ_{B} B (m_{ℓ} + g_{s} m_{s})$ .

Counterexamples to common slips

The linear Stark effect is not a generic atomic phenomenon. It requires the exact parity degeneracy of the pure Coulomb spectrum; fine structure, the Lamb shift, and any non-Coulombic core lift that degeneracy and convert the leading shift back to quadratic except at fields strong enough to overwhelm the splitting.
The Zeeman triplet $Δ E = μ_{B} B m_{ℓ}$ (the normal effect) is the exception, not the rule. It appears only for singlet states ( $S = 0$ ), where $g_{J} = 1$ . Every state with $S \neq = 0$ shows the anomalous pattern, historically the puzzle that resisted explanation until electron spin was introduced.
Diagonalising $\hat{W}_{Zeeman}$ in the ${m_{ℓ}, m_{s}}$ basis when the field is weak gives wrong shifts: at weak field the spin-orbit coupling has already fixed $J$ as a good quantum number, and one must project onto $∣ J, m_{J} ⟩$ first. Using the wrong basis for the field regime is the standard error.

Key derivation Intermediate+

Theorem (linear Stark effect in hydrogen). Let $\hat{H}_{0} = p^{2} /2 m_{e} - e^{2} / r$ be the hydrogen Hamiltonian (Gaussian units) with the $n^{2}$ -fold degenerate level of principal quantum number $n$ , and let $\hat{W} = e E z$ . Separating the problem in parabolic coordinates, the first-order energy shifts are $$ \Delta E^{(1)}_{n_1 n_2 m} = \tfrac{3}{2}, e a_0 E, n,(n_1 - n_2), $$ where $a_{0} = ℏ^{2} / m_{e} e^{2}$ is the Bohr radius and the parabolic quantum numbers satisfy $n_{1}, n_{2} \geq 0$ , $∣ m ∣ \geq 0$ , and $n = n_{1} + n_{2} + ∣ m ∣ + 1$ . For $n = 2$ the shifts are ${+ 3 e a_{0} E, 0, 0, - 3 e a_{0} E}$ .

Proof. The Coulomb plus uniform-field problem separates in parabolic coordinates $$ \xi = r + z, \qquad \eta = r - z, \qquad \varphi = \arctan(y/x), $$ because $z = \frac{1}{2} (ξ - η)$ is a sum of a function of $ξ$ and a function of $η$ , and the Coulomb $1/ r = 2/ (ξ + η)$ together with the field term separates the Schrödinger equation $\hat{H}_{0} + e E z$ into two ordinary differential equations. Writing $ψ = f_{1} (ξ) f_{2} (η) e^{im φ}$ and introducing separation constants $β_{1} + β_{2} = Z$ (here $Z = 1$ ), each factor obeys a one-dimensional equation of the hydrogen-radial type with an added linear potential. To first order in $E$ the bound-state energies acquire the parabolic quantum numbers $n_{1}, n_{2}$ counting the nodes of $f_{1}, f_{2}$ , and the principal quantum number is $n = n_{1} + n_{2} + ∣ m ∣ + 1$ .

The first-order shift is the expectation of $\hat{W}$ in the unperturbed parabolic state. Treating the field term perturbatively, the linear-in- $E$ correction is the diagonal matrix element $⟨ e E z ⟩$ evaluated on the parabolic eigenfunctions. The parabolic states are eigenstates of the operator that, within the degenerate $n$ -shell, represents $z$ ; this is precisely why degenerate perturbation theory selects them. The required expectation follows from the hydrogenic relation $$ \langle z\rangle_{n_1 n_2 m} = \tfrac{3}{2}, a_0, n,(n_1 - n_2), $$ which is the parabolic analogue of the Runge-Lenz expectation. Multiplying by $e E$ gives the stated shift. For $n = 2$ the allowed triples $(n_{1}, n_{2}, m)$ are $(1, 0, 0), (0, 1, 0), (0, 0, \pm 1)$ , giving $n_{1} - n_{2} = + 1, - 1, 0, 0$ , hence shifts $3 e a_{0} E, - 3 e a_{0} E, 0, 0$ . $□$

The same answer arrives by direct degenerate perturbation theory in the ${∣ 2 ℓ m ⟩}$ basis. Parity forbids all diagonal elements and connects only $ℓ = 0$ to $ℓ = 1$ with $Δ m = 0$ . The single nonzero element is $⟨ 200 ∣ e E z ∣ 210 ⟩ = - 3 e a_{0} E$ ; diagonalising the resulting $4 \times 4$ matrix, block-diagonal in $m$ , reproduces eigenvalues $\pm 3 e a_{0} E$ in the $m = 0$ block and $0$ in the $m = \pm 1$ sector.

Bridge. This derivation builds toward the general theory of atomic level splitting in external fields, and the parabolic separation appears again in the scattering and ionization problems of strong-field atomic physics, where the field-distorted $η$ -channel becomes the tunnelling coordinate. The foundational reason hydrogen alone has a linear Stark effect is the exact $ℓ$ -degeneracy of the Coulomb spectrum: this is exactly the extra $SO (4)$ symmetry generated by the Runge-Lenz vector, the same hidden symmetry that makes the parabolic separation possible, so the linear shift is dual to a statement about the Runge-Lenz expectation. Putting these together, the degenerate-perturbation secular equation 12.07.01 and the parabolic separation are two computations of one matrix, and the central insight is that the right basis inside a degenerate shell is fixed by the perturbation, not chosen for convenience. The bridge is that the operator $\overset{z}{^}$ restricted to the $n$ -shell is, up to a constant, a component of the Runge-Lenz vector, whose eigenvalues are the integers $n_{1} - n_{2}$ .

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that the only nonzero matrix element of $\hat{W} = e E z$ within the $n = 2$ hydrogen shell is $⟨ 200 ∣ e E z ∣ 210 ⟩$ , and explain why parity and the magnetic quantum number forbid the rest.

Hint

$z = r cos θ$ is odd under parity and carries $Δ m = 0$ . Combine the parity selection rule $Δ ℓ = \pm 1$ with conservation of $m$ .

Answer

The operator $z$ is a rank-one spherical tensor component with $q = 0$ , so it connects $∣ ℓ m ⟩$ to $∣ ℓ^{'} m^{'} ⟩$ only when $m^{'} = m$ and $ℓ^{'} = ℓ \pm 1$ (Wigner-Eckart plus parity). Within $n = 2$ the states are $∣ 200 ⟩, ∣ 210 ⟩, ∣ 21, \pm 1 ⟩$ . Diagonal elements vanish because $ℓ^{'} = ℓ$ is forbidden. Elements with the $m = \pm 1$ states vanish because they would need $m^{'} = m \neq = 0$ matched to $∣ 200 ⟩$ with $m = 0$ . The single survivor is $ℓ = 0 \leftrightarrow ℓ = 1$ at $m = 0$ , namely $⟨ 200 ∣ z ∣ 210 ⟩$ . Its value is $- 3 a_{0}$ , so $⟨ 200 ∣ e E z ∣ 210 ⟩ = - 3 e a_{0} E$ . Rubric: full credit for the two selection rules plus identification of the unique element.

Exercise 6 (hard, symbolic).

Derive the quadratic Stark polarizability of the hydrogen ground state by a variational (Dalgarno-Lewis style) argument, and show the second-order energy is $E^{(2)} = - \frac{9}{4} a_{0}^{3} E^{2}$ in Gaussian units, i.e. $α_{pol} = \frac{9}{2} a_{0}^{3}$ .

Hint

Seek the first-order wavefunction correction $ψ^{(1)} = (A r + B r^{2}) cos θ ψ_{100}$ solving $(\hat{H}_{0} - E_{0}) ψ^{(1)} = - (e E z - ⟨ e E z ⟩) ψ_{100}$ exactly; then $E^{(2)} = ⟨ ψ_{100} ∣ e E z ∣ ψ^{(1)} ⟩$ . Cross-check against the variational bound 12.07.03.

Answer

The inhomogeneous equation $(\hat{H}_{0} - E_{0}) ψ^{(1)} = - e E z ψ_{100}$ admits the closed-form solution $ψ^{(1)} = - \frac{m _{e} e E}{ℏ ^{2}} (\frac{a _{0} r}{2} + \frac{r ^{2}}{2}) cos θ ψ_{100}$ up to normalisation, found by substituting the polynomial ansatz $(A r + B r^{2}) cos θ ψ_{100}$ and matching powers of $r$ . The second-order energy is $E^{(2)} = ⟨ ψ_{100} ∣ e E z ∣ ψ^{(1)} ⟩$ . Evaluating the radial integral with $ψ_{100} = (π a_{0}^{3})^{- 1/2} e^{- r / a_{0}}$ and $⟨ cos^{2} θ ⟩ = \frac{1}{3}$ gives $E^{(2)} = - \frac{9}{4} a_{0}^{3} E^{2}$ , so $α_{pol} = - 2 E^{(2)} / E^{2} = \frac{9}{2} a_{0}^{3}$ . The Dalgarno-Lewis exact-summation method avoids summing over the full intermediate spectrum, including the continuum. The variational principle of 12.07.03 guarantees the trial estimate is an upper bound on the magnitude, consistent with the exact coefficient. Rubric: full credit for $E^{(2)} = - \frac{9}{4} a_{0}^{3} E^{2}$ via the closed-form $ψ^{(1)}$ .

Exercise 7 (hard, short-answer).

Explain the crossover from the anomalous Zeeman regime to the Paschen-Back regime as the field increases, naming what changes about the good quantum numbers and writing the limiting energy expressions on each side.

Hint

Compare the Zeeman term $μ_{B} B$ to the spin-orbit splitting $ζ ⟨ L \cdot S ⟩$ ; the larger term decides which operators commute with the dominant Hamiltonian.

Answer

When $μ_{B} B ≪ ζ$ (weak field), the spin-orbit coupling dominates and $L, S$ precess rapidly about $J$ ; ${J, m_{J}}$ are good and the shift is $Δ E = g_{J} μ_{B} B m_{J}$ with the Landé factor. When $μ_{B} B ≫ ζ$ (strong field, Paschen-Back), the magnetic term dominates and $L, S$ decouple, precessing independently about $B$ ; ${m_{ℓ}, m_{s}}$ become good and the shift is $Δ E = μ_{B} B (m_{ℓ} + g_{s} m_{s}) \approx μ_{B} B (m_{ℓ} + 2 m_{s})$ , with the residual spin-orbit term $ζ m_{ℓ} m_{s}$ as a small correction. In between, neither set is exactly good and one diagonalises the full $2 \times 2$ (or larger) matrix combining both terms, which interpolates continuously. The crossover field is set by $μ_{B} B \sim ζ$ . Rubric: full credit for naming the good-quantum-number switch and both limiting formulas.

Advanced results Master

The clean separation of hydrogen in parabolic coordinates extends to all orders. Writing the field term exactly, the $ξ$ and $η$ equations remain decoupled, and a systematic expansion produces the higher Stark corrections. The second-order shift of a general parabolic level is $$ \Delta E^{(2)}{n_1 n_2 m} = -\frac{1}{16},n^4,\big[17 n^2 - 3(n_1 - n_2)^2 - 9 m^2 + 19\big],a_0^3 E^2, $$ in Gaussian units, the result first obtained by Epstein and by Wentzel in the old quantum theory and confirmed by Schrödinger's wave-mechanical treatment. The ground state $n = 1$ ( $n_{1} = n_{2} = m = 0$ ) gives $- \frac{9}{4} a_{0}^{3} E^{2}$ , recovering $\alpha{\mathrm{pol}} = \tfrac92 a_0^3$. The linear term dominates for excited states; the quadratic term is the whole effect only when the linear one vanishes, as it must for the parity-paired ground state.

The Stark expansion is, strictly, asymptotic rather than convergent: a static electric field renders every bound state metastable, since the potential $- e^{2} / r + e E z$ has no true minimum and the electron can tunnel out along the down-field direction. The bound-state energy acquires a small imaginary part, the ionization rate, exponentially small in $1/ E$ , $$ \Gamma_{1s} \sim \frac{4}{E_{\mathrm{a.u.}}}\exp!\left(-\frac{2}{3 E_{\mathrm{a.u.}}}\right)\quad(\text{atomic units}), $$ computed by matching the bound wavefunction to the WKB tunnelling tail in the $η$ -channel. The real part of the energy is the Stark series; the imaginary part is invisible to any finite order of perturbation theory. This is the prototype of a Borel-summable divergent series in quantum mechanics.

Theorem (Wigner-Eckart projection and the Landé formula). Within a multiplet of fixed total angular momentum $J$ , the expectation of any vector operator $V$ equals the expectation of its projection onto $J$ : $$ \langle J m_J\mid \mathbf V\mid J m_J'\rangle = \frac{\langle\mathbf V\cdot\mathbf J\rangle}{J(J+1)\hbar^2},\langle J m_J\mid\mathbf J\mid J m_J'\rangle. $$ Applying this to $μ = - (μ_{B} /ℏ) (L + g_{s} S)$ with $g_{s} = 2$ yields the Zeeman shift $Δ E = g_{J} μ_{B} B m_{J}$ with $$ g_J = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}. $$

The projection theorem reduces the anomalous Zeeman problem to a single number per multiplet. Its content is that $L + 2 S = J + S$ has, within fixed $J$ , the same matrix elements as a scalar multiple of $J$ , the scalar being exactly $g_{J}$ . The normal Zeeman triplet is the special case $S = 0$ , $g_{J} = 1$ ; the doublet and quartet patterns of alkali and transition spectra are the $S \neq = 0$ cases that defied classical explanation and forced the introduction of spin.

In the Paschen-Back limit the spin-orbit term $ζ L \cdot S$ is the small perturbation on top of the magnetic Hamiltonian. The zeroth-order states are $∣ m_{ℓ}, m_{s} ⟩$ with energy $μ_{B} B (m_{ℓ} + 2 m_{s})$ , and the spin-orbit correction is the diagonal element $ζ ⟨ L \cdot S ⟩ = ζ m_{ℓ} m_{s}$ (the off-diagonal $\hat{L}_{+} \hat{S}_{-}$ pieces shift only at second order). The full field-dependence of any level, interpolating between the weak-field Landé pattern and the strong-field Paschen-Back pattern, follows from diagonalising the $2 \times 2$ matrix $μ_{B} B (\hat{L}_{z} + 2 \hat{S}_{z}) + ζ L \cdot S$ in each $m_{J} = m_{ℓ} + m_{s}$ block — the Breit-Rabi formula for a single valence electron.

Synthesis. The unifying object is the secular matrix of degenerate perturbation theory 12.07.01, and the foundational reason both effects yield clean formulas is that the perturbation, restricted to a degenerate shell, is a single tensor operator whose matrix elements the angular-momentum algebra fixes completely. This is exactly the Wigner-Eckart content: the Stark operator $z$ is the $q = 0$ component of a rank-one tensor, and its restriction to the hydrogen $n$ -shell is a component of the Runge-Lenz vector, so the linear shifts $\frac{3}{2} e a_{0} E n (n_{1} - n_{2})$ are dual to the Runge-Lenz eigenvalues; the Zeeman operator $L + 2 S$ is a vector whose multiplet expectation the projection theorem identifies with $g_{J} J$ , so the Landé formula is the same algebra applied to a magnetic rather than electric coupling. Putting these together, the linear Stark effect, the anomalous Zeeman effect, and the Paschen-Back crossover are three instances of one computation — diagonalise a small operator inside a degenerate or near-degenerate space — and the central insight is that the right basis is dictated by which interaction dominates: parabolic states for the field-dominated Coulomb shell, $∣ J m_{J} ⟩$ for the spin-orbit-dominated weak-field multiplet, $∣ m_{ℓ} m_{s} ⟩$ for the field-dominated strong-field limit. The quadratic Stark polarizability $\frac{9}{2} a_{0}^{3}$ generalises the same machinery to second order, where the variational bound 12.07.03 controls the sign and magnitude, and the metastability of every Stark state builds toward the strong-field ionization and resonance theory that the parabolic separation continues to organize.

Full proof set Master

Proposition (the linear Stark matrix element in hydrogen $n = 2$ ). The perturbation $\hat{W} = e E z$ within the $n = 2$ shell has the single independent nonzero matrix element $⟨ 200 ∣ \hat{W} ∣ 210 ⟩ = - 3 e a_{0} E$ , and its eigenvalues on the four-dimensional shell are ${+ 3 e a_{0} E, 0, 0, - 3 e a_{0} E}$ .

Proof. The four states are $∣ 200 ⟩, ∣ 210 ⟩, ∣ 211 ⟩, ∣ 21, - 1 ⟩$ . The operator $z = r cos θ$ is odd under spatial inversion and carries azimuthal index $q = 0$ , hence connects states with $Δ m = 0$ and $Δ ℓ$ odd; parity forbids $Δ ℓ = 0$ , so all diagonal elements vanish and the only candidates are the $m = 0$ pair $∣ 200 ⟩, ∣ 210 ⟩$ . The states $∣ 21, \pm 1 ⟩$ have no partner of opposite parity at $m = \pm 1$ within $n = 2$ and so are unshifted to first order. The surviving element is $$ \langle 210\mid z\mid 200\rangle = \int \psi_{210}^*, r\cos\theta,\psi_{200}, d^3r. $$ Using $ψ_{200} = (32 π a_{0}^{3})^{- 1/2} (2 - r / a_{0}) e^{- r /2 a_{0}}$ and $ψ_{210} = (32 π a_{0}^{3})^{- 1/2} (r / a_{0}) e^{- r /2 a_{0}} cos θ$ , the angular integral $\int cos^{2} θ d Ω = 4 π /3$ and the radial integral $\int_{0}^{\infty} (2 - r / a_{0}) (r / a_{0}) r \cdot r^{2} e^{- r / a_{0}} d r$ combine to give $⟨ 210 ∣ z ∣ 200 ⟩ = - 3 a_{0}$ . Thus $⟨ 200 ∣ \hat{W} ∣ 210 ⟩ = - 3 e a_{0} E$ . In the ordered basis $(∣ 200 ⟩, ∣ 210 ⟩)$ the $m = 0$ block is $\begin{psmallmatrix}0 & -3 e a_0 E\\ -3 e a_0 E & 0\end{psmallmatrix}$ , with eigenvalues $\pm 3 e a_{0} E$ ; the $m = \pm 1$ block is the zero matrix. The full spectrum is ${+ 3 e a_{0} E, 0, 0, - 3 e a_{0} E}$ . $□$

Proposition (Landé $g$ -factor from the projection theorem). For a multiplet $∣ L S J m_{J} ⟩$ the diagonal Zeeman shift is $Δ E = g_{J} μ_{B} B m_{J}$ with $g_{J} = 1 + [J (J + 1) + S (S + 1) - L (L + 1)] / [2 J (J + 1)]$ .

Proof. The magnetic moment is $μ = - (μ_{B} /ℏ) (L + 2 S) = - (μ_{B} /ℏ) (J + S)$ . By the projection (Wigner-Eckart for a vector) theorem, within fixed $J$ , $$ \langle\mathbf J + \mathbf S\rangle = \frac{\langle(\mathbf J + \mathbf S)\cdot\mathbf J\rangle}{J(J+1)\hbar^2}\langle\mathbf J\rangle = \left(1 + \frac{\langle\mathbf S\cdot\mathbf J\rangle}{J(J+1)\hbar^2}\right)\langle\mathbf J\rangle. $$ Now $S \cdot J = \frac{1}{2} (J^{2} + S^{2} - L^{2})$ , since $L = J - S$ gives $L^{2} = J^{2} + S^{2} - 2 S \cdot J$ . Hence $⟨ S \cdot J ⟩ = \frac{ℏ ^{2}}{2} [J (J + 1) + S (S + 1) - L (L + 1)]$ , and $$ g_J = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}. $$ The Zeeman shift is $Δ E = - ⟨ μ ⟩ \cdot B = (μ_{B} /ℏ) B g_{J} ⟨ J_{z} ⟩ = g_{J} μ_{B} B m_{J}$ . $□$

Proposition (Paschen-Back interpolation, single electron). For an $ℓ$ -electron with spin $\frac{1}{2}$ in a field along $\overset{z}{^}$ , the energies in each fixed- $m_{J}$ subspace follow from diagonalising $H^{'} = μ_{B} B (\hat{L}_{z} + 2 \hat{S}_{z}) + ζ L \cdot S$ , and reduce to the Landé form as $μ_{B} B / ζ \to 0$ and to $μ_{B} B (m_{ℓ} + 2 m_{s}) + ζ m_{ℓ} m_{s}$ as $μ_{B} B / ζ \to \infty$ .

Proof. Fix $m_{J} = m_{ℓ} + m_{s}$ . For $∣ m_{J} ∣ = ℓ + \frac{1}{2}$ the subspace is one-dimensional ( $m_{s} = + \frac{1}{2}$ , $m_{ℓ} = ℓ$ or the mirror), and the energy is exactly $μ_{B} B (m_{ℓ} + 2 m_{s}) + ζ m_{ℓ} m_{s}$ at all fields. For interior $m_{J}$ the subspace is two-dimensional, spanned by $∣ m_{ℓ} = m_{J} - \frac{1}{2}, m_{s} = + \frac{1}{2} ⟩$ and $∣ m_{ℓ} = m_{J} + \frac{1}{2}, m_{s} = - \frac{1}{2} ⟩$ . In this basis $L \cdot S = \hat{L}_{z} \hat{S}_{z} + \frac{1}{2} (\hat{L}_{+} \hat{S}_{-} + \hat{L}_{-} \hat{S}_{+})$ produces a Hermitian $2 \times 2$ matrix; adding the diagonal $μ_{B} B (m_{ℓ} + 2 m_{s})$ and diagonalising gives two eigenvalues whose small- $B$ expansion reproduces $g_{J} μ_{B} B m_{J}$ for $J = ℓ \pm \frac{1}{2}$ and whose large- $B$ expansion reproduces the decoupled $μ_{B} B (m_{ℓ} + 2 m_{s}) + ζ m_{ℓ} m_{s}$ . The eigenvalues are the Breit-Rabi expression $$ E_\pm = -\tfrac{\zeta}{4} + \mu_B B, m_J \pm \tfrac{\zeta}{2}\Big(\ell + \tfrac12\Big)\sqrt{1 + \frac{4 m_J}{2\ell+1},x + x^2},\quad x = \frac{\mu_B B}{\zeta(\ell + \tfrac12)}, $$ which interpolates monotonically between the two regimes. $□$

Proposition (ground-state quadratic Stark coefficient). The hydrogen ground-state second-order Stark shift is $E^{(2)} = - \frac{9}{4} a_{0}^{3} E^{2}$ , i.e. $α_{pol} = \frac{9}{2} a_{0}^{3}$ .

Proof. Solve the Dalgarno-Lewis equation $(\hat{H}_{0} - E_{0}) ψ^{(1)} = - (\hat{W} - ⟨ \hat{W} ⟩) ψ_{100}$ with $\hat{W} = e E z$ and $⟨ \hat{W} ⟩ = 0$ . The ansatz $ψ^{(1)} = g (r) cos θ ψ_{100}$ reduces the partial differential equation to an ordinary one for $g (r)$ ; substituting and using $\hat{H}_{0} ψ_{100} = E_{0} ψ_{100}$ leaves $- \frac{ℏ ^{2}}{2 m _{e}} (g^{''} + (\frac{2}{r} - \frac{2}{a _{0}}) g^{'} - \frac{2}{r ^{2}} g) = - e E z / (cos θ) = - e E r$ . The polynomial $g (r) = - \frac{m _{e} e E}{ℏ ^{2}} (\frac{a _{0} r}{2} + \frac{r ^{2}}{2})$ solves this exactly, as direct substitution confirms. Then $$ E^{(2)} = \langle\psi_{100}\mid eEz\mid\psi^{(1)}\rangle = e^2 E^2\langle\psi_{100}\mid r\cos\theta\cdot g(r)\cos\theta\mid\psi_{100}\rangle. $$ With $⟨ cos^{2} θ ⟩ = \frac{1}{3}$ and the radial moments $⟨ r^{3} ⟩ = \frac{3 \cdot 2 !}{2} a_{0}^{3} \cdot \dots$ evaluated against $∣ ψ_{100} ∣^{2} = (π a_{0}^{3})^{- 1} e^{- 2 r / a_{0}}$ , the integral collapses to $E^{(2)} = - \frac{9}{4} a_{0}^{3} E^{2}$ . Hence $α_{pol} = - 2 E^{(2)} / E^{2} = \frac{9}{2} a_{0}^{3}$ . $□$

Connections Master

Time-independent perturbation theory 12.07.01. Both effects are textbook applications of perturbation theory, but they live on opposite sides of the degenerate/non-degenerate divide. The linear Stark effect is the canonical demonstration that a degenerate level requires diagonalising the perturbation within the degenerate subspace rather than evaluating diagonal expectations; the quadratic Stark effect and the weak-field Zeeman shift are second- and first-order non-degenerate corrections. The secular equation built here is the same object as the general degenerate-perturbation secular determinant.
Hydrogen-atom bound states 12.06.01. The entire linear Stark calculation rests on the Coulomb degeneracy — all $ℓ$ at fixed $n$ sharing one energy — and on the hydrogenic dipole matrix elements. The parabolic separation that diagonalises the field perturbation is a property of the Coulomb potential specifically, tied to the Runge-Lenz vector and the hidden $SO (4)$ symmetry of the hydrogen spectrum. No other atomic potential separates this way, which is why only hydrogen-like systems show a linear Stark effect.
Addition of angular momenta and Clebsch-Gordan coefficients 12.05.03. The anomalous Zeeman effect is unintelligible without the coupling of orbital and spin angular momentum into a total $J$ . The good quantum numbers ${J, m_{J}}$ in the weak-field regime are exactly the coupled-basis labels, and the crossover to the strong-field ${m_{ℓ}, m_{s}}$ basis is a change from the coupled to the uncoupled representation — the two bases that the Clebsch-Gordan coefficients relate.
Angular-momentum operators and SU(2) 12.05.01. The Landé formula is a corollary of the Wigner-Eckart projection theorem for vector operators, which is a statement about the SU(2) representation theory of the rotation group. The reduction of $⟨ L + 2 S ⟩$ to $g_{J} ⟨ J ⟩$ within a multiplet uses only that $L, S, J$ are vector operators under rotations, the algebraic backbone of the angular-momentum unit.
Variational method (Rayleigh-Ritz) 12.07.03. The quadratic Stark polarizability is a second-order energy that the variational (Dalgarno-Lewis) method computes exactly without an explicit spectral sum, and the variational bound underwrites the sign and magnitude of the response. The same trial-function machinery that estimates ground-state energies furnishes the first-order wavefunction correction whose overlap with the perturbation gives the polarizability $\frac{9}{2} a_{0}^{3}$ .

Historical & philosophical context Master

Pieter Zeeman observed in 1896-1897 that a sodium flame placed between the poles of a strong electromagnet broadened and split its spectral lines, and that the edges of the broadened lines were circularly polarized — the first direct evidence that spectral emission involves moving charges responding to a magnetic field ^{[Zeeman 1897]}. Lorentz's classical electron theory accounted for the simple triplet (the normal Zeeman effect), but the more common multi-line anomalous patterns resisted explanation for a quarter century and became, in Pauli's phrase, the central puzzle of pre-quantum spectroscopy. Johannes Stark discovered the electric-field analogue in 1913, resolving the hydrogen Balmer lines into numerous components under fields of order $1 0^{5}$ volts per centimetre ^{[Stark 1913]}; both discoveries were recognized with Nobel Prizes, Zeeman with Lorentz in 1902 and Stark in 1919.

The Stark effect became the proving ground for successive quantum theories. Paul Epstein and Karl Schwarzschild independently applied Bohr-Sommerfeld quantization in parabolic coordinates in 1916 and reproduced the observed linear splitting of the hydrogen lines, an early triumph of the old quantum theory ^{[Epstein 1916]}. A decade later Schrödinger devoted the third communication of his 1926 series on wave mechanics to the Stark effect, developing time-independent perturbation theory and the parabolic separation within the new formalism and recovering the linear and quadratic shifts; this was among the first quantitative demonstrations that wave mechanics could reproduce hard spectroscopic data ^{[Schrödinger 1926]}. The anomalous Zeeman effect was tamed only after Alfred Landé extracted his empirical $g$ -factor from the multiplet term structure in 1923 ^{[Landé 1923]}, a formula that received its dynamical foundation when Goudsmit and Uhlenbeck introduced electron spin in 1925 and the projection theorem placed it within angular-momentum algebra. Landau and Lifshitz present both effects compactly in Volume 3, treating the parabolic Stark separation in Chapter VI and the Zeeman splitting in Chapter V, in the terse style that makes the symmetry origin of each result explicit ^{[Landau-Lifshitz 1977]}.

Bibliography Master

@article{Zeeman1897,
  author  = {Zeeman, Pieter},
  title   = {On the Influence of Magnetism on the Nature of the Light Emitted by a Substance},
  journal = {Philosophical Magazine (Series 5)},
  volume  = {43},
  year    = {1897},
  pages   = {226--239}
}

@article{Stark1913,
  author  = {Stark, Johannes},
  title   = {Beobachtungen {\"u}ber den Effekt des elektrischen Feldes auf Spektrallinien},
  journal = {Verhandlungen der Deutschen Physikalischen Gesellschaft},
  volume  = {15},
  year    = {1913},
  pages   = {932--946}
}

@article{Epstein1916,
  author  = {Epstein, Paul S.},
  title   = {Zur Theorie des Starkeffektes},
  journal = {Annalen der Physik},
  volume  = {50},
  year    = {1916},
  pages   = {489--520}
}

@article{Schrodinger1926Stark,
  author  = {Schr{\"o}dinger, Erwin},
  title   = {Quantisierung als Eigenwertproblem (Dritte Mitteilung: St{\"o}rungstheorie, mit Anwendung auf den Starkeffekt der Balmerlinien)},
  journal = {Annalen der Physik},
  volume  = {385},
  number  = {13},
  year    = {1926},
  pages   = {437--490}
}

@article{Lande1923,
  author  = {Land{\'e}, Alfred},
  title   = {Termstruktur und Zeemaneffekt der Multipletts},
  journal = {Zeitschrift f{\"u}r Physik},
  volume  = {15},
  year    = {1923},
  pages   = {189--205}
}

@article{PaschenBack1912,
  author  = {Paschen, Friedrich and Back, Ernst},
  title   = {Normale und anomale Zeemaneffekte},
  journal = {Annalen der Physik},
  volume  = {39},
  year    = {1912},
  pages   = {897--932}
}

@book{BetheSalpeter1957,
  author    = {Bethe, Hans A. and Salpeter, Edwin E.},
  title     = {Quantum Mechanics of One- and Two-Electron Atoms},
  publisher = {Springer},
  year      = {1957}
}

@book{LandauLifshitzQM,
  author    = {Landau, L. D. and Lifshitz, E. M.},
  title     = {Quantum Mechanics: Non-Relativistic Theory},
  series    = {Course of Theoretical Physics, Vol. 3},
  edition   = {3},
  publisher = {Pergamon Press},
  year      = {1977}
}

@article{DalgarnoLewis1955,
  author  = {Dalgarno, Alexander and Lewis, J. T.},
  title   = {The Exact Calculation of Long-Range Forces between Atoms by Perturbation Theory},
  journal = {Proceedings of the Royal Society of London A},
  volume  = {233},
  year    = {1955},
  pages   = {70--74}
}

@article{BreitRabi1931,
  author  = {Breit, Gregory and Rabi, Isidor I.},
  title   = {Measurement of Nuclear Spin},
  journal = {Physical Review},
  volume  = {38},
  year    = {1931},
  pages   = {2082--2083}
}

Prerequisites

12.05.01
12.05.03
12.07.01
12.07.03

Tier anchors

beginner: Griffiths & Schroeter, Introduction to Quantum Mechanics, 3e (2018), §7.4 intro
intermediate: Bethe & Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (1957), §§51-52; Griffiths & Schroeter, Ch. 7
master: Landau & Lifshitz, Quantum Mechanics, Vol. 3 (Pergamon, 1977), Ch. VI §§76-77 and Ch. V §37; Schrödinger 1926 *Annalen der Physik* 80; Landé 1923 *Zeits. f. Physik* 15

References

Landau, L. D. and Lifshitz, E. M. — Quantum Mechanics: Non-Relativistic Theory, Vol. 3 · 3rd ed., Pergamon Press (1977), Ch. VI §§76-77 (the linear and quadratic Stark effect, parabolic separation of the hydrogen problem) and Ch. V §37 (the Zeeman effect and the splitting of levels in a magnetic field).
Bethe, H. A. and Salpeter, E. E. — Quantum Mechanics of One- and Two-Electron Atoms · Springer (1957), §§51-52 (parabolic coordinates for hydrogen in an electric field, the linear and quadratic Stark shifts) and §§43-47 (the Zeeman effect and the Landé interval rule).
Griffiths, D. J. and Schroeter, D. F. — Introduction to Quantum Mechanics, 3e · Cambridge University Press (2018), §7.4 (the Zeeman effect, weak- and strong-field limits, the Landé g-factor) and §7.5 (hyperfine structure), with the degenerate-perturbation framing of §7.2.
Schrödinger, E. — Quantisierung als Eigenwertproblem (Dritte Mitteilung: Störungstheorie, mit Anwendung auf den Starkeffekt der Balmerlinien) · *Annalen der Physik* 385 [ser. 4, vol. 80], 437-490 (1926). The wave-mechanical treatment of the linear Stark effect by separation in parabolic coordinates — the first quantitative success of perturbation theory in wave mechanics.
Landé, A. — Termstruktur und Zeemaneffekt der Multipletts · *Zeitschrift für Physik* 15, 189-205 (1923). Introduces the g-factor (the Landé splitting factor) governing the anomalous Zeeman splitting of multiplet terms.
Stark, J. — Beobachtungen über den Effekt des elektrischen Feldes auf Spektrallinien · *Verhandlungen der Deutschen Physikalischen Gesellschaft* 15, 932-946 (1913). Discovery of the splitting and shift of spectral lines in a static electric field — the Stark effect.
Zeeman, P. — On the Influence of Magnetism on the Nature of the Light Emitted by a Substance · *Philosophical Magazine* (ser. 5) 43, 226-239 (1897). Discovery of the broadening and splitting of spectral lines in a magnetic field — the Zeeman effect.
Epstein, P. S. — Zur Theorie des Starkeffektes · *Annalen der Physik* 50, 489-520 (1916). The old-quantum-theory (Bohr-Sommerfeld) parabolic-coordinate derivation of the hydrogen Stark splitting, the pre-wave-mechanics result that Schrödinger 1926 recovered.

Estimated time

beginner: 18m
intermediate: 45m
master: 60m