MATHEMATICAL TRIPOS Part II PAPER 3 Before you begin read ...

1934D Applications of Quantum MechanicsWrite down the classical Hamiltonian for a particle of mass m, electric charge −eand momentum p moving in the background of an electromagnetic field with vector andscalar potentials A(x, t) and φ(x, t).Consider the case of a constant uniform magnetic field, B = (0, 0, B) and E = 0.Working in the gauge with A = (−By, 0, 0) and φ = 0, show that Hamilton’s equations,ẋ = ∂H∂p ,ṗ = −∂H ∂x ,admit solutions corresponding to circular motion in the x-y plane with angular frequencyω B = eB/m.Show that, in the same gauge, the coordinates (x 0 , y 0 , 0) of the centre of the circleare related to the instantaneous position x = (x, y, z) and momentum p = (p x , p y , p z ) ofthe particle byx 0 = x − p yeB ,y 0 = p xeB . (1)Write down the quantum Hamiltonian Ĥ for the system. In the case of a uniformconstant magnetic field discussed above, find the allowed energy levels. Working inthe gauge specified above, write down quantum operators corresponding to the classicalquantities x 0 and y 0 defined in (1) above and show that they are conserved.[In this question you may use without derivation any facts relating to the energyspectrum of the quantum harmonic oscillator provided they are stated clearly.]Part II, Paper 3[TURN OVER