#import "template.typ": *

// Take a look at the file `template.typ` in the file panel
// to customize this template and discover how it works.
#show: project.with(
  title: "Symmetry Groups and Degeneracy",
  authors: (
    "<NAME>",
  ),
  date: "10 Août, 2024",
)

#set heading(numbering: "1.1.")

== Symmetry Groups and Degeneracy in Advanced Quantum Mechanics
<symmetry-groups-and-degeneracy-in-advanced-quantum-mechanics>
=== 1. Rotational Symmetry:
<rotational-symmetry>
- Rotational symmetry is a fundamental concept in quantum mechanics that
  arises from the isotropy of space, i.e., the laws of physics remain
  unchanged under rotations.
- Rotational symmetry is closely related to angular momentum, which
  quantifies the rotational motion of a system.
- Angular momentum operators, denoted by $J$, are defined by the
  commutation relations
  $[J_i , J_j] = i planck.reduce epsilon.alt_(i j k) J_k$, where $i$,
  $j$, and $k$ are the Cartesian components and $epsilon.alt_(i j k)$ is
  the Levi-Civita symbol.

=== 2. Angular Momentum:
<angular-momentum>
- Angular momentum is a conserved quantity in quantum mechanics and
  plays a crucial role in the description of particles and their
  interactions.
- In quantum mechanics, angular momentum is quantized, meaning it can
  only take certain discrete values.
- The total angular momentum operator, denoted by $J$, is the sum of the
  orbital angular momentum operator ($L$) and the spin angular momentum
  operator ($S$).

=== 3. Commutator:
<commutator>
- The commutator between two operators $A$ and $B$, denoted by
  $[A , B]$, is defined as $[A , B] = A B - B A$.
- The commutator quantifies the non-commutativity of operators and
  determines the order in which they act.
- In quantum mechanics, the commutator between two observables
  represents the uncertainty relation between them.

=== 4. Degeneracy:
<degeneracy>
- Degeneracy refers to the phenomenon where multiple states of a quantum
  system have the same energy.
- Degeneracy arises due to the existence of symmetry in the system,
  leading to multiple states with indistinguishable energies.
- Degenerate states form a subspace within the larger Hilbert space
  associated with the system.
- Degeneracy can have profound implications for the behavior and
  properties of quantum systems.

=== 5. Symmetry Generators:
<symmetry-generators>
- Symmetry generators are operators that generate symmetry
  transformations on a quantum system.
- For rotational symmetry, the symmetry generators are the components of
  angular momentum operators ($J_x , J_y , J_z$).
- Symmetry generators act on quantum states to produce transformed
  states that belong to the same symmetry class.

=== 6. Symmetry Groups:
<symmetry-groups>
- Symmetry groups are mathematical structures that describe the
  collection of all symmetry transformations that leave a physical
  system invariant.
- In quantum mechanics, symmetry groups play a fundamental role in
  determining the properties and behaviors of quantum systems.
- The symmetry group associated with rotational symmetry is the special
  unitary group in three dimensions, denoted by $S U (2)$.
- Symmetry groups provide a powerful framework for understanding the
  degeneracy and symmetry-related properties of quantum systems.

=== 7. Lie Algebra:
<lie-algebra>
- The Lie algebra of a symmetry group is a vector space that captures
  the algebraic properties of the group’s generators.
- In the case of rotational symmetry, the Lie algebra associated with
  the special unitary group in three dimensions ($S U (2)$) is the
  algebra of Pauli spin matrices.
- The Lie algebra provides a mathematical foundation for studying the
  symmetry transformations and their algebraic relations.

=== 8. Raising and Lowering Operators:
<raising-and-lowering-operators>
- Raising and lowering operators are operators that allow the generation
  of new states with different angular momentum quantum numbers from a
  given state.
- In the context of angular momentum, raising operators ($J_(+)$)
  increase the angular momentum quantum number, while lowering operators
  ($J_(-)$) decrease it.
- The action of raising and lowering operators on angular momentum
  eigenstates leads to the construction

of multiplets and facilitates the understanding of degeneracy.

#link("Maths.pdf")[Maths]
