ParticleMagneticField - Charged Particle in a Magnetic...

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Charged Particle in a Magnetic Field Michael Fowler 1/16/08 Introduction Classically, the force on a charged particle in electric and magnetic fields is given by the Lorentz force law: vB FqE c ⎛⎞ × =+ ⎜⎟ ⎝⎠ G G G G This velocity-dependent force is quite different from the conservative forces from potentials that we have dealt with so far, and the recipe for going from classical to quantum mechanics— replacing momenta with the appropriate derivative operators—has to be carried out with more care. We begin by demonstrating how the Lorentz force law arises classically in the Lagrangian and Hamiltonian formulations. Laws of Classical Mechanics Recall first (or look it up in Shankar, Chapter 2) that the Principle of Least Action leads to the Euler-Lagrange equations for the Lagrangian L : (,) 0, , being coordinates and velocities. ii Lq q d qq dt q q ∂∂ −= ±± ± ± The canonical momentum p i is defined by the equation i i L p q = ± and the Hamiltonian is defined by performing a Legendre transformation of the Lagrangian: (, ) i i Hq p pq Lqq =− It is straightforward to check that the equations of motion can be written: , HH qp pq == These are known as Hamilton’s Equations . Note that if the Hamiltonian is independent of a particular coordinate q i , the corresponding momentum p i remains constant. (Such a coordinate is termed cyclic , because the most common example is an angular coordinate in a spherically symmetric Hamiltonian, where angular momentum remains constant.)
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2 For the conservative forces we have been considering so far, L = T V , H = T + V , with T the kinetic energy, V the potential energy. Poisson Brackets Any dynamical variable f in the system is some function of the q i ’s and p i ’s and (assuming it does not depend explicitly on time) its development is given by: (, ) {, } . ii i i i i df f f H f H fqp q p fH d t q p qp pq ∂∂ =+ = = ∂∂ ∂∂ ±± The curly brackets are called Poisson Brackets , and are defined for any dynamical variables as: {,} . A BA B AB =− We have shown from Hamilton’s equations that for any variable . f = ± It is easy to check that for the coordinates and canonical momenta, {, } 0 {, } , .
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This note was uploaded on 12/07/2011 for the course PHYSICS 751 taught by Professor Michaelfowler during the Fall '07 term at UVA.

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ParticleMagneticField - Charged Particle in a Magnetic...

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