T18 – Rotation operator

12 Novembro 2024, 11:00 • Ana Maria Formigal de Arriaga

• rotation operator in E (quantum states space).
• rotation generators – cartesian components of the total angular momentum; commutation laws
• consequences of the application of the rotation operator in an eigenstate of S² and S_z
• invariance under rotations and commutation relations of the Hamiltonian with the angular momentum; conservation of the angular momentum

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TP 8 - Coupling of Angular Momenta

7 Novembro 2024, 12:00 • Ana Maria Formigal de Arriaga

Set 3

Coupling of Angular Momenta

• Reading CG coefficients from a table
• Resolution of problem: 10a)
• Suggestions for resolution of problems: 8 and 9

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T17 – Rotation operator

7 Novembro 2024, 11:00 • Ana Maria Formigal de Arriaga

Transformed state and transformed coordinates under a rotation

Parameters of the rotation.

Rotation operator in R₃ (coordinates space).

Rotation operator in E (quantum states space)

• for infinitesimal rotations about the z-axis
• for finite rotations about the z-axis
• for finite rotations about an arbitrary axis

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T16 - Coupling of angular momenta

5 Novembro 2024, 11:00 • Ana Maria Formigal de Arriaga

Hydrogen atom with the Coulomb potential

• Symmetries and degeneracies

Supplementary terms in the Hamiltonian of the Hydrogen atom: fine and hyperfine structures

Hydrogen atom fine structure

• mass-velocity and Darwin terms: brief reference
• spin-orbit term : change of basis, operator action and calculation of the energy in 1st order perturbation theory
• split of the energy levels: reducing energy degeneracy

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TP 7 - Coupling of Angular Momenta

31 Outubro 2024, 12:00 • Ana Maria Formigal de Arriaga

Set 3 - Coupling of Angular Momenta

Resolution of problem: problems 2 and 7

Suggestions for resolution of problems: problems 3, 4, 5 and 6

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