11.2 Lie algebras in quantum mechanics and angular momentum

Lie algebras in quantum mechanics reveal the deep connection between symmetries and observables. They provide a powerful framework for understanding angular momentum, a fundamental concept in quantum physics.

The su(2) Lie algebra, formed by angular momentum operators, is crucial for describing spin and rotational symmetry. It showcases how Lie algebras help us grasp quantum systems' behavior and conservation laws.

Lie Algebras and Quantum Observables

Representing Observables as Hermitian Operators

Top images from around the web for Representing Observables as Hermitian Operators

• 

  Angular Momentum and Its Conservation | Physics View original

  Is this image relevant?

• 

  Lie Algebras [The Physics Travel Guide] View original

  Is this image relevant?

• 

  quantum mechanics - Calculating values related to angular momentum and then their uncertainties ... View original

  Is this image relevant?

• 

  Angular Momentum and Its Conservation | Physics View original

  Is this image relevant?

• 

  Lie Algebras [The Physics Travel Guide] View original

  Is this image relevant?

1 of 3

Top images from around the web for Representing Observables as Hermitian Operators

• 

  Angular Momentum and Its Conservation | Physics View original

  Is this image relevant?

• 

  Lie Algebras [The Physics Travel Guide] View original

  Is this image relevant?

• 

  quantum mechanics - Calculating values related to angular momentum and then their uncertainties ... View original

  Is this image relevant?

• 

  Angular Momentum and Its Conservation | Physics View original

  Is this image relevant?

• 

  Lie Algebras [The Physics Travel Guide] View original

  Is this image relevant?

1 of 3

• In quantum mechanics, observables are represented by Hermitian operators acting on a Hilbert space
  • Hermitian operators ensure real eigenvalues, which correspond to measurable quantities
  • Examples of observables include position, momentum, energy, and angular momentum
• The generators of the Lie algebra are the basis elements of the space of observables
  • They satisfy specific commutation relations that characterize the Lie algebra
  • For example, the angular momentum operators (Jx, Jy, Jz) are generators of the su(2) Lie algebra

Commutator and Non-Commutativity

• The commutator of two operators A and B is defined as [A, B] = AB - BA
  • It captures the non-commutativity of quantum mechanical observables
  • Non-commuting observables cannot be measured simultaneously with arbitrary precision (Heisenberg uncertainty principle)
• The commutation relations of quantum mechanical observables define the structure of the corresponding Lie algebra
  • The structure constants of the Lie algebra are determined by these commutation relations
  • For example, the commutation relations for angular momentum operators are [Jx, Jy] = iħJz, [Jy, Jz] = iħJx, and [Jz, Jx] = iħJy, where ħ is the reduced Planck's constant

Symmetries and Conservation Laws

• The Lie algebra of observables determines the symmetries and conservation laws of the quantum mechanical system
  • Observables that commute with the Hamiltonian are conserved quantities
  • For example, if the Hamiltonian is rotationally invariant, the angular momentum operators commute with the Hamiltonian, and angular momentum is conserved
• Casimir operators, which commute with all generators of the Lie algebra, are related to the invariants of the system under symmetry transformations
  • The eigenvalues of Casimir operators label the irreducible representations of the Lie algebra and provide a complete set of quantum numbers characterizing the system

Lie Algebras for Angular Momentum

Angular Momentum Operators and su(2) Lie Algebra

• The angular momentum operators (Jx, Jy, Jz) form a Lie algebra known as the su(2) Lie algebra or the angular momentum algebra
  • They satisfy the commutation relations [Jx, Jy] = iħJz, [Jy, Jz] = iħJx, and [Jz, Jx] = iħJy, where ħ is the reduced Planck's constant
  • These commutation relations are isomorphic to the Lie algebra of the special unitary group SU(2), allowing the use of SU(2) representation theory in studying angular momentum
• The total angular momentum operator, J^2 = Jx^2 + Jy^2 + Jz^2, commutes with all the individual angular momentum operators: [J^2, Jx] = [J^2, Jy] = [J^2, Jz] = 0
  • This commutation relation implies that the total angular momentum and one of its components (usually chosen to be Jz) can be measured simultaneously

Raising and Lowering Operators

• The raising and lowering operators, J+ and J-, are defined as linear combinations of the angular momentum operators: J+ = Jx + iJy and J- = Jx - iJy
  • They satisfy the commutation relations [Jz, J+] = ħJ+ and [Jz, J-] = -ħJ-
  • These operators are used to construct the ladder of angular momentum states within an irreducible representation
• The action of the raising and lowering operators on the basis states is given by J+|j, m⟩ = ħ√(j(j+1)-m(m+1)) |j, m+1⟩ and J-|j, m⟩ = ħ√(j(j+1)-m(m-1)) |j, m-1⟩
  • These relations allow the construction of the representation matrix elements and the study of transitions between different angular momentum states

Representation Theory in Quantum Mechanics

Irreducible Representations and Quantum Numbers

• Representation theory of Lie algebras plays a crucial role in understanding the quantum mechanical properties of a system with a given symmetry
• An irreducible representation of a Lie algebra is a vector space on which the Lie algebra acts, and which cannot be decomposed into smaller invariant subspaces
  • Each irreducible representation is characterized by its dimension and a set of quantum numbers
  • For the angular momentum algebra, the irreducible representations are labeled by the total angular momentum quantum number j, which can take integer or half-integer values, and the dimension of the representation is 2j+1

Basis States and Matrix Elements

• The basis states of an irreducible representation are eigenstates of the Casimir operator (J^2) and one of the generators (usually Jz)
  • These eigenstates are denoted as |j, m⟩, where m is the eigenvalue of Jz and ranges from -j to j in integer steps
  • The action of the raising and lowering operators on these basis states allows the construction of the representation matrix elements
• The Wigner-Eckart theorem simplifies the calculation of matrix elements of spherical tensor operators between states in different irreducible representations
  • It separates the matrix element into a geometric factor (Clebsch-Gordan coefficient) and a reduced matrix element that depends on the specific tensor operator
  • This theorem is particularly useful in the study of selection rules and transition probabilities between different angular momentum states

Casimir Operators and Eigenvalues

Properties of Casimir Operators

• Casimir operators are special elements of the universal enveloping algebra of a Lie algebra that commute with all the generators of the Lie algebra
  • In the case of the angular momentum algebra, the Casimir operator is the total angular momentum operator J^2 = Jx^2 + Jy^2 + Jz^2
  • The eigenvalues of the Casimir operator label the irreducible representations of the Lie algebra
• The eigenvalues of the Casimir operator are related to the invariants of the system under the symmetry transformations generated by the Lie algebra
  • For the angular momentum algebra, the eigenvalues of J^2 are j(j+1)ħ^2, where j is the total angular momentum quantum number
  • These eigenvalues are invariant under rotations in three-dimensional space

Physical Interpretation and Applications

• The Casimir operator and its eigenvalues play a crucial role in the classification of states and the determination of selection rules for transitions between different states in a quantum mechanical system
  • The eigenvalues of the Casimir operator provide a complete set of quantum numbers characterizing the irreducible representations of the Lie algebra
  • Selection rules for transitions between states are determined by the matrix elements of relevant operators, which are constrained by the symmetries of the system and the corresponding Lie algebra
• In more general Lie algebras, there may be multiple Casimir operators, each with its own physical interpretation
  • For example, in the case of the su(3) Lie algebra, which is relevant for the description of quarks and the strong interaction, there are two Casimir operators related to the electric charge and the color charge of the particles
  • The eigenvalues of these Casimir operators provide a complete set of quantum numbers characterizing the irreducible representations of the su(3) Lie algebra and the properties of the corresponding particles