---
title: Symmetries, Generators, and Conservation Laws
module: The Oscillator Algebraically, and Symmetry
moduleNumber: 5
lessonNumber: 3
order: 503
summary: >
  Every continuous symmetry of a quantum system is a unitary operator built by
  exponentiating a Hermitian generator: momentum generates translations, angular
  momentum generates rotations, the Hamiltonian generates time evolution. When a
  generator commutes with the Hamiltonian, the transformation leaves the dynamics
  unchanged and the generator is conserved — the quantum form of Noether's theorem
  — and any symmetry that mixes states within a level forces degeneracy.
topics: [The Oscillator Algebraically, and Symmetry]
sources:
  - book: Sakurai & Napolitano
    ref: "Ch. 4 — Symmetry in Quantum Mechanics; §4.1 Symmetries, Conservation Laws, and Degeneracies"
  - book: Shankar
    ref: "Ch. 11 — Symmetries and Their Consequences; §11.2–§11.4"
draft: false
---

A conservation law in classical mechanics comes from a symmetry of the action:
translation invariance conserves momentum, rotational invariance conserves angular
momentum, time-translation invariance conserves energy. This is Noether's theorem.
Quantum mechanics realizes the same correspondence more directly. A symmetry is a
unitary operator, that operator is the exponential of a Hermitian **generator**, and
the generator is conserved exactly when it commutes with the Hamiltonian.[^sn-sym]
The conserved quantities of the theory and the transformations that leave it
invariant are two views of one algebraic object.

## Symmetries are unitary

A physical symmetry is a mapping of states that preserves every measurable
prediction, which means it preserves inner-product magnitudes:
$|\langle\phi'\vert\psi'\rangle| = |\langle\phi\vert\psi\rangle|$ for the transformed
states. Wigner's theorem identifies the operators that can do this.

> **Theorem (Wigner).** Any transformation of the Hilbert space that preserves all
> transition probabilities $|\langle\phi\vert\psi\rangle|^{2}$ is represented, up to
> a phase, by an operator that is either **unitary** ($\hat U^{\dagger}\hat U = 1$)
> or **antiunitary**. Continuous symmetries connected to the identity are unitary;
> antiunitary operators are needed only for
> [time reversal](/quantum-mechanics/oscillator-and-symmetry/parity-time-reversal-and-discrete-symmetries).

A one-parameter family of symmetries $\hat U(\varepsilon)$ with
$\hat U(0) = 1$ is fixed by its behavior near the identity. Expanding to first order,

$$
\hat U(\varepsilon) = 1 - \frac{i}{\hbar}\,\varepsilon\,\hat G + O(\varepsilon^{2}),
$$

and unitarity $\hat U^{\dagger}\hat U = 1$ to first order forces
$\hat G = \hat G^{\dagger}$: the **generator** is Hermitian, hence an observable.
Composing $N$ infinitesimal steps of size $\varepsilon/N$ and taking $N\to\infty$
exponentiates the generator,

$$
\hat U(\varepsilon) = \lim_{N\to\infty}\left(1 - \frac{i}{\hbar}\frac{\varepsilon}{N}\hat G\right)^{N}
= \exp\!\left(-\frac{i}{\hbar}\,\varepsilon\,\hat G\right).
$$

The factor of $\hbar$ makes the exponent dimensionless and gives $\hat G$ the units
of the conjugate observable: an action per unit $\varepsilon$.

$$
% caption: A finite symmetry is built from infinitesimal steps: each factor
% $1 - i\varepsilon\hat G/\hbar$ advances the parameter by a sliver, and the product
% of infinitely many exponentiates the generator to $\exp(-i\varepsilon\hat G/\hbar)$.
\begin{tikzpicture}[>=stealth, font=\scriptsize,
  st/.style={draw, minimum width=13mm, minimum height=8mm, align=center}]
  \definecolor{acc}{HTML}{4A6FA5}
  \node[st] (a) at (0,0) {$1$};
  \node[st] (b) at (2.3,0) {step};
  \node[st] (c) at (4.6,0) {step};
  \node[anchor=west] at (6.4,0) {...};
  \node[st, acc, fill=acc!12] (d) at (8.6,0) {total};
  \draw[->, black!70, thick] (a) -- (b);
  \draw[->, black!70, thick] (b) -- (c);
  \draw[->, black!70, thick] (7.6,0) -- (d);
  \node[black, anchor=north, font=\tiny] at (2.3,-0.7) {infinitesimal};
  \node[acc, anchor=north, font=\tiny] at (8.6,-0.7) {exponentiated};
\end{tikzpicture}
$$

## Momentum generates translations

Take the spatial translation $\hat T(a)$ that shifts a wave packet by $a$,
$\hat T(a)\psi(x) = \psi(x-a)$. For an infinitesimal shift $\delta$, Taylor expansion
gives

$$
\psi(x-\delta) = \psi(x) - \delta\,\frac{\partial\psi}{\partial x}
= \left(1 - \delta\,\frac{\partial}{\partial x}\right)\psi(x).
$$

Substituting $\partial/\partial x = (i/\hbar)\hat p$ from
$\hat p = -i\hbar\,\partial/\partial x$ turns this into the generator form
$\hat T(\delta) = 1 - (i/\hbar)\delta\,\hat p$. The generator of translations is
**momentum**, and finite translations exponentiate it:

$$
\hat T(a) = \exp\!\left(-\frac{i}{\hbar}\,a\,\hat p\right).
$$

Two immediate consequences follow. The commutator of the generator with position,
$[\hat x,\hat p] = i\hbar$, is the statement that $\hat p$ moves $\hat x$:
$\hat T^{\dagger}(a)\,\hat x\,\hat T(a) = \hat x + a$. And a Hamiltonian invariant
under all translations, $\hat T^{\dagger}(a)\hat H\hat T(a) = \hat H$, must commute
with the generator, $[\hat p,\hat H] = 0$ — momentum is conserved. A free particle
is the archetype; adding any position-dependent potential $V(\hat x)$ breaks
translation invariance and $\hat p$ is no longer conserved.

$$
% caption: An infinitesimal translation carries $\psi(x)$ to $\psi(x-\delta)$,
% shifting the packet rightward. The momentum operator $\hat p$ is the generator:
% $\hat T(\delta) = 1 - i\delta\hat p/\hbar$.
\begin{tikzpicture}[>=stealth, font=\scriptsize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (-0.4,0) -- (6.2,0) node[right, font=\scriptsize, text=black] {$x$};
  % original packet
  \draw[black, thick] (0.4,0.02) .. controls (1.4,0.05) and (1.7,1.6) .. (2.3,1.6)
    .. controls (2.9,1.6) and (3.2,0.05) .. (4.2,0.02);
  \node[black, anchor=south, font=\tiny] at (2.0,1.6) {before};
  % shifted packet
  \draw[acc, thick] (1.7,0.02) .. controls (2.7,0.05) and (3.0,1.6) .. (3.6,1.6)
    .. controls (4.2,1.6) and (4.5,0.05) .. (5.5,0.02);
  \node[acc, anchor=south, font=\tiny] at (3.9,1.6) {after};
  % shift arrow
  \draw[->, black!70, thick] (2.3,0.75) -- (3.6,0.75);
  \node[black!70, anchor=south, font=\tiny] at (2.95,0.78) {shift};
\end{tikzpicture}
$$

## Angular momentum generates rotations

The identical construction in the angular variable gives rotations. A rotation by
angle $\phi$ about the axis $\hat n$ is generated by the component of orbital
angular momentum along that axis,

$$
\hat U_{\hat n}(\phi) = \exp\!\left(-\frac{i}{\hbar}\,\phi\,\hat n\cdot\hat{\vec L}\right),
\qquad \hat{\vec L} = \hat{\vec r}\times\hat{\vec p}.
$$

Because rotations about different axes do not commute, their generators do not
commute either. Working out $[\hat L_i,\hat L_j]$ from
$\hat{\vec L} = \hat{\vec r}\times\hat{\vec p}$ and $[\hat x_i,\hat p_j] = i\hbar\delta_{ij}$
reproduces the rotation group's structure:

$$
[\hat L_i,\hat L_j] = i\hbar\,\epsilon_{ijk}\,\hat L_k .
$$

This non-abelian algebra is the entire content of
[angular-momentum quantization](/quantum-mechanics/angular-momentum/the-angular-momentum-algebra):
the eigenvalues and multiplet structure follow from these commutators alone, exactly
as the oscillator spectrum followed from $[\hat a,\hat a^\dagger] = 1$. A Hamiltonian
invariant under all rotations, such as any central potential $V(r)$, commutes with
every $\hat L_i$, so all three components of angular momentum are conserved.

$$
% caption: A rotation by $\phi$ about the $z$ axis, generated by $\hat L_z$, carries
% a state's expectation vector around a cone; infinitesimally, $\hat L_z$ rotates
% $\hat x$ into $\hat y$.
\begin{tikzpicture}[>=stealth, font=\scriptsize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (-2.2,0) -- (2.4,0) node[right, font=\scriptsize, text=black] {$x$};
  \draw[black, ->] (0,-2.0) -- (0,2.2) node[above, font=\scriptsize, text=black] {$y$};
  \draw[black, dashed] (0,0) circle (1.6);
  % initial vector
  \draw[->, black, thick] (0,0) -- (1.6,0);
  \node[black, anchor=west, font=\tiny] at (1.55,-0.25) {before};
  % rotated vector
  \draw[->, acc, thick] (0,0) -- (1.13,1.13);
  \node[acc, anchor=south west, font=\tiny] at (1.05,1.05) {after};
  % rotation arc
  \draw[->, black!70] (1.9,0) arc (0:45:1.9);
  \node[black!70, anchor=west, font=\tiny] at (1.85,0.85) {rotate};
\end{tikzpicture}
$$

## Conservation from commuting with the Hamiltonian

The link between symmetry and conservation is the Heisenberg equation of motion. For
any observable $\hat G$ with no explicit time dependence,

$$
\frac{\d}{\d t}\langle\hat G\rangle
= \frac{i}{\hbar}\,\langle[\hat H,\hat G]\rangle .
$$

If $\hat G$ generates a symmetry of the Hamiltonian, then
$\hat U^{\dagger}(\varepsilon)\hat H\hat U(\varepsilon) = \hat H$ for all
$\varepsilon$; differentiating at $\varepsilon = 0$ gives $[\hat G,\hat H] = 0$.
The right side vanishes, and $\langle\hat G\rangle$ is constant — as is the entire
probability distribution of $\hat G$, since $[\hat G,\hat H] = 0$ makes
$\hat U(\varepsilon)$ commute with the evolution operator $e^{-i\hat H t/\hbar}$.

> **Theorem (Symmetry and conservation).** If a Hermitian generator $\hat G$
> commutes with the Hamiltonian, $[\hat G,\hat H] = 0$, then the unitary family
> $\hat U(\varepsilon) = e^{-i\varepsilon\hat G/\hbar}$ is a symmetry of the dynamics
> and $\hat G$ is a constant of the motion: $\d\langle\hat G\rangle/\d t = 0$ in
> every state. This is the quantum form of Noether's theorem — one conserved
> observable per continuous symmetry.

Time evolution itself fits the same pattern. The evolution operator
$e^{-i\hat H t/\hbar}$ is generated by the Hamiltonian, so **energy** is the
conserved quantity associated with time-translation invariance, provided $\hat H$
has no explicit time dependence.

$$
% caption: A generator commuting with $\hat H$ has a time-independent expectation
% (flat, conserved); one that does not commute evolves. Conservation is the direct
% signature of the corresponding symmetry.
\begin{tikzpicture}[>=stealth, font=\scriptsize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (0,0) -- (6.0,0) node[right, font=\scriptsize, text=black] {$t$};
  \draw[black, ->] (0,-1.6) -- (0,1.9) node[above, font=\scriptsize, text=black] {expectation};
  % conserved: flat line
  \draw[acc, very thick] (0.2,1.1) -- (5.6,1.1);
  \node[acc, anchor=west, font=\tiny] at (4.4,1.35) {conserved};
  % not conserved: oscillating
  \draw[black, thick, dashed] (0.2,-0.2)
    .. controls (0.9,-1.2) and (1.6,-1.2) .. (2.3,-0.2)
    .. controls (3.0,0.8) and (3.7,0.8) .. (4.4,-0.2)
    .. controls (5.0,-1.0) and (5.4,-0.9) .. (5.6,-0.5);
  \node[black, anchor=west, font=\tiny] at (3.8,0.55) {not conserved};
\end{tikzpicture}
$$

## Symmetry forces degeneracy

A symmetry does more than conserve a quantity; it organizes the spectrum. Suppose
$[\hat U,\hat H] = 0$ and $\lvert\psi\rangle$ is an energy eigenstate,
$\hat H\lvert\psi\rangle = E\lvert\psi\rangle$. Then

$$
\hat H\bigl(\hat U\lvert\psi\rangle\bigr) = \hat U\hat H\lvert\psi\rangle
= E\bigl(\hat U\lvert\psi\rangle\bigr),
$$

so $\hat U\lvert\psi\rangle$ is another eigenstate at the same energy. If it is not
simply a phase times $\lvert\psi\rangle$, the level is **degenerate**. A single
generator that commutes with $\hat H$ but whose transformations rotate one state into
a genuinely different one produces a whole family of states sharing an energy.

The size of the degenerate multiplet is fixed by the symmetry group. For a central
potential the three components $\hat L_x,\hat L_y,\hat L_z$ all commute with $\hat H$
but not with each other, and this non-abelian structure forces the $2\ell+1$ states
of fixed $\ell$ to be degenerate: the $\hat L_\pm = \hat L_x \pm i\hat L_y$ ladder
maps any one onto the others without changing the energy. An abelian symmetry (a
single commuting generator, like translation) does not force degeneracy; a
non-abelian one does. A degeneracy not explained by the obvious geometric symmetry
signals a **hidden symmetry** with extra commuting generators, as for the
$1/r$ Coulomb potential and the isotropic
[oscillator](/quantum-mechanics/central-potentials/the-isotropic-oscillator-and-hidden-symmetry).

$$
% caption: A rotation commuting with $\hat H$ maps every state of a level onto
% another state at the same energy. The $2\ell+1$ members of an angular-momentum
% multiplet are degenerate because the non-abelian rotation generators connect them.
\begin{tikzpicture}[>=stealth, font=\scriptsize]
  \definecolor{acc}{HTML}{4A6FA5}
  % a degenerate level: three dots on one horizontal line
  \draw[acc, thick] (0,2.0) -- (5.0,2.0);
  \node[acc, anchor=west] at (5.1,2.0) {$E$ (level)};
  \fill[acc] (1.0,2.0) circle (2.4pt);
  \fill[acc] (2.5,2.0) circle (2.4pt);
  \fill[acc] (4.0,2.0) circle (2.4pt);
  % symmetry arrows mapping states onto each other
  \draw[->, black!70] (1.15,2.2) to[bend left=40] (2.35,2.2);
  \draw[->, black!70] (2.65,2.2) to[bend left=40] (3.85,2.2);
  \node[black!70, anchor=south, font=\tiny] at (2.5,2.55) {symmetry maps within the level};
  % a nondegenerate level below for contrast
  \draw[black, thick] (0,0.5) -- (5.0,0.5);
  \node[black, anchor=west] at (5.1,0.5) {nondegenerate};
  \fill[black] (2.5,0.5) circle (2.4pt);
\end{tikzpicture}
$$

## The dictionary of symmetries and conserved quantities

The construction is uniform: read off the generator of a symmetry, and it is the
conserved observable when the Hamiltonian respects that symmetry.

| Symmetry | Unitary operator | Generator | Conserved when |
| --- | --- | --- | --- |
| Spatial translation | $e^{-i a\hat p/\hbar}$ | momentum $\hat p$ | $V$ is uniform |
| Rotation | $e^{-i\phi\,\hat n\cdot\hat{\vec L}/\hbar}$ | angular momentum $\hat{\vec L}$ | $V$ is central |
| Time translation | $e^{-i\hat H t/\hbar}$ | Hamiltonian $\hat H$ | $\hat H$ has no explicit $t$ |
| Global phase | $e^{-i\lambda\hat Q}$ | charge $\hat Q$ | interactions are phase-invariant |

Each row is one instance of the same theorem. The continuous symmetries treated here
are connected to the identity and represented by unitary operators. The remaining
symmetries — parity and time reversal — are **discrete**, cannot be reached by
exponentiating a generator, and one of them requires the antiunitary case of
Wigner's theorem. Those are the subject of the
[next lesson](/quantum-mechanics/oscillator-and-symmetry/parity-time-reversal-and-discrete-symmetries).

[^sn-sym]: **Sakurai & Napolitano**, _Modern Quantum Mechanics_ (3rd ed., Cambridge, 2021), §4.1 — symmetries, conservation laws, and degeneracies: the unitary generator, $[\hat G,\hat H]=0$ implying conservation, and non-abelian symmetry forcing degeneracy. Cambridge listing: https://www.cambridge.org/highereducation/books/modern-quantum-mechanics/6C8BB37F5B120694E9AB9DF6EBFD6D48. Group-theoretic development: **Shankar**, _Principles of Quantum Mechanics_ (2nd ed., Springer, 1994), Ch. 11 — https://link.springer.com/book/10.1007/978-1-4757-0576-8.