Problem 10.1: Hamiltonian with vector potential. (20 points) a) Show that the Lorentz force law, F...

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Problem 10.1: Hamiltonian with vector potential. (20 points) a) Show that the Lorentz force law, F = q + qu B, can be written as mra-q +q. ra ra dt (1) (2) where a = x, y, z is one of the components of the vector, and r, ry and r, just mean x, y and z as usual. A helpful vector identity is [ ( )]=. (V)A. ra (3) You will also need to think carefully about the difference between the partial derivative 9/t and the total derivative d/dt. b) The Hamiltonian for a charged particle in a vector potential and a scalar potential up is H= (p-q) +94. 2m Show that Hamilton's first set of equations, (4) reproduce the relationship dra dt p=m+q. (5) (6) Notice in the presence of the vector potential the "momentum" used by the Hamiltonian formalism, called the canonical momentum, is not just the mass times the velocity (called the kinetic or kinematic momentum). It is the canonical momentum that is conjugate to position in the sense of the Hamiltonian formalism, and in the quantum theory, the canonical momentum which has the standard commutation relation with position [ra, pb] = iab c) Using the result of part b), show that Hamilton's other set of equations, dpa dt ra' 1 (7) reproduce the Lorentz force law as you expressed it in part a). Recall that in the Hamiltonian formalism, it is p, that is kept constant when one differentiates with respect to ra, and vice versa. Problem 10.1: Hamiltonian with vector potential. (20 points) a) Show that the Lorentz force law, F = q + qu B, can be written as mra-q +q. ra ra dt (1) (2) where a = x, y, z is one of the components of the vector, and r, ry and r, just mean x, y and z as usual. A helpful vector identity is [ ( )]=. (V)A. ra (3) You will also need to think carefully about the difference between the partial derivative 9/t and the total derivative d/dt. b) The Hamiltonian for a charged particle in a vector potential and a scalar potential up is H= (p-q) +94. 2m Show that Hamilton's first set of equations, (4) reproduce the relationship dra dt p=m+q. (5) (6) Notice in the presence of the vector potential the "momentum" used by the Hamiltonian formalism, called the canonical momentum, is not just the mass times the velocity (called the kinetic or kinematic momentum). It is the canonical momentum that is conjugate to position in the sense of the Hamiltonian formalism, and in the quantum theory, the canonical momentum which has the standard commutation relation with position [ra, pb] = iab c) Using the result of part b), show that Hamilton's other set of equations, dpa dt ra' 1 (7) reproduce the Lorentz force law as you expressed it in part a). Recall that in the Hamiltonian formalism, it is p, that is kept constant when one differentiates with respect to ra, and vice versa.