Symmetry and Selection Rules for Transitions

1. The General Principle of Selection Rules

The probability of a transition from an initial state $|{\psi }_i⟩$ to a final state $|{\psi }_f⟩$, induced by some interaction described by an operator $\hat{O}$, is proportional to the square of the transition matrix element, $|⟨{\psi }_f|\hat{O}|{\psi }_i⟩{|}^2$. A selection rule is a principle that tells us when this matrix element must be exactly zero, rendering the transition "forbidden."

The core idea is beautifully simple: if the integrand ${\psi }_f^{\ast }\hat{O}{\psi }_i$ is "odd" under a symmetry transformation of the system, its integral over all space must vanish.

Let's formalize this. Suppose a Hamiltonian $H$ has a symmetry represented by a unitary operator $\hat{S}$, such that $[\hat{S},H]=0$. As we've seen, this allows us to choose energy eigenstates that are also eigenstates of the symmetry operator. Let $|{\psi }_i⟩$ and $|{\psi }_f⟩$ be two such states:

where $s_i$ and $s_f$ are the corresponding eigenvalues (which are phase factors, since $\hat{S}$ is unitary).

Now consider the matrix element $M=⟨{\psi }_f|\hat{O}|{\psi }_i⟩$. We can insert the identity ${\hat{S}}^{†}\hat{S}=I$ into this expression:

Using the eigenstate property of the final state, $⟨{\psi }_f|{\hat{S}}^{†}=(\hat{S}|{\psi }_f⟩{)}^{†}=(s_f|{\psi }_f⟩{)}^{†}=s_f^{\ast }⟨{\psi }_f|$. This gives:

To proceed, we need to understand how the operator $\hat{O}$ itself behaves under the symmetry transformation. Let's rewrite the operator as $\hat{O}={\hat{S}}^{†}(\hat{S}\hat{O}{\hat{S}}^{†})\hat{S}$. The term in the parenthesis, ${\hat{O}}^{\prime }=\hat{S}\hat{O}{\hat{S}}^{†}$, is the transformed operator. Inserting this into the matrix element:

This equation contains the essence of all selection rules. It states that the original matrix element $M$ is proportional to the matrix element of the transformed operator ${\hat{O}}^{\prime }$. If ${\hat{O}}^{\prime }$ is related to $\hat{O}$ in a simple way (e.g., ${\hat{O}}^{\prime }=\lambda \hat{O}$), we can derive powerful constraints.

2. Parity and Selection Rules

The most common and instructive application of this principle is with the parity operator, $\hat{\pi }$. The parity operator performs a spatial inversion through the origin, $\mathbf{r}\to -\mathbf{r}$.

To refresh your understanding of the parity operator, its properties, and its connection to selection rules, the following video provides an excellent summary.

Let's assume our system's Hamiltonian is invariant under parity, $[H,\hat{\pi }]=0$, which is true for central potentials and for all fundamental interactions except the weak force. The energy eigenstates can thus be assigned a definite parity eigenvalue, $p=\pm 1$.

Operators can also be classified by their behavior under parity. The transformation is ${\hat{O}}^{\prime }=\hat{\pi }\hat{O}{\hat{\pi }}^{†}$. Since $\hat{\pi }$ is its own inverse and adjoint, this simplifies to ${\hat{O}}^{\prime }=\hat{\pi }\hat{O}\hat{\pi }$.

• Even Operator: $\hat{\pi }\hat{O}\hat{\pi }=+\hat{O}$. Example: The Hamiltonian $H$ for a central potential.
• Odd Operator: $\hat{\pi }\hat{O}\hat{\pi }=-\hat{O}$. Example: The position operator $\hat{\mathbf{r}}$ or momentum operator $\hat{\mathbf{p}}$.

Now we can derive the selection rules. Let $|{\psi }_i⟩$ and $|{\psi }_f⟩$ be states with parities $p_i$ and $p_f$.

Case 1: $\hat{O}$ is an odd operator

The transition matrix element is $M=⟨{\psi }_f|\hat{O}|{\psi }_i⟩$. We can use the fact that $\hat{O}=-\hat{\pi }\hat{O}\hat{\pi }$ and insert this into the expression for $M$:

Since $|{\psi }_i⟩$ and $|{\psi }_f⟩$ are parity eigenstates, this becomes:

This implies $M(1+p_f{p}_i)=0$. For the transition to be allowed ($M\ne 0$), we must have $1+p_f{p}_i=0$, which means:

Selection Rule: A transition induced by an odd operator is only allowed if the initial and final states have opposite parity.

Case 2: $\hat{O}$ is an even operator

Here, $\hat{O}=+\hat{\pi }\hat{O}\hat{\pi }$. Following the same steps:

This implies $M(1-p_f{p}_i)=0$. For the transition to be allowed, we must have $1-p_f{p}_i=0$, which means:

Selection Rule: A transition induced by an even operator is only allowed if the initial and final states have the same parity.

3. Application: Atomic Electric Dipole Transitions

The most prominent application of these rules is in atomic physics. When an atom interacts with an electromagnetic field, the dominant process for emitting or absorbing a photon is the electric dipole transition. In the long-wavelength approximation, the operator responsible for this transition is the electric dipole operator, $\hat{\mathbf{d}}=e\hat{\mathbf{r}}$, which is proportional to the position operator.

Since $\hat{\mathbf{r}}$ is an odd operator under parity, electric dipole transitions are governed by the selection rule we derived in Case 1: the parity of the atomic state must change. This famous result is known as Laporte's rule.

For an electron in a central potential (like the hydrogen atom), the eigenstates are $|n,l,m⟩$ and their wavefunctions are ${\psi }_{nlm}(\mathbf{r})=R_{nl}(r)Y_{lm}(\theta ,\varphi )$. The parity of such a state is determined entirely by the orbital angular momentum quantum number $l$:

Thus, the parity of the state is $p=(-1{)}^l$.

Laporte's rule, $p_f{p}_i=-1$, translates to:

This requires that $\Delta l=l_f-l_i$ must be an odd integer. This immediately forbids transitions where $\Delta l=0,\pm 2,\pm 4,\dots$.

The following reading from J.J. Sakurai's notes provides a concise derivation of this crucial result.

Note that symmetry under rotations also imposes selection rules, giving $\Delta l=0,\pm 1$. Combining the parity rule ($\Delta l=\text{odd}$) with the rotational symmetry rule leaves only $\Delta l=\pm 1$ as the allowed electric dipole transitions. This is a classic example of how different symmetries work together to constrain the dynamics of a system.

4. A Broader Perspective on Symmetries and Matrix Elements

The principle we've used for parity is completely general and forms the basis of the Wigner-Eckart theorem, a topic we will cover in detail later. The theorem provides a general formula for matrix elements based entirely on symmetry properties.

The key insight is that symmetry dramatically simplifies calculations. Instead of computing every single matrix element $⟨{\psi }_f|\hat{O}|{\psi }_i⟩$ by performing a complicated integral, we can first use symmetry to determine which ones are guaranteed to be zero. For the non-zero ones, symmetry can relate many different matrix elements to a smaller set of fundamental quantities called "reduced matrix elements."

This flowchart illustrates the two approaches to calculating matrix elements. The "Elementary approach" on the left involves direct, brute-force integration for every single element. The "Wigner-Eckart approach" on the right leverages symmetry from the very beginning.

The derivation of selection rules is the first and most important step in this more sophisticated, symmetry-adapted approach. By simply checking the parity of the initial state, the final state, and the transition operator, we can immediately predict whether a transition is possible, saving immense computational effort. In your work in condensed matter and QFT, where systems can have many interacting particles and complex states, this kind of simplification is not just a convenience—it is often the only tractable way to make predictions.

Conclusion

Today, we have seen how symmetry principles, which we previously used to understand static properties like energy level degeneracy, also provide powerful rules governing system dynamics.

Key Takeaways:

• Selection rules determine whether a transition matrix element $⟨{\psi }_f|\hat{O}|{\psi }_i⟩$ is allowed or forbidden (non-zero or zero).
• These rules arise from analyzing the transformation properties of the initial state, final state, and transition operator under a symmetry of the Hamiltonian.
• For a parity-invariant system, transitions induced by odd operators (like the electric dipole operator $\hat{\mathbf{r}}$) require the system's parity to change.
• Transitions induced by even operators require the system's parity to be conserved.
• This leads to Laporte's rule for atomic transitions, which states that for an electric dipole transition, the change in orbital angular momentum, $\Delta l$, must be an odd integer.

You now possess two fundamental tools from symmetry analysis: using continuous symmetries to find conserved quantities (Noether's theorem) and using discrete or continuous symmetries to find selection rules.

In the next lesson, we will turn our attention to another crucial discrete symmetry: time reversal. Unlike parity, the time-reversal operator has some surprising and non-intuitive properties, especially for particles with spin, leading to one of the most important results in condensed matter physics: Kramers' degeneracy.