Physics:Quantum Symmetry in quantum mechanics

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In quantum mechanics, a symmetry is a transformation that leaves the physical properties of a system unchanged. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating 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.[1]

Symmetry in quantum mechanics: angular momentum, parity, rotational symmetry, and conservation laws.

Template:Lie groups

Symmetry principles

Mathematically, symmetries are represented by operators acting on the Hilbert space ℋ. A transformation is a symmetry if it preserves transition probabilities:

|⟨ϕ∣ψ⟩|2=|⟨Uϕ∣Uψ⟩|2.

Such transformations are represented by unitary (or antiunitary) operators.[2]

Unitary transformations

A unitary operator U satisfies

U†U=I.

Under such a transformation, a state changes as

ψ→Uψ,

while expectation values of observables remain invariant.

If an observable A^ is invariant under the symmetry, then

U†A^U=A^.

Continuous symmetries

Continuous symmetries are generated by operators through exponentiation. A transformation depending on a parameter α can be written as

U(α)=e−iαG^,

where G^ 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 G^ commutes with the Hamiltonian,

[H^,G^]=0,

then the corresponding observable is conserved in time.[3]

This is the quantum analogue of 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.[4]

Rotational symmetry

A system is rotationally symmetric if its Hamiltonian H^ commutes with the generators of rotations:

[H^,L^i]=0,i=x,y,z.

This implies that angular momentum is conserved.

Rotations are represented by unitary operators

U(R)=e−iθ⋅𝐋^/ℏ,

where 𝐋^ is the angular momentum operator.

Angular momentum algebra

The components of angular momentum satisfy the commutation relations

[L^x,L^y]=iℏL^z,[L^y,L^z]=iℏL^x,[L^z,L^x]=iℏL^y.

These relations define the Lie algebra of SO(3).[5]

Eigenvalues and quantum numbers

The operators L^2 and L^z commute and can be simultaneously diagonalized:

L^2|l,m⟩=ℏ2l(l+1)|l,m⟩,L^z|l,m⟩=ℏm|l,m⟩.

The quantum numbers satisfy:

  • l=0,1,2,…
  • m=−l,−l+1,…,l

Each value of l corresponds to a representation of the rotation group.

Spherical harmonics

In position space, the eigenfunctions of angular momentum are the spherical harmonics Ylm(θ,ϕ), 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 r, states with the same l but different m 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.[6]

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.[7]

Spin operators

Spin is described by operators S^x,S^y,S^z that satisfy the same commutation relations as orbital angular momentum:

[S^x,S^y]=iℏS^z,[S^y,S^z]=iℏS^x,[S^z,S^x]=iℏS^y.

The total spin operator is

S^2=S^x2+S^y2+S^z2.

Spin-½ systems

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

S^i=ℏ2σi.

The Pauli matrices are

σx=(0110),σy=(0−ii0),σz=(100−1).

The eigenvalues of S^z are

±ℏ2.

Spinors

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

χ=(ab).

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

χ→−χ.

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.[8]

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.[9]

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

Symmetry and generators

A continuous symmetry transformation can be written as

U(α)=e−iαG^,

where G^ is the generator of the symmetry.

If the Hamiltonian H^ is invariant under this transformation, then

U†(α)H^U(α)=H^.

This implies

[H^,G^]=0.

Conserved quantities

If a generator G^ commutes with the Hamiltonian, its expectation value is conserved in time:

ddt⟨G^⟩=0.

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 G^ in the Heisenberg picture is given by

dG^dt=iℏ[H^,G^].

If [H^,G^]=0, then G^ 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.[10]

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.[11]

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

⟨f∣O^∣i⟩,

where O^ is the operator corresponding to the interaction (e.g. electric dipole operator), and |i⟩, |f⟩ 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:

  • Δl=±1
  • Δm=0,±1

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:

ψ(𝐫)→ψ(−𝐫).

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.[12]

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

Table of contents (185 articles)

Index

Full contents

9. Quantum optics and experiments (5) ↑ Back to index
14. Plasma and fusion physics (8) ↑ Back to index
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

References

  1. ↑ Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley. 
  2. ↑ Wigner, Eugene (1959). Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Academic Press. 
  3. ↑ Ballentine, Leslie E. (1998). Quantum Mechanics: A Modern Development. World Scientific. 
  4. ↑ Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley. 
  5. ↑ Ballentine, Leslie E. (1998). Quantum Mechanics: A Modern Development. World Scientific. 
  6. ↑ Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley. 
  7. ↑ Ballentine, Leslie E. (1998). Quantum Mechanics: A Modern Development. World Scientific. 
  8. ↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. 
  9. ↑ Ballentine, Leslie E. (1998). Quantum Mechanics: A Modern Development. World Scientific. 
  10. ↑ Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley. 
  11. ↑ Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley. 
  12. ↑ Hamermesh, Morton (1962). Group Theory and Its Application to Physical Problems. Addison-Wesley. 
Author: Harold Foppele

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