Parity: Symmetry, Eigenstates, and Selection Rules

Introduction

In our previous lesson, we established a deep connection between the continuous symmetry of rotation and the conservation of angular momentum. We saw that this connection leads to the necessary degeneracy of energy levels in a rotationally invariant system. Today, we shift our focus from continuous symmetries to a fundamental discrete symmetry: parity.

Parity corresponds to a spatial inversion through the origin, the transformation $r \to - r$ . While it might seem like a simple mirror reflection, its consequences in quantum mechanics are profound. Unlike continuous symmetries, which have associated infinitesimal generators, discrete symmetries impose powerful constraints on the system in a different way.

In this lesson, we will define the parity operator, explore its fundamental properties, and determine its eigenstates. We will then analyze how it constrains the form of physical laws and leads to the conservation of parity in certain interactions. The culmination of our analysis will be the derivation of parity selection rules, which allow us to determine whether certain quantum transitions are allowed or forbidden purely based on symmetry, without solving any dynamical equations.

The Parity Operator $Π$

Let's begin by formalizing the concept of spatial inversion in the language of quantum mechanics. We postulate the existence of a linear operator, $Π$ , called the parity operator, which implements the inversion of coordinates.

parity.pdf

These notes by Robert Littlejohn provide a rigorous introduction to the parity operator, starting from its classical analogue and building up its quantum mechanical properties.

First, read Section 2, Spatial Inversion, to understand the classical operation. Then, in Section 3, Parity in Quantum Mechanics, focus on the postulates that define the quantum operator \Pi, particularly its unitarity, the property \Pi^2=1, and its commutation with rotation operators. Finally, read the first part of Section 4, which gives the explicit definition of \Pi through its action on position eigenstates.

As the notes establish, the parity operator $Π$ is defined by its core properties and its action on the basis states of the Hilbert space. The key postulates are:

Unitarity: $Π^{†} Π = I$ . This ensures that the norm of a state is preserved under a parity transformation.
Involutory Property: $Π^{2} = I$ . Applying the inversion twice returns you to the original state.
Commutation with Rotations: $[Π, U (R)] = 0$ for any rotation $U (R)$ . This means an inversion followed by a rotation is the same as a rotation followed by an inversion. This implies $[Π, J] = 0$ , meaning parity is a scalar operator.

From the first two properties, we can immediately deduce that $Π$ is also Hermitian. Since $Π^{2} = I$ , its inverse is itself: $Π^{- 1} = Π$ . And since it is unitary, its inverse is its adjoint: $Π^{- 1} = Π^{†}$ . Therefore, $Π = Π^{†}$ .

The action of the parity operator is most transparently defined on the position eigenstates:

Π∣ r ⟩ = ∣ - r ⟩

This directly leads to its effect on a state $∣ ψ ⟩$ represented by a wave function $ψ (r) = ⟨ r ∣ ψ ⟩$ . The wave function of the transformed state, $Π∣ ψ ⟩$ , is:

⟨ r ∣Π∣ ψ ⟩ = ⟨ - r ∣ ψ ⟩ = ψ (- r)

So, the parity operator acts on a wave function by inverting the coordinates of its argument.

This short video provides a very clear walkthrough of these definitions and properties.

The parity operator in quantum mechanics

This video from "Professor M does Science" clearly defines the parity operator and its fundamental properties.

First, watch the segment that defines the operator via its action on position states and its effect on wave functions. Then, review the derivation of its properties, confirming that it is involutory, Hermitian, and unitary.

Eigenstates of Parity

Since $Π$ is a Hermitian operator, its eigenvalues are real. Furthermore, since $Π^{2} = I$ , its eigenvalues $λ$ must satisfy $λ^{2} = 1$ , which means the only possible eigenvalues are $λ = + 1$ and $λ = - 1$ .

An eigenstate of the parity operator is a state that has a definite parity.

Even States: If $Π∣ ψ ⟩ = + ∣ ψ ⟩$ , the state is said to have even parity. In the position representation, this corresponds to an even function: $ψ (- r) = ψ (r)$ .
Odd States: If $Π∣ ψ ⟩ = - ∣ ψ ⟩$ , the state is said to have odd parity. This corresponds to an odd function: $ψ (- r) = - ψ (r)$ .

Any arbitrary state can be decomposed into a superposition of an even and an odd part using the projection operators $P_{\pm} = \frac{1}{2} (I \pm Π)$ . The state $P_{+} ∣ ψ ⟩$ is guaranteed to be even, and $P_{-} ∣ ψ ⟩$ is guaranteed to be odd.

Transformation of Observables and Parity Conservation

The behavior of a physical system under parity is determined by how its Hamiltonian transforms. This, in turn, depends on how the fundamental observables like position, momentum, and angular momentum behave.

Let's find the transformation rule for an operator $A^$ under parity, which is given by $A^^{'} = Π A^Π^{†}$ . Since $Π$ is its own inverse, this is $Π A^Π$ .

Position: $Π x Π = - x$ . Position is odd under parity. It is a polar vector (or true vector).
Momentum: Since $p = - i ℏ\nabla$ , and the gradient operator flips sign under $r \to - r$ , we find $Π p Π = - p$ . Momentum is also odd under parity and is a polar vector.
Angular Momentum: Using the transformation of $x$ and $p$ , we can find the transformation of orbital angular momentum $L = x \times p$ . $Π L Π = Π (x \times p) Π = (Π x Π) \times (Π p Π) = (- x) \times (- p) = + (x \times p) = + L$ Angular momentum is even under parity. It is an axial vector (or pseudovector). Spin angular momentum $S$ transforms in the same way.

The following notes provide a clear classification of operators based on their behavior under parity.

Quantum Theory I, Lecture 22 Notes

These notes concisely define the standard terminology for classifying observables based on their parity.

Please read the subsection 22.1.1 Some Standard Terminology. This provides the definitions for scalar, pseudoscalar, vector, and pseudovector, which are essential for discussing parity conservation.

Parity is conserved if the Hamiltonian is invariant under parity, meaning $[Π, H] = 0$ . This requires the Hamiltonian to be a true scalar (even under parity). For example, a typical non-relativistic Hamiltonian for a particle in a central potential is:

H = \frac{p ^{2}}{2 m} + V (∣ r ∣)

The kinetic energy term $p^{2}$ is even under parity since $(- p) \cdot (- p) = p^{2}$ . The potential energy $V (∣ r ∣)$ depends only on the magnitude of $r$ , which is also invariant. Thus, this Hamiltonian commutes with $Π$ , and parity is a conserved quantity.

However, parity is not a universal symmetry. The strong and electromagnetic interactions conserve parity, but the weak interaction does not. This was a shocking discovery in 1956, proposed by Lee and Yang and experimentally confirmed by Wu.

This diagram illustrates the decay of a positive pion ($\pi^+$) into a positive muon ($\mu^+$) and a muon neutrino ($\nu_\mu$). This is a weak interaction process. (a) shows the observed decay where the neutrino is left-handed (spin anti-aligned with momentum). (b) shows the mirror image (parity transformation), which would produce a right-handed neutrino. This mirror-image process is *not observed* in nature, demonstrating that the weak interaction violates parity symmetry.

Parity of Energy Eigenstates

If parity is conserved ( $[Π, H] = 0$ ), we can choose a basis of energy eigenstates that are also eigenstates of parity.

For a non-degenerate energy eigenstate $∣ E ⟩$ , it must be an eigenstate of parity. This is a direct consequence of the theorem stating that if two operators commute, a non-degenerate eigenstate of one must also be an eigenstate of the other.

For a degenerate energy level, things are slightly more subtle. However, for a rotationally invariant system, all $2 j + 1$ states belonging to an angular momentum multiplet (a given $j$ ) can be shown to have the same parity.

A crucial example is the set of eigenstates of orbital angular momentum, the spherical harmonics $Y_{l m} (θ, ϕ)$ . Under a parity transformation $(r \to - r)$ , the spherical coordinates transform as $(θ, ϕ) \to (π - θ, ϕ + π)$ . One can show that:

Π Y_{l m} (θ, ϕ) = Y_{l m} (π - θ, ϕ + π) = (- 1)^{l} Y_{l m} (θ, ϕ)

Therefore, an eigenstate of orbital angular momentum with quantum number $l$ has a definite parity of $(- 1)^{l}$ .

Cross-sections of hydrogen atom probability densities for various (n, l, m) states. The parity of each state is determined by the orbital angular momentum quantum number, $l$, as $(-1)^l$. For example, all s-orbitals ($l=0$, first column) are even under parity. All p-orbitals ($l=1$, second column) are odd. All d-orbitals ($l=2$, third column) are even again. This visual pattern of symmetry reflects the mathematical property of the spherical harmonics.

Parity Selection Rules

The conservation of parity leads to powerful selection rules, which dictate which transitions or processes are allowed. Let's derive the general rule for a matrix element $⟨ ψ_{f} ∣ O^∣ ψ_{i} ⟩$ . This quantity is central to calculating transition rates and expectation values.

Quantum Theory I, Lecture 22 Notes

This section of the notes provides a clear and concise derivation of the general selection rule based on parity.

Read subsection 22.1.4 Selection Rules. Follow the derivation that starts by inserting the identity \Pi^2 into the matrix element to find the condition under which it can be non-zero.

Let's retrace the elegant derivation. We assume the initial and final states, $∣ ψ_{i} ⟩$ and $∣ ψ_{f} ⟩$ , are eigenstates of parity with eigenvalues $s_{i}$ and $s_{f}$ (where $s = \pm 1$ ). We also assume the operator $O^$ has a definite parity $λ$ , meaning $Π O^Π = λ O^$ .

Consider the matrix element:

⟨ ψ_{f} ∣ O^∣ ψ_{i} ⟩

Since $Π^{2} = I$ and $Π = Π^{†}$ , we can insert $Π^{2}$ without changing the value:

⟨ ψ_{f} ∣ O^∣ ψ_{i} ⟩ = ⟨ ψ_{f} ∣ΠΠ O^ΠΠ∣ ψ_{i} ⟩ = ⟨ Π ψ_{f} ∣ (Π O^Π) ∣Π ψ_{i} ⟩ = (s_{f} ⟨ ψ_{f} ∣) (λ O^) (s_{i} ∣ ψ_{i} ⟩) = λ s_{f} s_{i} ⟨ ψ_{f} ∣ O^∣ ψ_{i} ⟩

Rearranging gives:

(1 - λ s_{f} s_{i}) ⟨ ψ_{f} ∣ O^∣ ψ_{i} ⟩ = 0

This equation tells us that for the matrix element to be non-zero, we must have $λ s_{f} s_{i} = 1$ . This is the parity selection rule. It can be stated as follows:

If the operator $O^$ is even ( $λ = + 1$ ), the matrix element is non-zero only if the initial and final states have the same parity ( $s_{f} = s_{i}$ ).
If the operator $O^$ is odd ( $λ = - 1$ ), the matrix element is non-zero only if the initial and final states have opposite parity ( $s_{f} = - s_{i}$ ).

A prime application is in atomic physics. The dominant mechanism for atoms to emit or absorb light is through electric dipole transitions. The operator for this interaction is proportional to the electric dipole moment operator, $d = - e x$ , which is odd under parity. Therefore, electric dipole transitions are only allowed between states of opposite parity. This gives rise to Laporte's rule: $Δ l$ must be odd. Since we will later see that angular momentum conservation also restricts $Δ l = \pm 1, 0$ , the combination of both symmetries leads to the famous selection rule $Δ l = \pm 1$ for electric dipole transitions.

Conclusion

In this lesson, we have explored the rich consequences of a simple discrete symmetry: spatial inversion.

Key Takeaways:

Parity ( $Π$ ) is a discrete symmetry corresponding to $r \to - r$ . The operator is linear, unitary, and Hermitian, with eigenvalues $\pm 1$ .
Eigenstates of parity are either even ( $ψ (- r) = ψ (r)$ ) or odd ( $ψ (- r) = - ψ (r)$ ).
Observables can be classified by their behavior under parity: polar vectors (like $x$ and $p$ ) are odd, while axial vectors (like $L$ and $S$ ) are even.
If a Hamiltonian is a true scalar (i.e., commutes with $Π$ ), parity is a conserved quantity, and its energy eigenstates can be assigned a definite parity. States with orbital angular momentum $l$ have parity $(- 1)^{l}$ .
Parity conservation leads to powerful selection rules: a matrix element $⟨ ψ_{f} ∣ O^∣ ψ_{i} ⟩$ can only be non-zero if the product of the parities of the operator, the initial state, and the final state is +1 ( $λ s_{f} s_{i} = 1$ ).

By analyzing symmetries, we can deduce which physical processes are allowed or forbidden without performing complex calculations. This is one of the most powerful paradigms in modern physics.

Having now studied both continuous and discrete spatial symmetries, we are well-prepared for the next module. We will take the generator of rotations, angular momentum, and dissect its algebraic structure in great detail. This will allow us to understand the properties of spin, the addition of angular momenta, and ultimately, the structure of matter.

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