A proton (charge

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch Charged particles in electromagnetic fields

Problem

A proton (charge +e+e, mass mpm_p) moves in a region with a uniform magnetic field B⃗=Bz^\vec B = B\hat z and no electric field. Using the Landau gauge A⃗=(0,Bx,0)\vec A = (0, Bx, 0), write the minimally coupled Hamiltonian H^=12mp(p⃗^−eA⃗)2\hat H = \frac{1}{2m_p}(\hat{\vec p} - e\vec A)^2 and expand the square into an explicit operator expression in p^x,p^y,p^z,x^\hat p_x,\hat p_y,\hat p_z,\hat x, separating the field-free kinetic term, the term linear in BB, and the term quadratic in BB. Then show that p^y\hat p_y commutes with H^\hat H, replace it by its eigenvalue ℏk\hbar k, and read off the centre x0x_0 and the angular frequency of the oscillator that now governs the motion in xx.

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