19 - Physics 6210/Spring 2007/Lecture 19 Lecture 19...

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Physics 6210/Spring 2007/Lecture 19 Lecture 19 Relevant sections in text: § 2.6 Charged particle in an electromagnetic field We now turn to another extremely important example of quantum dynamics. Let us describe a non-relativistic particle with mass m and electric charge q moving in a given electromagnetic field. This system has obvious physical significance. We use the same position and momentum operators (in the Schr¨ odinger picture) ~ X and ~ P (although there is a subtlety concerning the meaning of momentum, to be mentioned later). To describe the electromagnetic field we need to use the electromagnetic scalar and vector potentials φ ( ~x, t ) , ~ A ( ~x, t ). They are related to the familiar electric and magnetic fields ( ~ E, ~ B ) by ~ E = -∇ φ - 1 c ~ A ∂t , ~ B = ∇ × ~ A. The dynamics of a particle with mass m and charge q is determined by the Hamiltonian H = 1 2 m ± ~ P - q c ~ A ( ~ X, t )) ² 2 + ( ~ X, t ) . This Hamiltonian takes the same form as the classical expression in Hamiltonian mechanics. We can see that this is a reasonable form for H by computing the Heisenberg equations of motion, and seeing that they are equivalent to the Lorentz force law, which we shall now demonstrate. For simplicity we assume that the potentials are time independent, so that the Heisen- berg and Schr¨odinger picture Hamiltonians are the same, taking the form H = 1 2 m ± ~ P - q c ~ A ( ~ X )) ² 2 + ( ~ X ) . For the positions we get (exercise) d dt ~ X ( t ) = 1 i ¯ h [ ~ X ( t ) , H ] = 1 m { ~ P ( t ) - q c ~ A ( ~ X ( t )) } . We see that (just as in classical mechanics) the momentum – defined as the generator of translations – is not necessarily given by the mass times the velocity, but rather ~ P ( t ) = m d ~ X ( t ) dt + q c ~ A ( ~ X ( t )) . As in classical mechanics we sometimes call ~ P the canonical momentum , to distinguish it from the mechanical momentum ~ Π = m d ~ X ( t ) dt = ~ P - q c ~ A ( ~ X ( t )) 1
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Physics 6210/Spring 2007/Lecture 19 Note that the mechanical momentum has a direct physical meaning, while the canonical momentum depends upon the non-unique form of the potentials. We will discuss this in detail soon.
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This note was uploaded on 02/18/2012 for the course PHYSICS 6210 taught by Professor M during the Spring '07 term at AIU Online.

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19 - Physics 6210/Spring 2007/Lecture 19 Lecture 19...

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