Understanding Classical Mechanics and Fields in Physics

Explore the concepts of radiation, matter, fields, Hamiltonian, and more in classical mechanics with a focus on velocity-dependent potentials and conservative systems. Dive into the detailed discussions on gauge theory, Coulomb gauge, magnetic fields, and Hamiltonian operators within the framework of physics.

• Classical Mechanics
• Fields
• Radiation
• Hamiltonian
• Quantum Physics

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1. 1.RADIATION J. Planelles

11. 2.Matter and Field J. Planelles

12. Classical Mechanics: an overview Newton s Law Conservative systems Lagrange equation

13. Velocity-dependent potentials Time-independent field No magnetic monopoles

14. Velocity-dependent potentials: cont. Define:

15. Velocity-dependent potentials: cont. kinematic momentum: canonical momentum: Hamiltonian:

16. Conservative systems: V(x,y,z) & L = T - V canonical momentum: kinematic momentum: Hamiltonian:

17. Gauge ; We may select : Coulomb Gauge :

18. Hamiltonian (coulomb gauge) ;

19. Magnetic field: summary No magnetic monopoles: vector potential velocity-dependent potential: No conservative field: Lagrangian: kinematic momentum Canonical momentum: Hamiltonian: Coulomb gauge: Hamiltonian operator:

20. 3. Electron-photon interaction (selection rules and resonanace) J. Planelles

21. Hamiltonian of a charge in an electromagnetic field Set of charges {qi}:

22. Time-dependent perturbation theory

23. Time-dependent perturbation theory (cont.) Transition probability: Taylor expansion:

24. Time-dependent perturbation theory (cont.)

25. Time-dependent perturbation theory (cont.) resonance: resonance: selection rule:

26. Time-dependent perturbation theory (cont.) First (and larger) perturbation term: (with q = - e, m of electron) Integral involved: Single photon transition Spontaneous emission

27. Time-dependent perturbation theory (cont.) After some algebra:

28. Time-dependent perturbation theory (cont.) Second (smaller) perturbation term: (q = - e, m of electron) terms : two-photon transitions two-photon transitions: weak ( use LASER). e.g. Raman