Lecture Notes on Spherical Tensor Operators

Introduction

• Definition of spherical tensor operator (T_{k}^{(q)}) with rank (K): a collection of (2K + 1) operators indexed by (q), transforming under rotations like spherical harmonics.
• Overview of the lecture structure:
  • Part 1: Tensor operators
  • Part 2: Spherical basis
  • Part 3: Calculation of components from Cartesian components._

Part 1: Tensor Operators

• Characterization of a Tensor: Transform behavior under rotations.
• Types of Rotations:
  • Active Rotations: Rotation of state vector (S_y) using unitary operator (D).
  • Passive Rotations: Operators are rotated in the opposite direction.
• Mathematical Observations:
  • Expectation values are the same for both rotation types:
    • (D^{ ext{dagger}} O D) for active rotation.
    • (D O D^{ ext{dagger}}) for passive rotation.
• Vector Transformation: Classical vector (V_i) transforms as (R_{ij} V_j). Quantum mechanics holds similar relations:
  • (D^{ ext{dagger}} V_i D = R_{ij} V_j).
• Cartesian Tensor Operators: Generalization of vector transformation using multiples of the rotation matrix (R).

Part 2: Spherical Basis

• Definition: Spherical basis vectors transform under rotations in the same way as spherical harmonics.
• Basis Vectors: Spherical harmonics (Y_{1}^{1}, Y_{1}^{0}, Y_{1}^{-1}) proportional to Cartesian coordinates.
• Orthonormality Condition: Used to find proportionality factors (\alpha, \beta, \gamma).
• Contravariant and Covariant Components:
  • Spherical components defined via Cartesian components:
    • Contravariant: Index upstairs.
    • Covariant: Index downstairs (dual basis).
• Useful Relations in Spherical Basis:
  • Cross-product and inner product relations defined._

Spherical Tensors

• Transformation Behavior: Spherical tensors transform like spherical harmonics.
• Eigenstate Definition: Spherical harmonics are eigenstates of angular momentum operator projected onto coordinate basis.
• Transformation Equations:
  • Using completeness relation and rotation matrices for spherical harmonics under rotation.
• Definition of Spherical Tensor Operator: A set of (2K + 1) operators transforming as spherical harmonics do.
• Commutator Relations: Derived from unitary rotation operator; can be simplified using ladder operators.
• Hermitian Adjoint: Relations for Hermitian conjugate defined.

Part 3: Examples

• Rank 0 Tensor: Scalar is the same in both bases.
• Rank 1 Tensor (Vector): Covariant components are spherical tensor operators, transforming similarly to basis vectors.
• Rank 2 Tensor:
  • Total of 9 components; however, rank 2 tensor is reducible.
  • Use direct products of rank 1 tensors for calculation.
• Clebsch-Gordon Coefficients: Used to couple spherical tensor operators, providing a way to combine states of definite angular momentum.
• Example Calculation of (T_{1}^{0}) component:
  • Derived from covariant components and leads to final result involving Cartesian tensor operators.
• Higher-Rank Tensors: Can be calculated similarly, using the technique established for lower ranks._

Conclusion

• Summary of how to switch from Cartesian to spherical tensor components.
• Emphasis on use of Clebsch-Gordon coefficients for determining ranks and components.