Parity and Time-Reversal Symmetry in Quantum Systems

Hello! Welcome to your third lesson.

In our previous lesson, we explored the continuous symmetry of rotation, uncovering the deep connection between the algebraic structure of SO(3) and SU(2) and the quantization of angular momentum. We saw how the SU(2) group provides the mathematical framework for half-integer spin.

Today, we shift our focus from continuous transformations to two fundamental discrete symmetries: parity (space inversion) and time reversal. These symmetries, while seemingly simple, have profound and sometimes counter-intuitive consequences in the quantum realm. This lesson will address the learning outcome: Apply parity and time-reversal symmetry operators to quantum systems. We will define the operators for these transformations, investigate their unique mathematical properties, and explore their physical implications, including selection rules and the important concept of Kramers degeneracy, which has a central role in your field of interest, condensed matter physics.

1. Parity: The Mirror Symmetry of Space

Parity, at its core, is the operation of inverting all spatial coordinates through the origin: $\vec{r}\to -\vec{r}$. In quantum mechanics, this transformation is implemented by a linear operator, $\hat{\Pi }$.

To begin, let's watch a video that introduces the parity operator, $\hat{\Pi }$, and establishes its fundamental properties.

As the video explains, the parity operator is defined by its action on the position basis states:

$$\hat{\Pi }|\vec{r}⟩=|-\vec{r}⟩$$

This definition leads to several crucial properties:

• Action on Wavefunctions: For a state $|\psi ⟩$ with wavefunction $\psi (\vec{r})=⟨\vec{r}|\psi ⟩$, the transformed state $\hat{\Pi }|\psi ⟩$ has the wavefunction $\psi (-\vec{r})$.
• Properties: The parity operator is:
  • Hermitian: ${\hat{\Pi }}^{†}=\hat{\Pi }$
  • Unitary: ${\hat{\Pi }}^{†}\hat{\Pi }=\hat{I}$
  • Involutory: ${\hat{\Pi }}^2=\hat{I}$
• Eigenvalues: Since ${\hat{\Pi }}^2=\hat{I}$, its eigenvalues, often denoted by ${\eta }_P$, must be $\pm 1$.
• Eigenstates: States with eigenvalue +1 are said to have even parity, while those with eigenvalue -1 have odd parity. Any state can be decomposed into a sum of an even and an odd part.

Action on Operators and Parity Conservation

The physical significance of parity becomes clear when we consider its effect on other operators and its commutation with the Hamiltonian. For a more detailed look, we'll turn to a set of excellent lecture notes.

As the notes detail, operators can be classified by how they transform under parity:

• Vectors (like $\vec{r}$ and $\vec{p}$) are odd: $\hat{\Pi }\hat{O}{\hat{\Pi }}^{†}=-\hat{O}$
• Pseudo-vectors (or axial vectors, like $\vec{L}=\vec{r}\times \vec{p}$ and $\vec{S}$) are even: $\hat{\Pi }\hat{O}{\hat{\Pi }}^{†}=+\hat{O}$
• Scalars are even: $\hat{\Pi }\hat{O}{\hat{\Pi }}^{†}=+\hat{O}$
• Pseudo-scalars (like $\vec{p}\cdot \vec{S}$) are odd: $\hat{\Pi }\hat{O}{\hat{\Pi }}^{†}=-\hat{O}$

If a Hamiltonian is invariant under parity (i.e., $[\hat{\Pi },\hat{H}]=0$), then parity is a conserved quantity. This means that the energy eigenstates of the system can be chosen to be eigenstates of parity as well. For a central potential $V(r)$, the Hamiltonian is parity-invariant. The resulting energy eigenfunctions, ${\psi }_{n,l,m}(\vec{r})\propto R_{n,l}(r)Y_{l,m}(\theta ,\varphi )$, have a definite parity determined by the orbital angular momentum quantum number $l$:

$$\hat{\Pi }|n,l,m⟩=(-1{)}^l|n,l,m⟩$$

This has major consequences, such as forbidding certain transitions. For example, in atomic physics, the dominant electric dipole transitions are only allowed between states of opposite parity, leading to the selection rule $\Delta l=\pm 1$.

2. Time Reversal: A Peculiar Symmetry

Time reversal, the symmetry of "running the movie backwards" ($t\to -t$), is conceptually more subtle in quantum mechanics than parity. Its operator, $\hat{\Theta }$, has a unique mathematical structure that sets it apart from all other symmetries we have discussed.

The video highlights the central problem: the Schrödinger equation is first-order in time. A transformation $t\to -t$ changes the sign of the left-hand side ($i\hslash \frac{\partial }{\partial t}$) but not the right-hand side ($\hat{H}$). However, if we also complex-conjugate the equation, the sign of $i$ flips, restoring the equality. This suggests the time-reversal operator $\hat{\Theta }$ involves complex conjugation.

This leads to the conclusion that $\hat{\Theta }$ must be an anti-linear and anti-unitary operator.

• Anti-linear: It complex conjugates any scalar coefficients it acts on: $\hat{\Theta }(c|\psi ⟩)=c^{\ast }\hat{\Theta }|\psi ⟩$.
• Anti-unitary: It preserves the magnitude of inner products ($|⟨\varphi |\psi ⟩{|}^2$), but conjugates the value itself ($⟨\hat{\Theta }\varphi |\hat{\Theta }\psi ⟩=⟨\psi |\varphi ⟩=⟨\varphi |\psi {⟩}^{\ast }$). This is required to conserve probabilities. An anti-unitary operator satisfies ${\hat{\Theta }}^{†}\hat{\Theta }=\hat{I}$.

The anti-linear property is essential. Consider the canonical commutation relation $[{\hat{x}}_i,{\hat{p}}_j]=i\hslash {\delta }_{ij}$. We expect time reversal to leave position untouched but to flip momentum ($\vec{p}\to -\vec{p}$).
If $\hat{\Theta }$ were linear and unitary, we would have:

$$\hat{\Theta }[{\hat{x}}_i,{\hat{p}}_j]{\hat{\Theta }}^{†}=[\hat{\Theta }{\hat{x}}_i{\hat{\Theta }}^{†},\hat{\Theta }{\hat{p}}_j{\hat{\Theta }}^{†}]=[{\hat{x}}_i,-{\hat{p}}_j]=-[{\hat{x}}_i,{\hat{p}}_j]=-i\hslash {\delta }_{ij}$$

But also $\hat{\Theta }(i\hslash {\delta }_{ij}){\hat{\Theta }}^{†}=i\hslash {\delta }_{ij}$, a contradiction. If $\hat{\Theta }$ is anti-linear, it conjugates the $i$, resolving the paradox: $\hat{\Theta }(i\hslash {\delta }_{ij}){\hat{\Theta }}^{†}=-i\hslash {\delta }_{ij}$.

Kramers Degeneracy

The most stunning consequence of time-reversal symmetry arises when we consider its action on particles with spin. This leads to a powerful theorem with profound implications for condensed matter physics.

Let's summarize the key results from the reading:

1. Action on Angular Momentum: Just like classical angular momentum, both orbital ($\vec{L}$) and spin ($\vec{S}$) angular momentum are odd under time reversal:

   $$\hat{\Theta }\vec{L}{\hat{\Theta }}^{-1}=-\vec{L},\hat{\Theta }\vec{S}{\hat{\Theta }}^{-1}=-\vec{S}$$

2. The Operator ${\hat{\Theta }}^2$: The reading shows that for a general spin state, the action of $\hat{\Theta }$ twice has a remarkable property that depends on the type of particle. You may notice the deep connection to the SU(2) structure from our last lesson, where a $2\pi$ rotation also distinguishes these cases.

   • For integer spin systems (bosons), ${\hat{\Theta }}^2=+1$.
   • For half-integer spin systems (fermions), ${\hat{\Theta }}^2=-1$.

3. Kramers' Theorem: Now, consider a system with a time-reversal invariant Hamiltonian ($[\hat{\Theta },\hat{H}]=0$) and an odd number of fermions (so the total spin is half-integer and ${\hat{\Theta }}^2=-1$).

   • If $|\psi ⟩$ is an energy eigenstate with energy $E$, then $|{\psi }^{\prime }⟩=\hat{\Theta }|\psi ⟩$ must also be an energy eigenstate with the same energy $E$.
   • Is it the same state? Let's assume it is, i.e., $\hat{\Theta }|\psi ⟩=c|\psi ⟩$ for some phase $c$.
   • Then ${\hat{\Theta }}^2|\psi ⟩=\hat{\Theta }(c|\psi ⟩)=c^{\ast }\hat{\Theta }|\psi ⟩=c^{\ast }c|\psi ⟩=|c{|}^2|\psi ⟩=+1|\psi ⟩$.
   • But for this system, we know ${\hat{\Theta }}^2=-1$. This is a contradiction!
   • Therefore, the initial assumption must be wrong. $|\psi ⟩$ and $\hat{\Theta }|\psi ⟩$ cannot be the same state (they are, in fact, orthogonal).

This proves Kramers' Theorem: For any system with time-reversal symmetry containing an odd number of fermions, every energy level is at least doubly degenerate. This forced degeneracy is known as Kramers degeneracy.

This is not a mere theoretical curiosity. Since electrons are spin-1/2 fermions, this theorem dictates a fundamental feature of the electronic band structure of most materials. As long as there is no external magnetic field (which breaks time-reversal symmetry), every energy band must have this twofold degeneracy at every point in momentum space. This is a cornerstone for understanding phenomena from spin transport to topological insulators.

Test your understanding!

Consider a single electron in a crystal lattice potential $V(\vec{r})$ that is not spherically symmetric, so orbital angular momentum $\vec{L}$ is not conserved. However, there is no external magnetic field. Would you expect the electron's energy levels to be degenerate? Why or why not?

Show answer

Yes, you would expect the energy levels to be doubly degenerate. The system consists of a single electron (a fermion with spin-1/2). The Hamiltonian $H=\frac{{\vec{p}}^2}{2m}+V(\vec{r})$ is invariant under time reversal because ${\vec{p}}^2$ is even and $V(\vec{r})$ is invariant. Since the system has time-reversal symmetry and contains an odd number of fermions, Kramers' theorem applies. It guarantees that every energy level must be at least doubly degenerate (a "Kramers pair"), regardless of the lack of spatial symmetries.

Conclusion

In this lesson, we have moved beyond continuous symmetries to investigate the discrete symmetries of parity and time reversal.

Key Takeaways:

• Parity ($\hat{\Pi }$) is a linear, Hermitian, and unitary operator that inverts spatial coordinates. Its eigenvalues are $\pm 1$. When it commutes with the Hamiltonian, it gives rise to a conserved quantum number and governs selection rules for transitions.
• Time Reversal ($\hat{\Theta }$) is a unique anti-linear and anti-unitary operator. This structure is required to make the reversal of motion ($t\to -t,\vec{p}\to -\vec{p}$) compatible with the fundamental laws of quantum mechanics.
• The operator ${\hat{\Theta }}^2$ evaluates to $+1$ for integer-spin systems but $-1$ for half-integer-spin systems.
• Kramers' Theorem, a direct consequence of ${\hat{\Theta }}^2=-1$, states that any time-reversal invariant system with an odd number of fermions must have at least a twofold degeneracy for every energy level.

Preview of the next lesson:
We have now seen how continuous symmetries lead to conservation laws (Noether's Theorem) and discrete symmetries can enforce degeneracy (Kramers' Theorem). The next lesson will formalize and generalize this connection. We will begin to use the tools of group theory to systematically understand how the symmetry group of a Hamiltonian determines the degeneracy of its energy levels, providing a powerful framework for classifying quantum states.