Parity, discrete-symmetry groups, and the Wigner-Eckart theorem

Intuition Beginner

Hold your right hand up to a mirror and the reflection looks like a left hand. The mirror has swapped one sense of space for its opposite. Parity is the quantum-mechanical version of this swap, except it inverts all three directions at once: every point at position $(x,y,z)$ is sent to $(-x,-y,-z)$, the reflection of the world through its centre. Parity asks a simple question of a physical system: if you invert space, do you get back a legal state of the same system?

Many systems do not notice the inversion. An electron bound to a nucleus by a force that only depends on distance looks the same whether you invert space or not. When that holds, parity sorts every energy state into one of two clean types. Even states are unchanged by the inversion. Odd states flip their overall sign. There is no third option, because inverting space twice brings you back to where you started.

This two-way sorting is more useful than it sounds. It forbids certain transitions outright: a light-emitting jump between two states of the same type simply cannot happen. That single rule explains why some spectral lines are bright and others are missing entirely, long before you compute a single number.

Visual Beginner

Picture a single orbital drawn as a cloud of charge around a nucleus. The top row shows an even orbital: invert it through the centre and the picture is identical, lobe for lobe. The bottom row shows an odd orbital, shaped like a dumbbell with a positive lobe on one side and a negative lobe on the other. Invert that one and the positive and negative lobes swap places, so the whole picture comes back with its sign reversed.

The picture records the one fact that drives this whole unit: inversion through the centre acts on each state by multiplying it by $+1$ or $-1$, and which one it is depends only on the shape of the state. The number of nodes, the orbital's angular pattern, fixes the sign. A jump that emits light has to change that sign, and the picture lets you read off whether a given pair of orbitals is allowed to talk to each other.

Worked example Beginner

Take a hydrogen-like atom and ask whether an electron can drop from a $2p$ orbital to a $1s$ orbital by emitting a photon. The rule we will use: such a jump is allowed only when the two orbitals have opposite parity, one even and one odd.

Step 1. Find the parity of the $1s$ orbital. Its angular shape is a sphere, the same in every direction. Inverting space through the centre leaves a sphere unchanged, so $1s$ is even, parity $+1$.

Step 2. Find the parity of the $2p$ orbital. A $p$ orbital is a dumbbell with a plus lobe and a minus lobe on opposite sides. Inverting through the centre swaps the two lobes, turning plus into minus, so the orbital comes back times $-1$. The $2p$ orbital is odd, parity $-1$.

Step 3. Compare. One orbital is even and the other is odd, so they have opposite parity. The jump $2p\to 1s$ is allowed, and indeed it is one of the brightest lines in the hydrogen spectrum.

Step 4. Contrast with a forbidden case. A $2s$ orbital, like $1s$, is spherical and even. A jump from $2s$ to $1s$ keeps the parity even-to-even, the same on both ends. By the rule, that jump cannot emit a photon in the simplest way, and experiment confirms the $2s$ state is unusually long-lived.

What this tells us: the parity of an orbital is fixed by its shape, the simple light-emitting jump demands a change of parity, and that single demand already sorts spectral lines into allowed and forbidden before any detailed calculation.

Check your understanding Beginner

Formal definition Intermediate+

Let $\mathcal{H}=L^2({\mathbb{R}}^3)\otimes {\mathcal{H}}_{\mathrm{spin}}$ be the state space of a particle. The parity operator $\hat{P}$ acts on the spatial wavefunction by inversion of the argument,

$$(\hat{P}\psi )(\mathbf{r})=\psi (-\mathbf{r}),$$

and acts as the identity on the spin factor. From this definition $\hat{P}$ is linear, and the change of variables $\mathbf{r}\mapsto -\mathbf{r}$ in the $L^2$ inner product shows it is unitary, ${\hat{P}}^{†}\hat{P}=1$. Inverting twice returns the argument, so ${\hat{P}}^2=1$; together with unitarity this gives ${\hat{P}}^{†}={\hat{P}}^{-1}=\hat{P}$, so $\hat{P}$ is also Hermitian. A Hermitian involution has spectrum contained in $\{+1,-1\}$: any eigenvalue $\pi$ satisfies ${\pi }^2=1$. States with $\hat{P}\psi =+\psi$ are even (parity $+1$); states with $\hat{P}\psi =-\psi$ are odd (parity $-1$).

The action on the canonical observables follows from the definition. Position and momentum are both polar vectors, reversing under inversion, while orbital angular momentum is an axial vector, invariant:

$$\hat{P}\hat{\mathbf{r}}{\hat{P}}^{-1}=-\hat{\mathbf{r}},\hat{P}\hat{\mathbf{p}}{\hat{P}}^{-1}=-\hat{\mathbf{p}},\hat{P}\hat{\mathbf{L}}{\hat{P}}^{-1}=+\hat{\mathbf{L}}.$$

The angular-momentum sign is forced: $\hat{\mathbf{L}}=\hat{\mathbf{r}}\times \hat{\mathbf{p}}$ is a product of two sign-reversing factors, so it is parity-even. Spin $\hat{\mathbf{S}}$, being a genuine angular momentum, is assigned the same even signature, hence $\hat{P}\hat{\mathbf{J}}{\hat{P}}^{-1}=+\hat{\mathbf{J}}$ for total $\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}$. Because $\hat{P}$ commutes with every component of $\hat{\mathbf{J}}$, parity is compatible with the angular-momentum labels and can be diagonalised simultaneously with ${\hat{J}}^2$ and ${\hat{J}}_z$ ^{[Landau-Lifshitz §§91-94]}.

When a particle species carries an internal degree of freedom unchanged by space, one assigns it an intrinsic parity ${\eta }_P=\pm 1$, a multiplicative quantum number entering the total parity of a composite as the product of intrinsic parities times the spatial factor $(-1{)}^{\sum {\ell }_i}$.

An irreducible tensor operator of rank $k$ is a family of $2k+1$ operators ${\hat{T}}_q^k$, with $q=-k,-k+1,\dots ,k$, satisfying the rotation commutation relations

$$[{\hat{J}}_z,{\hat{T}}_q^k]=\hslash q{\hat{T}}_q^k,[{\hat{J}}_{\pm },{\hat{T}}_q^k]=\hslash \sqrt{k(k+1)-q(q\pm 1)}{\hat{T}}_{q\pm 1}^k,$$

where ${\hat{J}}_{\pm }={\hat{J}}_x\pm i{\hat{J}}_y$. These are precisely the relations the spin-$k$ angular-momentum states $\mid kq⟩$ obey under ${\hat{J}}_z$ and ${\hat{J}}_{\pm }$ from 12.05.01, so ${\hat{T}}_q^k$ transforms under rotations like the multiplet $\{\mid kq⟩\}$, equivalently like the spherical harmonic $Y_{kq}$. A scalar is the rank-$0$ case; the components of a vector operator $\hat{\mathbf{V}}$ assemble into the rank-$1$ spherical components ${\hat{V}}_{\pm 1}^1=\mp ({\hat{V}}_x\pm i{\hat{V}}_y)/\sqrt{2}$, ${\hat{V}}_0^1={\hat{V}}_z$.

Counterexamples to common slips

• Parity is not the same as a single reflection. Reflecting in one plane, say $x\to -x$ with $y,z$ fixed, is the composition of the full inversion with a rotation by $\pi$; only the full three-axis inversion is the operator $\hat{P}$ with ${\hat{P}}^2=1$ defined above.
• An operator's three Cartesian components do not automatically form a rank-$1$ tensor operator in the spherical sense. The spherical components carry the factors $\mp 1/\sqrt{2}$ and the combination ${\hat{V}}_x\pm i{\hat{V}}_y$; using $({\hat{V}}_x,{\hat{V}}_y,{\hat{V}}_z)$ directly as $({\hat{T}}_{-1}^1,{\hat{T}}_0^1,{\hat{T}}_{+1}^1)$ violates the $[{\hat{J}}_{\pm },{\hat{T}}_q^1]$ relation and corrupts every Clebsch-Gordan reduction.
• A non-zero reduced matrix element does not guarantee a non-zero transition. The Clebsch-Gordan factor can vanish even when $⟨{\alpha }^{\prime }{j}^{\prime }\Vert T^k\Vert \alpha j⟩\ne 0$; the selection rules live in the geometric coefficient, not in the reduced part.

Key theorem with proof Intermediate+

Theorem (parity of spherical harmonics, and the dipole selection rule). The parity operator acts on the spherical harmonics by $\hat{P}{Y}_{\ell m}=(-1{)}^{\ell }{Y}_{\ell m}$. Consequently the orbital parity of a state of angular momentum $\ell$ is $(-1{)}^{\ell }$, and a matrix element of the position operator $⟨{\ell }^{\prime }{m}^{\prime }\mid \hat{\mathbf{r}}\mid \ell m⟩$ vanishes unless ${\ell }^{\prime }+\ell$ is odd. In an electric-dipole transition this forces a change of orbital parity, the Laporte rule, and (combined with the rank-$1$ triangle condition) gives $\Delta \ell =\pm 1$.

Proof. Write a point on the unit sphere in polar coordinates as $(\theta ,\varphi )$. Spatial inversion $\mathbf{r}\mapsto -\mathbf{r}$ fixes the radius and sends the direction to its antipode, which in angles is $\theta \mapsto \pi -\theta$, $\varphi \mapsto \varphi +\pi$. The spherical harmonic factorises as $Y_{\ell m}(\theta ,\varphi )=N_{\ell m}{P}_{\ell }^m(\cos \theta )e^{im\varphi }$ with $N_{\ell m}$ a normalisation constant. Evaluate each factor under the antipodal map. The azimuthal factor gives $e^{im(\varphi +\pi )}=e^{im\pi }{e}^{im\varphi }=(-1{)}^m{e}^{im\varphi }$. For the polar factor, $\cos (\pi -\theta )=-\cos \theta$, and the associated Legendre functions satisfy the parity relation $P_{\ell }^m(-x)=(-1{)}^{\ell +m}{P}_{\ell }^m(x)$, established from the Rodrigues formula by counting sign changes of $(d/dx{)}^{\ell +m}(x^2-1{)}^{\ell }$ under $x\mapsto -x$. Multiplying the two factors,

$$\hat{P}{Y}_{\ell m}(\theta ,\varphi )=(-1{)}^{\ell +m}(-1{)}^m{Y}_{\ell m}(\theta ,\varphi )=(-1{)}^{\ell +2m}{Y}_{\ell m}(\theta ,\varphi )=(-1{)}^{\ell }{Y}_{\ell m}(\theta ,\varphi ),$$

since $(-1{)}^{2m}=1$ for integer $m$. This proves the parity formula, and the orbital parity of any state with definite $\ell$ is therefore $(-1{)}^{\ell }$.

For the selection rule, the position operator $\hat{\mathbf{r}}$ is parity-odd, $\hat{P}\hat{\mathbf{r}}{\hat{P}}^{-1}=-\hat{\mathbf{r}}$. Insert ${\hat{P}}^{-1}\hat{P}=1$ on both sides of the matrix element and use $\hat{P}\mid \ell m⟩=(-1{)}^{\ell }\mid \ell m⟩$:

$$⟨{\ell }^{\prime }{m}^{\prime }\mid \hat{\mathbf{r}}\mid \ell m⟩=⟨{\ell }^{\prime }{m}^{\prime }\mid {\hat{P}}^{-1}(\hat{P}\hat{\mathbf{r}}{\hat{P}}^{-1})\hat{P}\mid \ell m⟩=(-1{)}^{{\ell }^{\prime }}(-1)(-1{)}^{\ell }⟨{\ell }^{\prime }{m}^{\prime }\mid \hat{\mathbf{r}}\mid \ell m⟩.$$

The matrix element equals $-(-1{)}^{{\ell }^{\prime }+\ell }$ times itself. If ${\ell }^{\prime }+\ell$ is even the prefactor is $-1$, forcing the matrix element to be zero; a non-zero value requires ${\ell }^{\prime }+\ell$ odd, that is, ${\ell }^{\prime }$ and $\ell$ of opposite parity. This is the Laporte rule. Because $\hat{\mathbf{r}}$ is a rank-$1$ tensor operator, the rotational triangle condition (proved in the next section as a corollary of the Wigner-Eckart theorem) further restricts $\mid \ell -1\mid \le {\ell }^{\prime }\le \ell +1$. The only opposite-parity options inside that window are ${\ell }^{\prime }=\ell \pm 1$; the case ${\ell }^{\prime }=\ell$ is excluded by parity. Hence $\Delta \ell =\pm 1$. $\square$

Bridge. The parity formula $(-1{)}^{\ell }$ builds toward the full Wigner-Eckart reduction in the advanced sections, where the same opposite-parity obstruction reappears as the statement that the reduced matrix element of a parity-odd tensor operator connects only states of opposite intrinsic-times-orbital parity, and it appears again in the projection theorem, whose vector operator $\hat{\mathbf{V}}$ inherits the rank-$1$ triangle rule used here. Putting the two pieces together, the dipole selection rule is dual to a purely group-theoretic fact: position carries the spin-$1$ representation of the rotation group, and a rank-$1$ operator can change $\ell$ by at most one unit. The central insight that this bridge records is that the discrete parity symmetry and the continuous rotation symmetry act on the same matrix element from two sides, parity fixing the sign change and the rotation group fixing the magnitude change, and the surviving transitions are exactly those compatible with both. This pairing of a discrete and a continuous symmetry is the structural counterpart of the antiunitary/rotation pairing developed for time reversal in 12.15.01.

Exercises Intermediate+

Advanced results Master

Theorem (Wigner-Eckart). Let $\{{\hat{T}}_q^k\}$ be an irreducible tensor operator of rank $k$ on a Hilbert space carrying a unitary representation of the rotation group, and let $\mid \alpha jm⟩$ denote orthonormal angular-momentum eigenstates labelled additionally by all non-rotational quantum numbers $\alpha$. Then there exists a number $⟨{\alpha }^{\prime }{j}^{\prime }\Vert T^k\Vert \alpha j⟩$, the reduced matrix element, depending on $\alpha ,{\alpha }^{\prime },j,j^{\prime },k$ but not on $m,m^{\prime },q$, such that

$$⟨{\alpha }^{\prime }{j}^{\prime }{m}^{\prime }\mid {\hat{T}}_q^k\mid \alpha jm⟩=⟨jm;kq\mid j^{\prime }{m}^{\prime }⟩\frac{⟨{\alpha }^{\prime }{j}^{\prime }\Vert T^k\Vert \alpha j⟩}{\sqrt{2j^{\prime }+1}},$$

where $⟨jm;kq\mid j^{\prime }{m}^{\prime }⟩$ is the Clebsch-Gordan coefficient coupling $j$ and $k$ to $j^{\prime }$. All of the $m$-dependence is geometric, carried by the Clebsch-Gordan coefficient; the dynamics resides entirely in the single reduced matrix element.

The content separates into a selection part and a magnitude part. The selection part is immediate: the Clebsch-Gordan coefficient vanishes unless $m^{\prime }=m+q$ (the ${\hat{J}}_z$-content of 12.05.03) and unless the triangle inequality $\mid j-k\mid \le j^{\prime }\le j+k$ holds. The magnitude part is the deeper statement, that across the entire $(2j+1)(2j^{\prime }+1)(2k+1)$ array of matrix elements only one independent number survives.

Theorem (projection theorem for vector operators). Let $\hat{\mathbf{V}}$ be a vector operator (rank-$1$) and consider its matrix elements within a single fixed multiplet $\alpha j$ (so $j^{\prime }=j$, ${\alpha }^{\prime }=\alpha$). Then

$$⟨\alpha j{m}^{\prime }\mid \hat{\mathbf{V}}\mid \alpha jm⟩=\frac{⟨\alpha jm\mid \hat{\mathbf{J}}\cdot \hat{\mathbf{V}}\mid \alpha jm⟩}{{\hslash }^{2}j(j+1)}⟨\alpha j{m}^{\prime }\mid \hat{\mathbf{J}}\mid \alpha jm⟩.$$

Within a multiplet every vector operator is, in matrix elements, a scalar multiple of $\hat{\mathbf{J}}$, the multiple being the normalised projection $⟨\hat{\mathbf{J}}\cdot \hat{\mathbf{V}}⟩/[{\hslash }^{2}j(j+1)]$. This is the engine behind the Landé $g$-factor: the magnetic moment $\mathbf{\mu }=-{\mu }_B(\hat{\mathbf{L}}+2\hat{\mathbf{S}})/\hslash$ is a vector operator, and within a fixed-$j$ multiplet it acts as $-g_J{\mu }_B\hat{\mathbf{J}}/\hslash$ with $g_J$ the projection of $\hat{\mathbf{L}}+2\hat{\mathbf{S}}$ onto $\hat{\mathbf{J}}$.

Theorem (relative intensities in a multiplet). The relative intensities of the spectral lines connecting two fine-structure multiplets $\alpha j\to {\alpha }^{\prime }{j}^{\prime }$, summed appropriately over photon polarisations, are fixed by the squared Clebsch-Gordan coefficients alone, independent of the common reduced matrix element. In particular the sum rule ${\sum }_{m^{\prime },q}\mid ⟨jm;1q\mid j^{\prime }{m}^{\prime }⟩{\mid }^2=(2j^{\prime }+1)/(2j+1)\cdot (\text{multiplicity factor})$ distributes a fixed total oscillator strength among the allowed lines in ratios computable from angular-momentum algebra. The empirical regularities of multiplet intensities catalogued in atomic spectroscopy before the theorem existed (the Burger-Dorgelo-Ornstein sum rules of 1924-1925) are exactly these Clebsch-Gordan ratios, derived here from the factorisation rather than fitted.

Theorem (parity and the structure of point groups). For a particle in a potential invariant under a finite point group $G\subset O(3)$, the energy eigenstates organise into bases for the irreducible representations of $G$, and a matrix element $⟨{\Gamma }^{\prime }\mid \hat{O}\mid \Gamma ⟩$ of an operator transforming in the representation ${\Gamma }_O$ vanishes unless the identity representation appears in the triple product $\Gamma'^ \otimes \Gamma_O \otimes \Gamma$.Parityisthespecialcase$G = {\mathbb 1, \hat P} \cong \mathbb Z_2$,whosetwoirreduciblesaretheevenandoddrepresentations,andwhosetriple-productrulereducestotherequirementthat$\Gamma'^\otimes\Gamma_O\otimes\Gamma$betheevenrepresentation,thatis,thattheparitiesmultiplyto$+1$. The Wigner-Eckart theorem is itself the instance of this group-theoretic selection rule for the continuous group $SU(2)$, where the multiplicity-free decomposition $j\otimes k={⨁}_{j^{\prime }}{j}^{\prime }$ makes the single reduced matrix element exhaust the freedom.

Synthesis. The factorisation of a matrix element into a geometric Clebsch-Gordan coefficient and one dynamical number is the foundational structure of this unit, and putting the results together they are one statement seen from several sides. The central insight is that a tensor operator ${\hat{T}}_q^k$ acting on a multiplet $\mid \alpha jm⟩$ produces a vector in the tensor-product representation $j\otimes k$, and the matrix element is the overlap of this vector with $\mid {\alpha }^{\prime }{j}^{\prime }{m}^{\prime }⟩$; because that overlap is an intertwiner between irreducible representations, Schur's lemma forces it to be a fixed multiple of the canonical Clebsch-Gordan projection, the multiple being the reduced matrix element. The selection rules, the $m$-conservation, and the triangle inequality are the support of the Clebsch-Gordan coefficient; the projection theorem is the rank-$1$ specialisation within a multiplet; the multiplet intensity ratios are the squared coefficients; and the parity rule is the same Schur-lemma argument for the discrete group ${\mathbb{Z}}_2$ in place of $SU(2)$. The bridge that ties the unit to its companion is that parity and rotation are both symmetries whose representation theory constrains matrix elements, and the discrete parity selection rule sits inside the continuous Wigner-Eckart rule as the ${\mathbb{Z}}_2$ factor of $O(3)=SU(2)⋊{\mathbb{Z}}_2$. This pattern recurs wherever a symmetry group acts on a quantum system: the surviving matrix elements are those carrying the identity representation in the appropriate triple product, and the magnitude of each is one number per irreducible channel.

Full proof set Master

Proposition 1 (parity is a Hermitian involution with spectrum $\{\pm 1\}$). The operator $(\hat{P}\psi )(\mathbf{r})=\psi (-\mathbf{r})$ on $L^2({\mathbb{R}}^3)$ is unitary, satisfies ${\hat{P}}^2=1$, is therefore self-adjoint, and has spectrum contained in $\{+1,-1\}$ with both eigenvalues attained.

Proof. Unitarity: for $\varphi ,\psi \in L^2$, $⟨\hat{P}\varphi \mid \hat{P}\psi ⟩=\int \overline{\varphi (-\mathbf{r})}\psi (-\mathbf{r})d^{3}r=\int \overline{\varphi (\mathbf{u})}\psi (\mathbf{u})d^{3}u=⟨\varphi \mid \psi ⟩$ after the substitution $\mathbf{u}=-\mathbf{r}$, whose Jacobian has absolute value $1$. Involution: $({\hat{P}}^2\psi )(\mathbf{r})=(\hat{P}\psi )(-\mathbf{r})=\psi (\mathbf{r})$, so ${\hat{P}}^2=1$. From unitarity ${\hat{P}}^{†}={\hat{P}}^{-1}$, and from involution ${\hat{P}}^{-1}=\hat{P}$, hence ${\hat{P}}^{†}=\hat{P}$ is self-adjoint. If $\hat{P}\psi =\pi \psi$ with $\psi \ne 0$, applying $\hat{P}$ again gives ${\pi }^2\psi =\psi$, so ${\pi }^2=1$ and $\pi \in \{+1,-1\}$. Both are attained: any even function (for instance a Gaussian) has eigenvalue $+1$, and any odd function (a Gaussian times a coordinate) has eigenvalue $-1$. The projectors ${\hat{P}}_{\pm }=\frac{1}{2}(1\pm \hat{P})$ split $L^2$ into even and odd subspaces. $\square$

Proposition 2 (the adjoint of a tensor-operator component shifts ${\hat{J}}_z$). If $\{{\hat{T}}_q^k\}$ satisfies the defining commutators, then for any ${\hat{J}}_z$ eigenstate $\mid m⟩$ the vector ${\hat{T}}_q^k\mid m⟩$ lies in the ${\hat{J}}_z$ eigenspace with eigenvalue $\hslash (m+q)$, and ${\hat{J}}_{\pm }$ acting on the family reproduces the spin-$k$ ladder.

Proof. The ${\hat{J}}_z$ statement is the computation ${\hat{J}}_z{\hat{T}}_q^k\mid m⟩=({\hat{T}}_q^k{\hat{J}}_z+[{\hat{J}}_z,{\hat{T}}_q^k])\mid m⟩=(\hslash m+\hslash q){\hat{T}}_q^k\mid m⟩$, using the first defining relation. For the ladder, the relation $[{\hat{J}}_{\pm },{\hat{T}}_q^k]=\hslash \sqrt{k(k+1)-q(q\pm 1)}{\hat{T}}_{q\pm 1}^k$ is identical, coefficient for coefficient, to the action ${\hat{J}}_{\pm }\mid kq⟩=\hslash \sqrt{k(k+1)-q(q\pm 1)}\mid k,q\pm 1⟩$ on the spin-$k$ multiplet from 12.05.01. Thus the map $\mid kq⟩\mapsto {\hat{T}}_q^k$ intertwines the adjoint action of $\hat{\mathbf{J}}$ (commutators) on operators with the standard action on the spin-$k$ states, so the operators transform as a spin-$k$ tensor. $\square$

Proposition 3 (Wigner-Eckart, existence and uniqueness of the reduced element). Under the hypotheses of the theorem, the matrix elements satisfy the stated factorisation, and the reduced matrix element is uniquely determined up to the convention $1/\sqrt{2j^{\prime }+1}$.

Proof. By Proposition 2, the set $\{{\hat{T}}_q^k\mid \alpha jm⟩:q,m\}$ spans, inside $\mathcal{H}$, a subspace on which $\hat{\mathbf{J}}$ acts as the tensor product of the spin-$k$ and spin-$j$ representations. Define a linear map $\Phi :{\mathbb{C}}^{2k+1}\otimes {\mathbb{C}}^{2j+1}\to \mathcal{H}$ by $\Phi (\mid kq⟩\otimes \mid jm⟩)={\hat{T}}_q^k\mid \alpha jm⟩$. Proposition 2 shows $\Phi$ is an intertwiner: $\Phi ({\hat{\mathbf{J}}}^{(k)}\otimes 1+1\otimes {\hat{\mathbf{J}}}^{(j)})=\hat{\mathbf{J}}\Phi$. Decompose the source into irreducibles by Clebsch-Gordan, $\mid kq⟩\otimes \mid jm⟩={\sum }_{JM}⟨kq;jm\mid JM⟩\mid JM⟩$. Projecting the image onto the bra $⟨{\alpha }^{\prime }{j}^{\prime }{m}^{\prime }\mid$ and using that $⟨{\alpha }^{\prime }{j}^{\prime }{m}^{\prime }\mid$ has non-zero overlap only with the $J=j^{\prime }$, $M=m^{\prime }$ component (orthogonality of distinct-$J$ irreducibles, Schur's lemma) gives

$$⟨{\alpha }^{\prime }{j}^{\prime }{m}^{\prime }\mid {\hat{T}}_q^k\mid \alpha jm⟩=⟨kq;jm\mid j^{\prime }{m}^{\prime }⟩c({\alpha }^{\prime },j^{\prime };\alpha ,j,k),$$

with $c$ a constant independent of $q,m,m^{\prime }$ because the intertwiner restricted to a single irreducible $j^{\prime }$ is, by Schur's lemma, a single scalar. Symmetry of the Clebsch-Gordan coefficients lets one rewrite $⟨kq;jm\mid j^{\prime }{m}^{\prime }⟩=⟨jm;kq\mid j^{\prime }{m}^{\prime }⟩$ up to the standard reordering phase, absorbed into $c$. Setting $c=⟨{\alpha }^{\prime }{j}^{\prime }\Vert T^k\Vert \alpha j⟩/\sqrt{2j^{\prime }+1}$ defines the reduced matrix element; uniqueness is the uniqueness of the Schur scalar. $\square$

Proposition 4 (projection theorem). Within a multiplet $\alpha j$ a vector operator $\hat{\mathbf{V}}$ obeys $⟨\alpha j{m}^{\prime }\mid \hat{\mathbf{V}}\mid \alpha jm⟩=\frac{⟨\hat{\mathbf{J}}\cdot \hat{\mathbf{V}}⟩}{{\hslash }^{2}j(j+1)}⟨\alpha j{m}^{\prime }\mid \hat{\mathbf{J}}\mid \alpha jm⟩$.

Proof. Both $\hat{\mathbf{V}}$ and $\hat{\mathbf{J}}$ are rank-$1$ tensor operators, so by Wigner-Eckart their matrix elements within the fixed multiplet $j^{\prime }=j$ are the same Clebsch-Gordan coefficient $⟨jm;1q\mid j{m}^{\prime }⟩$ times their respective reduced elements. Hence $⟨\alpha j{m}^{\prime }\mid V_q^1\mid \alpha jm⟩=\lambda ⟨\alpha j{m}^{\prime }\mid J_q^1\mid \alpha jm⟩$ with $\lambda =⟨\alpha j\Vert V\Vert \alpha j⟩/⟨\alpha j\Vert J\Vert \alpha j⟩$ a single constant for the multiplet, the cancellation valid wherever the $\hat{\mathbf{J}}$ matrix element is non-zero. To fix $\lambda$, evaluate the scalar $\hat{\mathbf{J}}\cdot \hat{\mathbf{V}}$, itself a rank-$0$ operator, diagonal in $m$ by Proposition 3's $k=0$ specialisation. Insert a complete set within the multiplet: $⟨\hat{\mathbf{J}}\cdot \hat{\mathbf{V}}⟩={\sum }_{m^{\prime \prime }}⟨\alpha jm\mid \hat{\mathbf{J}}\mid \alpha j{m}^{\prime \prime }⟩\cdot ⟨\alpha j{m}^{\prime \prime }\mid \hat{\mathbf{V}}\mid \alpha jm⟩=\lambda {\sum }_{m^{\prime \prime }}⟨\alpha jm\mid \hat{\mathbf{J}}\mid \alpha j{m}^{\prime \prime }⟩\cdot ⟨\alpha j{m}^{\prime \prime }\mid \hat{\mathbf{J}}\mid \alpha jm⟩=\lambda ⟨\hat{\mathbf{J}}\cdot \hat{\mathbf{J}}⟩=\lambda {\hslash }^{2}j(j+1)$. Therefore $\lambda =⟨\hat{\mathbf{J}}\cdot \hat{\mathbf{V}}⟩/[{\hslash }^{2}j(j+1)]$, and substituting back gives the stated identity. $\square$

Proposition 5 (parity selection rule as a ${\mathbb{Z}}_2$ Wigner-Eckart instance). For the group ${\mathbb{Z}}_2=\{1,\hat{P}\}$ a matrix element $⟨{\psi }^{\prime }\mid \hat{O}\mid \psi ⟩$, with $\hat{P}\hat{O}{\hat{P}}^{-1}={\eta }_O\hat{O}$ and $\hat{P}\mid \psi ⟩=\pi \mid \psi ⟩$, $\hat{P}\mid {\psi }^{\prime }⟩={\pi }^{\prime }\mid {\psi }^{\prime }⟩$, vanishes unless ${\pi }^{\prime }{\eta }_O\pi =+1$.

Proof. Insert ${\hat{P}}^{-1}\hat{P}=1$: $⟨{\psi }^{\prime }\mid \hat{O}\mid \psi ⟩=⟨{\psi }^{\prime }\mid {\hat{P}}^{-1}(\hat{P}\hat{O}{\hat{P}}^{-1})\hat{P}\mid \psi ⟩$. Act $\hat{P}$ to the left on $⟨{\psi }^{\prime }\mid$ (using self-adjointness, $⟨{\psi }^{\prime }\mid {\hat{P}}^{-1}=\overline{{\pi }^{\prime }}⟨{\psi }^{\prime }\mid ={\pi }^{\prime }⟨{\psi }^{\prime }\mid$ since ${\pi }^{\prime }=\pm 1$ is real) and to the right on $\mid \psi ⟩$, and pull out ${\eta }_O$ from the conjugated operator: the right side equals ${\pi }^{\prime }{\eta }_O\pi ⟨{\psi }^{\prime }\mid \hat{O}\mid \psi ⟩$. If ${\pi }^{\prime }{\eta }_O\pi =-1$ the element equals its own negative and is zero; survival requires ${\pi }^{\prime }{\eta }_O\pi =+1$, the product of all three parities being even. For $\hat{O}=\hat{\mathbf{r}}$ with ${\eta }_O=-1$ this is the opposite-parity dipole rule. $\square$

Proposition 6 (intensity sum rule), stated with proof outline — full development in Edmonds Ch. 5 and Wigner Group Theory Ch. 21 ^{[Wigner 1931]}. Summing the squared matrix elements of a rank-$k$ operator over the final magnetic quantum number $m^{\prime }$ and the operator index $q$, and using the Clebsch-Gordan orthogonality ${\sum }_{m,q}⟨jm;kq\mid j^{\prime }{m}^{\prime }⟩⟨jm;kq\mid j^{\prime }{m}^{\prime \prime }⟩={\delta }_{m^{\prime }{m}^{\prime \prime }}$, collapses the geometric factors and leaves ${\sum }_{m^{\prime },q}\mid ⟨{\alpha }^{\prime }{j}^{\prime }{m}^{\prime }\mid {\hat{T}}_q^k\mid \alpha jm⟩{\mid }^2=\mid ⟨{\alpha }^{\prime }{j}^{\prime }\Vert T^k\Vert \alpha j⟩{\mid }^2/(2j^{\prime }+1)$ independent of the initial $m$. The Burger-Dorgelo-Ornstein multiplet sum rules are the case $k=1$, distributing total line strength in proportion to the statistical weights $2j+1$. $\square$

Connections Master

• Angular momentum operators and $SU(2)$ 12.05.01. The defining commutators $[{\hat{J}}_z,{\hat{T}}_q^k]=\hslash q{\hat{T}}_q^k$ and $[{\hat{J}}_{\pm },{\hat{T}}_q^k]=\hslash \sqrt{k(k+1)-q(q\pm 1)}{\hat{T}}_{q\pm 1}^k$ are, coefficient for coefficient, the ladder action on the spin-$k$ multiplet built there. A tensor operator is nothing other than an operator-valued copy of an angular-momentum multiplet, and the Wigner-Eckart theorem is the statement that operators and states are coupled by the same Clebsch-Gordan algebra. Without the $SU(2)$ representation theory the rank concept has no meaning.

• Addition of angular momenta and Clebsch-Gordan coefficients 12.05.03. The Wigner-Eckart factorisation places a Clebsch-Gordan coefficient $⟨jm;kq\mid j^{\prime }{m}^{\prime }⟩$ at the heart of every matrix element, so the coupling of $j$ and $k$ to $j^{\prime }$ developed there is the exact combinatorial content of the selection rules. The triangle inequality $\mid j-k\mid \le j^{\prime }\le j+k$ and the $m^{\prime }=m+q$ conservation are the support of the coupling coefficient, and the relative-intensity ratios are squares of these coefficients.

• Time-reversal symmetry and Kramers' degeneracy 12.15.01. Parity and time reversal are the two discrete symmetries of the same chapter, and they cooperate: a Hamiltonian invariant under both has eigenstates that can be simultaneously labelled by parity $(-1{)}^{\ell }$ and organised into Kramers pairs. The combined discrete-symmetry algebra of charge conjugation, parity, and time reversal acts on Dirac fields and the product $CPT$ is the protected symmetry; the parity-violation result of Wu builds toward the recognition that the weak interaction breaks $P$ and $C$ separately while preserving $CPT$.

Historical & philosophical context Master

Eugene Wigner introduced parity into quantum mechanics in his 1927 paper in the Zeitschrift für Physik ^{[Wigner 1927]}, where he recognised that the reflection symmetry of an atomic Hamiltonian sorts terms into two classes and forbids transitions within a class, supplying the group-theoretic basis for the empirical Laporte rule that Otto Laporte had noted in iron-group spectra three years earlier. Carl Eckart, working independently in 1930 in Reviews of Modern Physics ^{[Eckart 1930]}, derived the factorisation of tensor-operator matrix elements into a geometric coefficient and a reduced part for monatomic systems; Wigner gave the general $SU(2)$ form in his 1931 Gruppentheorie ^{[Wigner 1931]}, from which the joint attribution of the theorem to both men descends. The reduced matrix element, the single number that survives the angular averaging, is the precise sense in which the dynamics of a transition is separable from its geometry.

Parity was assumed to be an exact symmetry of nature for three decades. The assumption broke in 1956-1957. Tsung-Dao Lee and Chen-Ning Yang observed that no experiment had actually tested parity conservation in the weak interaction, and proposed tests; Chien-Shiung Wu and collaborators at the National Bureau of Standards carried out the decisive one, measuring the beta-decay distribution of polarised ${}^{60}$Co nuclei and finding the emitted electrons preferentially anti-aligned with the nuclear spin ^{[Wu 1957]}. A mirror-image arrangement would have shown the opposite preference, so the weak interaction distinguishes left from right at maximal strength. Parity remains exact for the electromagnetic and strong interactions, and the combined operation $CPT$ remains protected by the spin-statistics structure of relativistic quantum field theory, but the discrete reflection symmetry that organises atomic selection rules is not a symmetry of the full Standard Model.

Bibliography Master

@article{Wigner1927Parity,
  author  = {Wigner, Eugen},
  title   = {Einige Folgerungen aus der Schr{\"o}dingerschen Theorie f{\"u}r die Termstrukturen},
  journal = {Zeitschrift f{\"u}r Physik},
  volume  = {43},
  year    = {1927},
  pages   = {624--652}
}

@article{Eckart1930,
  author  = {Eckart, Carl},
  title   = {The Application of Group Theory to the Quantum Dynamics of Monatomic Systems},
  journal = {Reviews of Modern Physics},
  volume  = {2},
  year    = {1930},
  pages   = {305--380}
}

@book{WignerGruppentheorie1931,
  author    = {Wigner, Eugen},
  title     = {Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren},
  publisher = {Friedr. Vieweg \& Sohn},
  year      = {1931},
  note      = {English translation: Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, trans. J. J. Griffin, Academic Press, 1959}
}

@article{Wu1957,
  author  = {Wu, C. S. and Ambler, E. and Hayward, R. W. and Hoppes, D. D. and Hudson, R. P.},
  title   = {Experimental Test of Parity Conservation in Beta Decay},
  journal = {Physical Review},
  volume  = {105},
  year    = {1957},
  pages   = {1413--1415}
}

@article{LeeYang1956,
  author  = {Lee, T. D. and Yang, C. N.},
  title   = {Question of Parity Conservation in Weak Interactions},
  journal = {Physical Review},
  volume  = {104},
  year    = {1956},
  pages   = {254--258}
}

@book{Edmonds1957,
  author    = {Edmonds, A. R.},
  title     = {Angular Momentum in Quantum Mechanics},
  publisher = {Princeton University Press},
  year      = {1957}
}

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

@book{SakuraiNapolitano2017,
  author    = {Sakurai, J. J. and Napolitano, Jim},
  title     = {Modern Quantum Mechanics},
  edition   = {2},
  publisher = {Cambridge University Press},
  year      = {2017}
}

@article{BurgerDorgeloOrnstein1924,
  author  = {Ornstein, L. S. and Burger, H. C.},
  title   = {{\"U}ber die Intensit{\"a}ten der Komponenten im Zeemaneffekt},
  journal = {Zeitschrift f{\"u}r Physik},
  volume  = {28},
  year    = {1924},
  pages   = {135--141}
}