What a symmetry is · Quantum Mechanics

Intuition

A symmetry is a change of point of view that no experiment can detect: turn the laboratory, move it, or look at it in a mirror, and if every probability comes out the same, the change was a symmetry. In quantum mechanics that means a transformation of states that keeps every transition probability, and Wigner showed that such a transformation is always carried out by a unitary operator, or by an antiunitary one that conjugates numbers. When the transformation also commutes with the Hamiltonian, it is a symmetry of the motion as well: transformed states evolve exactly as the originals do, transformed.

A game of chess played on a board turned round, with every piece moved to match, is the same game: every rule works the same way. A symmetry of a quantum system is a turned board of that kind.

Contours of a potential that depends only on the distance from the centre. A rotation carries a particle at $\mathbf{r}$ to a point on the same contour, where it feels the same potential: nothing that then happens to it can tell the two apart.

Symmetries of states and of motion

A symmetry maps states to states and keeps every transition probability. By Wigner’s theorem it is carried out by a unitary or an antiunitary operator. It is a symmetry of the dynamics when it also commutes with the Hamiltonian.

\begin{aligned}&\bigl|\braket{\hat{U}\phi|\hat{U}\psi}\bigr|^{2}=\bigl|\braket{\phi|\psi}\bigr|^{2}\ \ \text{for all}\ \phi,\psi\\ &\hat{U}\hat{H}\hat{U}^{\dagger}=\hat{H}\iff[\hat{U},\hat{H}]=0\end{aligned}

What to know

Wigner’s theorem, used here without proof: every map of states that keeps all transition probabilities is, up to phases, a unitary or an antiunitary operator. Every symmetry reached continuously from doing nothing is unitary.
Observables transform with it: $\hat{A}\to\hat{U}\hat{A}\hat{U}^{\dagger}$ , so that .

A symmetry of the Hamiltonian carries motions to motions

[\hat{U},\hat{H}]=0\ \Longrightarrow\ \hat{U}e^{-i\hat{H}t/\hbar}=e^{-i\hat{H}t/\hbar}\hat{U}

A unitary operator keeps every inner product, so it keeps every probability. If it commutes with the Hamiltonian, apply it to a solution of the Schrödinger equation: the time derivative passes through it and the Hamiltonian can be moved in front, so the transformed state solves the same equation.

Proof steps

\braket{\hat{U}\phi|\hat{U}\psi}=\bra{\phi}\hat{U}^{\dagger}\hat{U}\ket{\psi}=\braket{\phi|\psi}

Unitarity keeps every inner product, and so every probability.

i\hbar\frac{d}{dt}\bigl(\hat{U}\ket{\psi(t)}\bigr)=\hat{U}\,i\hbar\frac{d}{dt}\ket{\psi(t)}=\hat{U}\hat{H}\ket{\psi(t)}

$\hat{U}$ does not depend on time, and $\ket{\psi(t)}$ solves the Schrödinger equation.

\hat{U}\hat{H}\ket{\psi(t)}=\hat{H}\bigl(\hat{U}\ket{\psi(t)}\bigr)

$\hat{U}$ commutes with $\hat{H}$ .

i\hbar\frac{d}{dt}\bigl(\hat{U}\ket{\psi(t)}\bigr)=\hat{H}\bigl(\hat{U}\ket{\psi(t)}\bigr)

So the transformed state solves the same equation, starting from $\hat{U}\ket{\psi(0)}$ .

\hat{U}e^{-i\hat{H}t/\hbar}\ket{\psi(0)}=e^{-i\hat{H}t/\hbar}\hat{U}\ket{\psi(0)}

Solutions with the same starting state agree, for every starting state.

Applications

Practice

A Change No Experiment Detects

A symmetry is a transformation of states that leaves every probability unchanged: the transition probability between any two transformed states equals the one between the originals.

\bigl|\braket{\hat{U}\phi|\hat{U}\psi}\bigr|^{2}=\bigl|\braket{\phi|\psi}\bigr|^{2}

Try it

What must a transformation of states preserve to be a symmetry?