Tensor Operators and Selection Rules

Tensor operators turn symmetry from a statement about states into a practical method for analyzing matrix elements. The central question is not merely whether an operator commutes with the Hamiltonian. It is:

How does the operator itself transform, and what does that transformation law force its matrix elements to do?

For rotations, the answer is organized by irreducible spherical tensors. A tensor component of rank $k$ carries a definite angular-momentum label, so its matrix elements obey the same coupling algebra used to add angular momenta. The Wigner–Eckart theorem then separates universal angular dependence from a reduced matrix element containing the system-specific dynamics.

The chapter follows three layers that should remain distinct:

1. Classification: determine the operator’s rotational rank, component, and discrete-symmetry character.
2. Symmetry constraints: identify matrix elements that must vanish and relate those that can be nonzero.
3. Dynamics: calculate reduced matrix elements, radial overlaps, rates, linewidths, and intensities.

A selection rule belongs to the first two layers. It does not, by itself, predict that every symmetry-allowed transition is strong.

Canonical Boundaries

This page owns the chapter map and the common reasoning workflow. Detailed proofs, convention choices, and applications remain at their canonical homes.

Topic	Canonical home	Role here
Cartesian operator classification	Scalar, Vector, and Tensor Operators	distinguishes scalars, vectors, and reducible Cartesian tensors
irreducible operator multiplets	Irreducible Spherical Tensors	defines $T_q^{(k)}$, spherical components, and tensor coupling
infinitesimal transformation tests	Commutators with Angular Momentum	derives the commutator criteria for each tensor rank
angular factorization theorem	Wigner–Eckart Theorem	separates angular coefficients from reduced matrix elements
general symmetry zeros	Selection Rules	states what a selection rule means and which assumptions it needs
leading light–matter example	Dipole Transitions	combines vector rank, parity, and polarization for $E1$ transitions
inversion constraints	Parity Selection Rules	owns the even- and odd-operator proof
electromagnetic hierarchy	Multipole Operators	classifies $E\lambda$ and $M\lambda$ operators by rank and parity
atomic application	Applications to Atomic Spectra	connects state labels and multipoles to atomic line assignments
molecular application	Applications to Molecular Rotations	derives rotor rules and explains the role of a permanent dipole

The numerical theory of transition probabilities belongs to Selection Rules in Transition Rates. Experimental resolution, line shapes, and instrumentation belong to Spectroscopy. This chapter determines the symmetry structure of the matrix elements that those subjects use.

The Matrix-Element Problem

Many observables and transition amplitudes reduce to

$$M_{fi} = \langle f|O|i\rangle.$$

Without symmetry, every pair of states and every component of $O$ might require a separate calculation. Symmetry supplies two kinds of information:

• exact zeros, when the state and operator transformation laws are incompatible;
• exact relations, when many component matrix elements share one reduced dynamical quantity.

For angular-momentum eigenstates, write

$$\begin{aligned} J^2|\alpha,j,m\rangle &= \hbar^2j(j+1)|\alpha,j,m\rangle, \\ J_z|\alpha,j,m\rangle &= \hbar m|\alpha,j,m\rangle. \end{aligned}$$

The label $\alpha$ collects everything not fixed by $j$ and $m$: radial quantum numbers, parity, spin-coupling labels, electronic configurations, vibrational labels, or degeneracy indices. Rotations organize the $j,m$ dependence. They do not remove the $\alpha$ dependence.

Operators Carry Rotation Representations

Let

$$U(R) = \exp\!\left( -\frac{i}{\hbar} \boldsymbol\theta\cdot\mathbf J \right)$$

implement a rotation. A collection of operators $O_a$ is closed under rotations when conjugation mixes only operators in that collection:

$$U^\dagger(R)O_aU(R) = \sum_b D_{ab}(R)O_b.$$

This is a representation of rotations on operator space. A rotational scalar forms a one-dimensional trivial representation. Three vector components form the ordinary three-dimensional representation. Higher Cartesian tensors are usually reducible and must be split into irreducible pieces.

Two conjugation conventions are common. Some authors display $U^\dagger O U$, while others define an active transformed operator as $UOU^\dagger$. Their finite-rotation matrices are related by inversion or complex conjugation. The invariant content is the representation carried by the operator; the commutators below fix the sign convention used for calculations.

Scalars

A scalar satisfies

$$U^\dagger(R)OU(R)=O$$

for every proper rotation, equivalently

$$[J_i,O]=0 \qquad (i=x,y,z).$$

Examples include $R^2$, $P^2$, $\mathbf L\cdot\mathbf S$, and a central Hamiltonian when $\mathbf J$ generates the rotation of all relevant degrees of freedom.

Scalar under rotations does not mean constant in time. Rotational invariance is tested with $\mathbf J$; conservation is tested with $H$:

$$[J_i,O]=0 \quad\hbox{and}\quad [H,O]=0$$

answer different questions.

Vectors

A vector operator $\mathbf V$ obeys

$$[J_i,V_j] = i\hbar \sum_k \epsilon_{ijk}V_k.$$

Position, momentum, and electric dipole moment are polar vectors. Orbital angular momentum, spin, and magnetic moment are axial vectors. Both types have rotational rank $1$, but they transform differently under parity. Rotational rank and parity must therefore be recorded as separate labels.

Cartesian tensors

A rank-two Cartesian tensor transforms with one rotation matrix on each index. Under rotations its nine components decompose as

$$3\otimes3 = 1\oplus3\oplus5.$$

Equivalently, in angular-momentum language,

$$1\otimes1 = 0\oplus1\oplus2.$$

The pieces are:

• a scalar trace, carrying rank $0$;
• an antisymmetric part, equivalent to an axial vector of rank $1$;
• a symmetric traceless part, carrying rank $2$.

This decomposition explains why the five independent components of a quadrupole tensor belong together. It also shows why the number of Cartesian indices is not, by itself, the irreducible angular-momentum rank.

Irreducible Spherical Tensors

An irreducible spherical tensor of rank $k$ is a set

$$\left\{ T_q^{(k)} \right\}_{q=-k}^{k}$$

with $2k+1$ components. Its defining commutators are

$$[J_z,T_q^{(k)}] = \hbar q\,T_q^{(k)},$$

and

$$\begin{aligned} [J_\pm,T_q^{(k)}] &= \hbar \sqrt{ k(k+1)-q(q\pm1) } \\ &\quad\times T_{q\pm1}^{(k)}. \end{aligned}$$

The operator components therefore form an angular-momentum multiplet under the adjoint action of rotations. Rank $k$ means the operator transforms like angular momentum $k$; it is not a statement about matrix rank.

For a vector, a standard spherical-component convention is

$$V_0=V_z, \qquad V_{\pm1} = \mp\frac{V_x\pm iV_y}{\sqrt2}.$$

The spherical basis is adapted to the ladder algebra: $q$ directly records the change in magnetic quantum number that the component can produce.

Tensor operators can themselves be coupled. If $A^{(k_1)}$ and $B^{(k_2)}$ are irreducible tensors, their coupled rank-$K$ product is built with Clebsch–Gordan coefficients:

$$\begin{aligned} \left[ A^{(k_1)} \otimes B^{(k_2)} \right]_Q^{(K)} &= \sum_{q_1,q_2} \mathcal C_{q_1q_2}^{KQ} \\ &\quad\times A_{q_1}^{(k_1)} B_{q_2}^{(k_2)}, \\ \mathcal C_{q_1q_2}^{KQ} &\equiv \langle k_1,q_1; k_2,q_2 |K,Q \rangle. \end{aligned}$$

This is the operator analogue of angular-momentum addition. The same triangle rules and phase conventions appear in both subjects.

From Commutators to Matrix-Element Rules

The commutators already reveal the magnetic selection rule. Insert angular-momentum eigenstates into

$$[J_z,T_q^{(k)}] = \hbar q\,T_q^{(k)}.$$

The left side gives

$$\begin{aligned} &\langle\alpha',j',m'| [J_z,T_q^{(k)}] |\alpha,j,m\rangle \\ &\quad = \hbar(m'-m) \langle\alpha',j',m'| T_q^{(k)} |\alpha,j,m\rangle. \end{aligned}$$

Comparison with the right side yields

$$\left( m'-m-q \right) \langle\alpha',j',m'| T_q^{(k)} |\alpha,j,m\rangle =0.$$

Hence a nonzero matrix element requires

$$m'=m+q.$$

This derivation gives one necessary condition. The complete rotational structure, including the allowed $j'$ values and relations among all magnetic components, is the content of the Wigner–Eckart theorem.

Wigner–Eckart Factorization

Using the reduced-matrix-element convention adopted in this chapter,

$$\begin{aligned} &\langle\alpha',j',m'| T_q^{(k)} |\alpha,j,m\rangle \\ &\quad = \mathcal A_{m'mq}^{j'kj} \langle \alpha',j' \lVert T^{(k)}\rVert \alpha,j \rangle, \\ \mathcal A_{m'mq}^{j'kj} &\equiv (-1)^{j'-m'} \begin{pmatrix} j'&k&j\\ -m'&q&m \end{pmatrix}. \end{aligned}$$

Equivalently,

$$\begin{aligned} &\langle\alpha',j',m'| T_q^{(k)} |\alpha,j,m\rangle \\ &\quad = \frac{ \langle j,m;k,q|j',m'\rangle }{ \sqrt{2j'+1} } \langle \alpha',j' \lVert T^{(k)}\rVert \alpha,j \rangle. \end{aligned}$$

The angular coefficient is fixed by symmetry. The reduced matrix element is independent of $m$, $m'$, and $q$, but it may depend on $j$, $j'$, $\alpha$, $\alpha'$, the physical operator, and the normalization convention.

Different references distribute factors such as $\sqrt{2j+1}$ and $\sqrt{2j'+1}$ differently. A reduced matrix element is meaningful only together with its stated Wigner–Eckart convention.

What follows immediately

The $3j$ symbol vanishes unless

$$m'=m+q$$

and

$$|j-k| \le j' \le j+k.$$

These are necessary rotational conditions. They do not include parity, particle exchange, point-group symmetry, charge-like quantum numbers, or accidental zeros of the reduced matrix element.

What does not follow

The theorem does not determine:

• the energy difference between the states;
• the numerical reduced matrix element;
• whether a radial integral is unusually small;
• whether an independent discrete symmetry forbids the transition;
• the density of final states or the observed linewidth;
• which weak mechanism dominates when the leading operator is forbidden.

Symmetry can reduce a large family of calculations to one reduced quantity. It does not replace the calculation of that quantity.

Selection Rules Are Conditional Zero Statements

A selection rule is an implication of the form

$$\begin{gathered} \text{stated symmetry assumptions} \\ \Longrightarrow \\ \langle f|O|i\rangle=0. \end{gathered}$$

Every application should therefore identify:

1. the Hamiltonian approximation and its symmetries;
2. the good quantum numbers of the states;
3. the transformation law of the operator;
4. the perturbative order or multipole being retained;
5. any additional symmetry-breaking terms or state mixing.

The word forbidden is shorthand for forbidden by a specified operator under specified assumptions. A line forbidden at electric-dipole order may occur through magnetic dipole, electric quadrupole, two-photon, hyperfine-mixing, or external-field mechanisms.

The General Discrete-Symmetry Test

Suppose a unitary symmetry has

$$U|i\rangle=u_i|i\rangle, \qquad U|f\rangle=u_f|f\rangle,$$

and the operator has a definite character,

$$UOU^\dagger=u_OO.$$

Then

$$\langle f|O|i\rangle = u_f^*u_Ou_i \langle f|O|i\rangle.$$

A nonzero matrix element therefore requires

$$u_f^*u_Ou_i=1.$$

This one-line test underlies parity rules and many charge-like or point-group selection rules. For a continuous symmetry the analogous information is often expressed through generator commutators and additive quantum numbers.

Parity Is Independent of Rotational Rank

Let

$$\Pi|i\rangle=\pi_i|i\rangle, \qquad \Pi|f\rangle=\pi_f|f\rangle,$$

with $\pi_i,\pi_f=\pm1$, and let

$$\Pi O\Pi^{-1} = \eta_OO.$$

A nonzero matrix element requires

$$\pi_f\eta_O\pi_i=1.$$

Thus:

Operator parity	Required relation between state parities
even, $\eta_O=+1$	$\pi_f=\pi_i$
odd, $\eta_O=-1$	$\pi_f=-\pi_i$

Rotational rank and parity answer different questions. Position $\mathbf R$ and angular momentum $\mathbf L$ are both rank-$1$ operators under rotations, but

$$\Pi\mathbf R\Pi^{-1} = -\mathbf R, \qquad \Pi\mathbf L\Pi^{-1} = \mathbf L.$$

The first is a polar vector and the second an axial vector. Treating every vector as parity odd is a common and consequential error.

For central-potential orbital states,

$$\pi_\ell=(-1)^\ell.$$

An operator of parity $\eta_O$ can connect $\ell_i$ and $\ell_f$ only if

$$(-1)^{\ell_f-\ell_i} = \eta_O.$$

This parity condition must then be intersected with the rotational triangle rule.

Electric Dipole Transitions as a Complete Audit

In the electric-dipole approximation,

$$\mathbf d = \sum_a q_a\mathbf r_a$$

couples to the electric field. Its spherical components $d_q$ have

$$k=1, \qquad q=-1,0,1, \qquad \eta_{E1}=-1.$$

The three pieces of the audit are:

Input	Consequence
rank $k=1$	$
component $q$	$m_f=m_i+q$
odd parity	$\pi_f=-\pi_i$

For total angular momentum, one often summarizes the triangle condition as

$$\Delta j=0,\pm1, \qquad 0\not\leftrightarrow0,$$

with the understanding that all values must satisfy the actual triangle inequalities.

For spinless central-potential orbital states, rotations alone allow

$$\ell_f=\ell_i-1,\ell_i,\ell_i+1.$$

Odd parity removes $\ell_f=\ell_i$, leaving

$$\Delta\ell=\pm1.$$

There is no universal electric-dipole rule for $\Delta n$. Radial overlap, energy conservation, and the detailed Hamiltonian determine the dependence on principal or vibrational quantum numbers.

Polarization

Relative to a chosen quantization axis,

Tensor component	Magnetic rule
$q=0$	$\Delta m=0$
$q=+1$	$\Delta m=+1$
$q=-1$	$\Delta m=-1$

The association of the two circular-polarization names with $q=\pm1$ depends on propagation direction and convention. The invariant statement is $\Delta m=q$ for the component appearing in the matrix element.

Multipoles Organize the Alternatives

Electromagnetic multipole operators carry both a rotational rank $\lambda$ and a parity:

$$\eta_{E\lambda} = (-1)^\lambda, \qquad \eta_{M\lambda} = (-1)^{\lambda+1}.$$

The lowest cases are:

Multipole	Rank	Parity	State-parity relation
$E1$	$1$	odd	opposite
$M1$	$1$	even	same
$E2$	$2$	even	same
$M2$	$2$	odd	opposite

For any rank-$\lambda$ multipole,

$$|J_i-\lambda| \le J_f \le J_i+\lambda,$$

and

$$M_f=M_i+q, \qquad q=-\lambda,\ldots,\lambda.$$

These conditions determine symmetry-allowed channels. They do not rank their strengths. In the long-wavelength regime, higher multipoles are usually suppressed by additional powers of a size-to-wavelength parameter such as $ka$, but that suppression is a dynamical scaling statement rather than a selection rule.

The hierarchy becomes especially useful when $E1$ vanishes. Instead of declaring the transition impossible, ask which next operator has compatible rank and parity.

Atomic Spectra: Labels Before Integrals

Atomic line positions are set by energy differences,

$$\hbar\omega_{fi} = E_f-E_i,$$

while line strengths depend on transition matrix elements. This distinction is fundamental: selection rules organize amplitudes, not spectra by themselves.

For a spinless hydrogenic state $|n,\ell,m\rangle$, the electric-dipole matrix element separates into radial and angular factors. Symmetry gives

$$\Delta\ell=\pm1, \qquad \Delta m=q,$$

while the radial integral controls the remaining magnitude.

With fine structure, the useful labels are instead often

$$|\alpha,J,M,\pi\rangle.$$

Then the robust $E1$ rules are

$$\Delta J=0,\pm1, \qquad 0\not\leftrightarrow0,$$ $$\Delta M=q, \qquad \pi_f=-\pi_i.$$

In an $LS$-coupling approximation, the leading electric dipole operator does not act directly on spin, giving the familiar approximate rule

$$\Delta S=0.$$

Spin–orbit coupling, configuration interaction, hyperfine mixing, and relativistic corrections can weaken that rule. Total $J$, parity, and the actual mixed eigenstates provide the safer description when pure $LS$ labels cease to be accurate.

Molecular Rotations: The Operator Must Exist

For an ideal linear rotor,

$$E_J=BJ(J+1), \qquad J=0,1,2,\ldots$$

but the energy ladder alone does not determine which transitions a probe can drive.

A polar linear molecule has a body-fixed permanent dipole

$$\boldsymbol\mu = \mu\hat{\mathbf n}.$$

Its laboratory components form a rank-$1$ tensor. The angular matrix element contains

$$\begin{pmatrix} J'&1&J\\ 0&0&0 \end{pmatrix},$$

whose zero structure removes the $\Delta J=0$ option for an ordinary pure rotational electric-dipole transition. The result is

$$\Delta J=\pm1, \qquad \Delta M=0,\pm1.$$

An ideal homonuclear diatomic molecule has no permanent electric dipole. Its rotational levels still exist, but the leading microwave $E1$ operator is absent. Rotational Raman scattering can instead probe the rank-$2$ part of the polarizability, producing the characteristic shifted branches

$$\Delta J=\pm2.$$

This example is a useful corrective: a selection rule is always a statement about states and an operator. Energy levels do not carry transition rules on their own.

A Reliable Selection-Rule Workflow

For a new problem, use the following order.

1. State the Hamiltonian regime

Identify which terms are retained and which symmetries are exact or approximate. Decide whether the useful labels are $\ell,m$, $j,m_j$, $L,S,J,M$, molecular rotor labels, or field-dressed labels.

2. Identify the physical operator

Specify whether the process is driven by $E1$, $M1$, $E2$, a static-field perturbation, a spin operator, a polarizability tensor, or another interaction. Different operators imply different rules between the same states.

3. Decompose into irreducible components

Determine $k$ and $q$. Remove reducible traces or antisymmetric pieces when necessary. Convert Cartesian components to spherical components before applying Wigner–Eckart machinery.

4. Apply the rotational tests

Check

$$|j_i-k| \le j_f \le j_i+k$$

and

$$m_f=m_i+q.$$

For special orbital or rotor matrix elements, inspect any additional zero in the relevant $3j$ coefficient rather than relying only on a shorthand $\Delta j$ list.

5. Apply independent symmetries

Check parity, exchange symmetry, charge-like quantum numbers, molecular point-group character, and any other exact symmetry of the approximation. Intersect the conditions; do not substitute one for another.

6. Inspect the reduced dynamics

If symmetry permits the matrix element, calculate or obtain the reduced matrix element. An allowed channel may still vanish accidentally or be strongly suppressed.

7. Convert amplitudes into observables

Rates and intensities require the squared amplitude together with phase space, populations, polarization, resonance conditions, and experimental geometry. This is where transition theory begins.

When the Rules Change

Selection rules can be weakened without becoming meaningless. If an exact-symmetry state $|a\rangle$ mixes with a small component of different symmetry,

$$|\widetilde a\rangle = |a\rangle +\epsilon|b\rangle, \qquad |\epsilon|\ll1,$$

then a formerly forbidden matrix element can appear at first order in $\epsilon$. External fields, spin–orbit coupling, hyperfine interactions, configuration mixing, lattice environments, and asymmetric boundaries all produce such effects in suitable systems.

The correct conclusion is not that the original rule failed. The rule identified the leading zero and predicts why the symmetry-breaking amplitude is small. The mixed-state calculation then quantifies the violation.

Strong fields can also change which quantum numbers are good. A rule written in weak-field $LS$ labels should not be applied mechanically after the Hamiltonian has crossed into a different coupling regime.

Chapter Map

The chapter is designed to be read in this order:

1. Scalar, Vector, and Tensor Operators develops the transformation viewpoint in Cartesian language.
2. Irreducible Spherical Tensors introduces the basis adapted to angular momentum.
3. Commutators with Angular Momentum gives the infinitesimal tests and fixes signs.
4. Wigner–Eckart Theorem factorizes matrix elements.
5. Selection Rules makes the assumptions behind a symmetry zero explicit.
6. Dipole Transitions works through the leading electromagnetic example.
7. Parity Selection Rules isolates the inversion argument and common operator parities.
8. Multipole Operators organizes higher electromagnetic mechanisms.
9. Applications to Atomic Spectra applies the workflow to orbital, fine-structure, and many-electron labels.
10. Applications to Molecular Rotations applies it to rotor states, permanent dipoles, and Raman alternatives.

For calculations, continue to Selection Rule Problems and the Wigner–Eckart Formula Card.

Reading Paths

First graduate pass

Read the operator classification, spherical tensors, Wigner–Eckart theorem, general selection rules, and dipole transitions. Then solve the exercises on this page before entering the application articles.

Atomic and optical physics

Focus on Wigner–Eckart factorization, parity, dipole transitions, multipoles, and atomic spectra. Keep the distinction between term labels and exact mixed eigenstates visible throughout.

Molecular spectroscopy

Review spherical tensors and $3j$ symbols, then move to molecular rotations. The permanent-dipole criterion and the tensor rank of the polarizability are as important as the rotor energy formula.

Reference use

Use the chapter map to locate the relevant proof, then use the formula cards for convention-checked lookup. Never import a reduced matrix element or Wigner symbol from another convention without checking its normalization and phases.

Distinctions Worth Keeping

Do not conflate	Why
rotational scalar and conserved observable	the first concerns $[J_i,O]$; the second concerns $[H,O]$
Cartesian tensor order and irreducible rank	a Cartesian rank-two tensor contains $k=0,1,2$ pieces
vector and polar vector	rotational rank does not determine parity
allowed and large	symmetry permits a channel but does not set its reduced matrix element
forbidden and impossible	another operator, higher order, or symmetry breaking may contribute
line position and line strength	energies set frequencies; matrix elements set amplitudes
$\Delta j$ shorthand and the triangle condition	boundary cases such as $0\leftrightarrow0$ require the full condition
exact and coupling-scheme-dependent spin rules	state mixing can invalidate pure $LS$ labels

Common Mistakes

• Applying Wigner–Eckart before verifying that the operator is an irreducible spherical tensor.
• Mixing finite-rotation conjugation conventions and then changing a commutator sign.
• Treating $q$ as the rank; $k$ labels the multiplet and $q$ labels one component.
• Forgetting that a reducible Cartesian tensor must be decomposed before assigning one rank.
• Deriving $\Delta\ell=\pm1$ from vector rank alone; parity removes the $\Delta\ell=0$ option for $E1$.
• Assuming every rotational vector is parity odd; axial vectors are even.
• Calling a transition forbidden without naming the operator and Hamiltonian approximation.
• Treating a nonzero Clebsch–Gordan coefficient as a prediction of a large rate.
• Using a reduced matrix element without checking the convention defining it.
• Applying weak-field or pure-coupling labels after state mixing has made them approximate.

Cross-Links

• Why Symmetry Matters
• Rotations and Orbital Angular Momentum
• Addition of Angular Momentum
• Discrete Symmetries
• Parity
• Clebsch–Gordan Coefficients
• Recoupling and Wigner Symbols
• Selection Rule Problems
• Wigner 3j, 6j, and 9j Symbols
• Wigner–Eckart Formula Card
• Selection Rules in Transition Rates
• Transition Rates in Light–Matter Interaction
• Spectroscopy
• Hydrogen Atom Angular Structure
• Rigid Rotor
• From Selection Rules to Ward Identities

References

• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• M. E. Rose, Elementary Theory of Angular Momentum, Wiley, 1957.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.
• D. M. Brink and G. R. Satchler, Angular Momentum, 3rd ed., Oxford University Press, 1993.
• R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Wiley, 1988.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions, Wiley, 1992.
• I. I. Sobelman, Atomic Spectra and Radiative Transitions, 2nd ed., Springer, 1992.

Exercises

1. A Cartesian operator $T_{ij}$ has no assumed symmetry in its indices. Decompose its rotational content and count the components in each irreducible piece.

Solution

Two vector indices transform in

$$1\otimes1 = 0\oplus1\oplus2.$$

The trace

$$T^{(0)} = \frac13 \sum_i T_{ii}$$

is one scalar component. The antisymmetric part

$$A_{ij} = \frac12 \left( T_{ij}-T_{ji} \right)$$

has three independent components and is equivalent to a rank-$1$ axial vector. The symmetric traceless part

$$Q_{ij} = \frac12 \left( T_{ij}+T_{ji} \right) - \frac13\delta_{ij} \sum_kT_{kk}$$

has five independent components and carries rank $2$. The count is

$$1+3+5=9,$$

matching the nine original Cartesian components.

2. A rank-$2$ tensor component with $q=+1$ acts on an initial state with $j_i=1/2$ and $m_i=1/2$. Which final $j_f,m_f$ values survive the basic rotational rules?

Solution

The magnetic rule gives

$$m_f=m_i+q = \frac12+1 = \frac32.$$

The triangle rule gives

$$\left| \frac12-2 \right| \le j_f \le \frac12+2,$$

so

$$j_f=\frac32 \quad\hbox{or}\quad j_f=\frac52.$$

Both multiplets contain $m_f=3/2$. Thus the basic rotational rules permit

$$\left( j_f,m_f \right) = \left( \frac32,\frac32 \right), \qquad \left( \frac52,\frac32 \right).$$

An additional symmetry or a zero reduced matrix element could still remove either channel.

3. Two states have the same parity, with $J_i=0$ and $J_f=2$. Among $E1$, $M1$, $E2$, and $M2$, which multipole first passes both the rank and parity tests?

Solution

A transition from $J_i=0$ to $J_f=2$ requires an operator whose rank can couple $0$ to $2$. Rank $1$ cannot do so, so both $E1$ and $M1$ fail the triangle rule.

Rank $2$ passes the angular test. The states have the same parity, so the operator must be parity even. $E2$ has parity

$$(-1)^2=+1,$$

whereas $M2$ has parity

$$(-1)^{2+1}=-1.$$

Therefore $E2$ is the first of the listed multipoles that passes both tests. This does not determine the size of its reduced matrix element.

4. Consider a central-potential electric-dipole transition

$$\begin{gathered} |n_i,\ell_i=2,m_i=1\rangle \\ \longrightarrow \\ |n_f,\ell_f=1,m_f=0\rangle \end{gathered}$$

Which spherical component can drive it, and do the rotational and parity rules permit the transition?

Solution

The magnetic change is

$$q = m_f-m_i = -1,$$

so the $d_{-1}$ component is required.

The orbital change is

$$\Delta\ell=-1,$$

which satisfies the rank-$1$ triangle rule. The initial parity is

$$(-1)^2=+1,$$

and the final parity is

$$(-1)^1=-1.$$

They are opposite, as required by the odd electric-dipole operator. The transition is therefore allowed by these symmetry tests. Its magnitude still depends on the radial reduced matrix element.

5. An ideal homonuclear diatomic molecule has rotational levels $E_J=BJ(J+1)$ but no permanent electric dipole. Explain why the energy spacing does not imply an ordinary microwave $E1$ spectrum, and name a tensor operator that can reveal rotational structure by another mechanism.

Solution

The level spacing determines possible photon energies, but an $E1$ transition also requires a nonzero electric-dipole matrix element. An ideal homonuclear diatomic molecule has no permanent body-fixed dipole, so the leading pure rotational $E1$ operator is absent even though the rotor levels exist.

The anisotropic polarizability contains a rank-$2$ irreducible tensor. It can produce rotational Raman transitions with characteristic shifted branches

$$\Delta J=\pm2.$$

This is a different operator and a different light–matter process, not a violation of the electric-dipole rule.