Quantum Mechanics · Chapter 08

Symmetry

A symmetry is a change of point of view that no experiment can detect. In quantum mechanics it is a unitary operator, a continuous one is generated by an observable, and whenever it commutes with the Hamiltonian that observable is conserved. Translations and momentum, rotations and angular momentum, parity, time reversal, and the degeneracy that symmetry forces on a spectrum.

U^(s) = e^{- \frac{i s G ^}{ℏ}}

[G^, H^] = 0

T^(a) ψ (x) = ψ (x - a)

P^ψ (x) = ψ (- x)

Lessons

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05Symmetry and conservation lawsPut the last three lessons together and a theorem appears. If a Hamiltonian is unchanged by a continuous family of transformations, it commutes with their generator, and a quantity that commutes with the Hamiltonian is conserved — not only its average, but the probability of each of its values. Time translations conserve energy, space translations momentum, rotations angular momentum, and a change of every charged particle’s phase conserves charge. In quantum mechanics the argument also runs backwards: every conserved quantity generates a symmetry.Read lesson →

06ParityParity reflects every state through the origin: the new wavefunction at $x$ is the old one at $-x$. Doing it twice changes nothing, so its eigenvalues are $+1$ and $-1$, the even and odd functions. When the potential is the same on both sides, parity commutes with the Hamiltonian and each level can be given a definite parity — and then a simple symmetry argument says which matrix elements must vanish. Selection rules for light, one of the most useful facts in atomic physics, start here.Read lesson →

07Time reversalFilm a quantum system and run the film backwards: is the reversed film also possible? For a particle in an ordinary potential it is, and the operation that reverses it — keeping positions and flipping momenta — is complex conjugation of the wavefunction. It is not a unitary operator but an antiunitary one, which conjugates the numbers in front of states, and it has to be: a unitary reversal would turn every energy $E$ into $-E$, and energies bounded below have no such mirror.Read lesson →

08Symmetry and degeneracyA symmetry carries each energy eigenstate to another state of the same energy. If the copy is a genuinely different state, the level is degenerate, and the symmetry has forced it. The cleanest version of the argument: two symmetries that do not commute with each other cannot both act as mere numbers on every level, so some level must hold several states. Break the symmetry and the degenerate level splits; the pattern of the splitting tells which symmetry was lost.Read lesson →

09Chapter practiceBring the chapter together: symmetries as probability-keeping maps, generators and their exponentials, translations and momentum, rotations and angular momentum, conservation laws, parity and its selection rule, time reversal and the degeneracies symmetry forces. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Symmetry

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