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{{Quantum book backlink|Mathematical structure and systems}}
In quantum mechanics, a '''symmetry''' is a transformation that leaves the physical properties of a system unchanged. In general, symmetry in physics, [[Wikipedia:invariant (physics)|invariance]], and [[Wikipedia:Conservation law (physics)|conservation law]]s, are fundamentally important constraints for formulating [[Wikipedia:Theoretical physics|physical theories]] and models. In practice, they are powerful methods for solving problems and predicting what can happen. Symmetries play a central role in determining conservation laws and the structure of quantum systems.<ref>{{cite book |last=Sakurai |first=J. J. |title=Modern Quantum Mechanics |publisher=Addison-Wesley |year=1994}}</ref>

[[File:Quantum_symmetry1.jpg|thumb|400px|Symmetry in quantum mechanics: angular momentum, parity, rotational symmetry, and conservation laws.]]
{{Lie groups |Physics}}
=Symmetry principles=
Mathematically, symmetries are represented by operators acting on the Hilbert space <math>\mathcal{H}</math>. A transformation is a symmetry if it preserves transition probabilities:

<math>
|\langle \phi \mid \psi \rangle|^2 = |\langle U\phi \mid U\psi \rangle|^2.
</math>

Such transformations are represented by '''unitary''' (or antiunitary) operators.<ref>{{cite book |last=Wigner |first=Eugene |title=Group Theory and its Application to the Quantum Mechanics of Atomic Spectra |publisher=Academic Press |year=1959}}</ref>

=== Unitary transformations ===

A unitary operator <math>U</math> satisfies

<math>
U^\dagger U = I.
</math>

Under such a transformation, a state changes as

<math>
\psi \rightarrow U \psi,
</math>

while expectation values of observables remain invariant.

If an observable <math>\hat{A}</math> is invariant under the symmetry, then

<math>
U^\dagger \hat{A} U = \hat{A}.
</math>

=== Continuous symmetries ===

Continuous symmetries are generated by operators through exponentiation. A transformation depending on a parameter <math>\alpha</math> can be written as

<math>
U(\alpha) = e^{-i \alpha \hat{G}},
</math>

where <math>\hat{G}</math> is the generator of the symmetry.

Examples include:

* translations → momentum operator
* rotations → angular momentum operator
* time evolution → Hamiltonian

=== Conservation laws ===

A fundamental result is that symmetries correspond to conserved quantities. If a generator <math>\hat{G}</math> commutes with the Hamiltonian,

<math>
[\hat{H}, \hat{G}] = 0,
</math>

then the corresponding observable is conserved in time.<ref>{{cite book |last=Ballentine |first=Leslie E. |title=Quantum Mechanics: A Modern Development |publisher=World Scientific |year=1998}}</ref>

This is the quantum analogue of [[Wikipedia:Noether's theorem|Noether’s theorem]].

=== Physical significance ===

Symmetry principles:

* determine allowed energy levels and degeneracies,
* simplify the solution of quantum systems,
* explain conservation laws such as energy, momentum, and angular momentum.

They provide a unifying framework connecting mathematics and physical observables in quantum theory.

=Rotation group SO(3)=

The '''rotation group''' SO(3) consists of all rotations in three-dimensional space that preserve distances and orientation. In quantum mechanics, it describes the symmetry of systems whose Hamiltonian is invariant under spatial rotations, such as atoms and isotropic potentials.<ref>{{cite book |last=Sakurai |first=J. J. |title=Modern Quantum Mechanics |publisher=Addison-Wesley |year=1994}}</ref>

=== Rotational symmetry ===

A system is rotationally symmetric if its Hamiltonian <math>\hat{H}</math> commutes with the generators of rotations:

<math>
[\hat{H}, \hat{L}_i] = 0, \quad i = x,y,z.
</math>

This implies that angular momentum is conserved.

Rotations are represented by unitary operators

<math>
U(R) = e^{-i \boldsymbol{\theta} \cdot \hat{\mathbf{L}}/\hbar},
</math>

where <math>\hat{\mathbf{L}}</math> is the angular momentum operator.

=== Angular momentum algebra ===

The components of angular momentum satisfy the commutation relations

<math>
[\hat{L}_x, \hat{L}_y] = i\hbar \hat{L}_z,
\quad
[\hat{L}_y, \hat{L}_z] = i\hbar \hat{L}_x,
\quad
[\hat{L}_z, \hat{L}_x] = i\hbar \hat{L}_y.
</math>

These relations define the Lie algebra of SO(3).<ref>{{cite book |last=Ballentine |first=Leslie E. |title=Quantum Mechanics: A Modern Development |publisher=World Scientific |year=1998}}</ref>

=== Eigenvalues and quantum numbers ===

The operators <math>\hat{L}^2</math> and <math>\hat{L}_z</math> commute and can be simultaneously diagonalized:

<math>
\hat{L}^2 |l,m\rangle = \hbar^2 l(l+1) |l,m\rangle,
\quad
\hat{L}_z |l,m\rangle = \hbar m |l,m\rangle.
</math>

The quantum numbers satisfy:

* <math>l = 0,1,2,\dots</math>
* <math>m = -l,-l+1,\dots,l</math>

Each value of <math>l</math> corresponds to a representation of the rotation group.

=== Spherical harmonics ===

In position space, the eigenfunctions of angular momentum are the [[Wikipedia:Spherical harmonics|spherical harmonics]] <math>Y_{lm}(\theta,\phi)</math>, which form an orthonormal basis on the sphere.

They arise naturally when solving the Schrödinger equation for systems with spherical symmetry, such as the hydrogen atom.

=== Degeneracy and symmetry ===

Rotational symmetry leads to degeneracy in energy levels. Since the Hamiltonian depends only on <math>r</math>, states with the same <math>l</math> but different <math>m</math> have the same energy.

This degeneracy reflects the invariance of the system under rotations.

=== Physical significance ===

The rotation group SO(3):

* explains conservation of angular momentum,
* determines the structure of atomic orbitals,
* provides the mathematical foundation for rotational symmetry in quantum systems.

It is a fundamental example of how symmetry groups shape the behavior of quantum systems.

=SU(2) and spin=

The group '''SU(2)''' is the group of 2×2 unitary matrices with determinant 1. In quantum mechanics, it plays a fundamental role in describing intrinsic angular momentum, or '''spin'''.<ref>{{cite book |last=Sakurai |first=J. J. |title=Modern Quantum Mechanics |publisher=Addison-Wesley |year=1994}}</ref>

Although classical rotations are described by the group SO(3), quantum systems require SU(2) to fully account for all possible angular momentum states.

=== Relation to SO(3) ===

The groups SU(2) and SO(3) are closely related: SU(2) is a '''double cover''' of SO(3). This means that each rotation in SO(3) corresponds to two elements in SU(2).

As a result:

* integer angular momentum states correspond to representations of SO(3),
* half-integer spin states (e.g. spin-½) arise naturally from SU(2).

This explains why particles such as electrons have spin values that have no classical analogue.<ref>{{cite book |last=Ballentine |first=Leslie E. |title=Quantum Mechanics: A Modern Development |publisher=World Scientific |year=1998}}</ref>

=== Spin operators ===

Spin is described by operators <math>\hat{S}_x, \hat{S}_y, \hat{S}_z</math> that satisfy the same commutation relations as orbital angular momentum:

<math>
[\hat{S}_x, \hat{S}_y] = i\hbar \hat{S}_z,
\quad
[\hat{S}_y, \hat{S}_z] = i\hbar \hat{S}_x,
\quad
[\hat{S}_z, \hat{S}_x] = i\hbar \hat{S}_y.
</math>

The total spin operator is

<math>
\hat{S}^2 = \hat{S}_x^2 + \hat{S}_y^2 + \hat{S}_z^2.
</math>

=== Spin-½ systems ===

For spin-½ particles, the spin operators can be written in terms of the [[Wikipedia:Pauli matrices|Pauli matrices]]:

<math>
\hat{S}_i = \frac{\hbar}{2} \sigma_i.
</math>

The Pauli matrices are

<math>
\sigma_x =
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}, \quad
\sigma_y =
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}, \quad
\sigma_z =
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}.
</math>

The eigenvalues of <math>\hat{S}_z</math> are

<math>
\pm \frac{\hbar}{2}.
</math>

=== Spinors ===

States of spin-½ particles are represented by two-component complex vectors called '''spinors''':

<math>
\chi =
\begin{pmatrix}
a \\
b
\end{pmatrix}.
</math>

Under rotations, spinors transform according to SU(2), not SO(3). A rotation by 360° changes the sign of the spinor:

<math>
\chi \rightarrow -\chi.
</math>

This property has no classical analogue and is a distinctive feature of quantum systems.

=== Physical significance ===

The SU(2) symmetry:

* explains intrinsic angular momentum (spin),
* governs the behavior of electrons and other fermions,
* underlies quantum statistics and magnetic interactions.

It is essential for understanding atomic structure, spin dynamics, and quantum information systems.<ref>{{cite book |last=Nielsen |first=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010}}</ref>

=Noether’s theorem=

'''Noether’s theorem''' establishes a fundamental connection between symmetries and conservation laws. In quantum mechanics, it states that every continuous symmetry of a system corresponds to a conserved observable.<ref>{{cite book |last=Ballentine |first=Leslie E. |title=Quantum Mechanics: A Modern Development |publisher=World Scientific |year=1998}}</ref>

Originally formulated in classical mechanics by [[Wikipedia:Emmy Noether|Emmy Noether]], the theorem extends naturally to quantum systems through operator methods.

=== Symmetry and generators ===

A continuous symmetry transformation can be written as

<math>
U(\alpha) = e^{-i \alpha \hat{G}},
</math>

where <math>\hat{G}</math> is the generator of the symmetry.

If the Hamiltonian <math>\hat{H}</math> is invariant under this transformation, then

<math>
U^\dagger(\alpha)\hat{H}U(\alpha) = \hat{H}.
</math>

This implies

<math>
[\hat{H}, \hat{G}] = 0.
</math>

=== Conserved quantities ===

If a generator <math>\hat{G}</math> commutes with the Hamiltonian, its expectation value is conserved in time:

<math>
\frac{d}{dt} \langle \hat{G} \rangle = 0.
</math>

Thus, symmetries correspond directly to conserved physical quantities.

Examples include:

* time translation → conservation of energy
* spatial translation → conservation of momentum
* rotational symmetry → conservation of angular momentum

=== Relation to quantum dynamics ===

The time evolution of an operator <math>\hat{G}</math> in the Heisenberg picture is given by

<math>
\frac{d\hat{G}}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{G}].
</math>

If <math>[\hat{H}, \hat{G}] = 0</math>, then <math>\hat{G}</math> is constant in time.

This provides a direct link between symmetry and conservation within the formalism of quantum mechanics.

=== Physical significance ===

Noether’s theorem:

* explains why conservation laws arise in quantum systems,
* provides a systematic way to identify conserved quantities,
* connects abstract symmetry transformations with measurable observables.

It is one of the most important principles linking mathematics and physics in both classical and quantum theories.<ref>{{cite book |last=Sakurai |first=J. J. |title=Modern Quantum Mechanics |publisher=Addison-Wesley |year=1994}}</ref>

=Selection rules=

'''Selection rules''' determine which transitions between quantum states are allowed or forbidden under a given interaction. They arise from the symmetry properties of the system and the operators involved in the transition.<ref>{{cite book |last=Sakurai |first=J. J. |title=Modern Quantum Mechanics |publisher=Addison-Wesley |year=1994}}</ref>

In quantum mechanics, transition probabilities are governed by matrix elements of the form

<math>
\langle f \mid \hat{O} \mid i \rangle,
</math>

where <math>\hat{O}</math> is the operator corresponding to the interaction (e.g. electric dipole operator), and <math>|i\rangle</math>, <math>|f\rangle</math> are the initial and final states.

A transition is allowed only if this matrix element is nonzero.

=== Angular momentum selection rules ===

For systems with rotational symmetry, selection rules follow from angular momentum conservation.

For electric dipole transitions, the typical rules are:

* <math>\Delta l = \pm 1</math>
* <math>\Delta m = 0, \pm 1</math>

These arise from the transformation properties of the dipole operator under rotations.

=== Parity selection rules ===

Parity describes the behavior of a wavefunction under spatial inversion:

<math>
\psi(\mathbf{r}) \rightarrow \psi(-\mathbf{r}).
</math>

For electric dipole transitions:

* the initial and final states must have opposite parity

If both states have the same parity, the transition is forbidden.

=== Origin from symmetry ===

Selection rules are a direct consequence of symmetry:

* conservation laws restrict allowed transitions,
* symmetry properties determine which matrix elements vanish,
* group theory provides a systematic framework for deriving rules.<ref>{{cite book |last=Hamermesh |first=Morton |title=Group Theory and Its Application to Physical Problems |publisher=Addison-Wesley |year=1962}}</ref>

=== Physical significance ===

Selection rules:

* explain spectral lines in atomic and molecular systems,
* determine allowed transitions in spectroscopy,
* reflect the underlying symmetry of quantum systems.

They provide a powerful connection between symmetry principles and observable physical phenomena.

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum Symmetry in quantum mechanics|1}}

Physics:Quantum Symmetry in quantum mechanics - Revision history

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