Symmetry Constraints on Hamiltonians

Symmetry is a filter on Hamiltonians. Instead of writing every operator allowed by dimension or convenience, one writes only terms compatible with the stated symmetries.

The basic criterion is:

$$SHS^{-1}=H.$$

For a unitary symmetry $U$, this is $UHU^\dagger=H$. For a continuous unitary symmetry generated by $G$, the infinitesimal condition is

$$[G,H]=0.$$

The Working Algorithm

To constrain a Hamiltonian by symmetry:

1. Choose the Hilbert space and degrees of freedom.
2. State how the symmetry acts on states and operators.
3. Write a general Hamiltonian in the relevant operator basis.
4. Impose $SHS^{-1}=H$ for each symmetry.
5. Remove terms whose coefficients must vanish or combine terms whose coefficients must be equal.
6. Interpret the remaining parameters physically.

The power of the method is that it works before solving the spectrum.

Spin-1/2 Example

Every Hermitian Hamiltonian for a two-level system can be written

$$H=aI+\mathbf b\cdot\boldsymbol\sigma,$$

with real $a$ and real vector $\mathbf b$. If there is no external vector selecting a direction and the Hamiltonian must be invariant under all spin rotations, then the vector term is forbidden:

$$H=aI.$$

If an external magnetic field $\mathbf B$ is present, a term

$$H=-\gamma\,\mathbf B\cdot\mathbf S$$

is allowed because $\mathbf B$ supplies a physical direction. If $\mathbf B$ is fixed along $z$, the Hamiltonian no longer has full rotational symmetry; it retains rotations about the $z$ axis.

Parity in One Dimension

For

$$H=\frac{p^2}{2m}+V(x),$$

parity acts as

$$PxP^{-1}=-x, \qquad PpP^{-1}=-p.$$

The kinetic term is invariant. The potential is invariant only when

$$V(-x)=V(x).$$

Thus parity symmetry constrains the potential to be even.

Translation Invariance

Spatial translations are generated by momentum:

$$U(a)=e^{-iaP/\hbar}.$$

For a Hamiltonian

$$H=\frac{P^2}{2m}+V(X),$$

translation invariance requires

$$[P,H]=0.$$

Since

$$[P,V(X)]=-i\hbar V'(X),$$

the potential must be constant on the translated region. This is why the free particle is translation invariant and a generic potential is not. The focused version of this test is Translation-Invariant Hamiltonians.

Central Potentials

For a particle in three dimensions,

$$H=\frac{\mathbf p^2}{2m}+V(r), \qquad r=\lvert\mathbf r\rvert,$$

the Hamiltonian is rotationally invariant. Therefore

$$[H,L_i]=0$$

for each component of orbital angular momentum. Energy eigenstates can be organized using angular momentum quantum numbers, even when the radial equation is still nontrivial.

Symmetry Breaking Terms

Adding a small term can break a symmetry and split degeneracies. For example,

$$H=aI+b_z\sigma_z$$

selects the $z$ direction. It commutes with $\sigma_z$ but not with $\sigma_x$ or $\sigma_y$. The symmetry has been reduced from full spin rotation to rotations about the $z$ axis.

This is not a failure of symmetry reasoning. It is one of its main uses: identifying what a perturbation breaks and which quantum numbers remain good.

Common Mistakes

• Imposing symmetry on a state when the question is about symmetry of the Hamiltonian.
• Forgetting to transform external fields or parameters when deciding whether a term is invariant.
• Treating full rotational invariance and axial symmetry as the same thing.
• Assuming a conserved quantity survives after adding a symmetry-breaking perturbation.
• Calling a Hamiltonian “generic” while silently imposing a symmetry.

Cross-Links

• Why Symmetry Matters
• Quantum Symmetries
• States, Observables, and Hamiltonians
• Symmetry Groups and Representations
• Translation-Invariant Hamiltonians
• Discrete Symmetries in Hamiltonians
• Degeneracy and Multiplets
• Unitary Symmetries
• Quantum Noether Principle
• Constants of Motion
• Exact Symmetry
• Explicit Symmetry Breaking
• Approximate Symmetry
• Accidental Symmetry
• Commutators
• Pauli Matrices
• Commutator Table
• Free Particle

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• H. F. Jones, Groups, Representations and Physics, 2nd ed., CRC Press, 1998.
• M. Tinkham, Group Theory and Quantum Mechanics, Dover, 2003.

Exercises

1. Which terms in $H=aI+b_x\sigma_x+b_y\sigma_y+b_z\sigma_z$ commute with $\sigma_z$?

Solution

The terms $aI$ and $b_z\sigma_z$ commute with $\sigma_z$. The Pauli commutators give

$$[\sigma_z,\sigma_x]=2i\sigma_y, \qquad [\sigma_z,\sigma_y]=-2i\sigma_x,$$

so nonzero $b_x$ or $b_y$ breaks the symmetry generated by $\sigma_z$.

2. Let $H=p^2/(2m)+V(x)$ and suppose $[P,H]=0$. What does this imply about $V(x)$ on a connected interval?

Solution

Since $[P,p^2/(2m)]=0$ and $[P,V(X)]=-i\hbar V'(X)$, the condition $[P,H]=0$ implies $V'(X)=0$ as an operator on the interval. Thus $V(x)$ is constant there.