3. The Hamiltonian of a particle of mass m and charge q in a static magnetic field may be written 2 where πί Pi-qAi(x). We shall assume that the magnetic field B is uniform, so that AiEijkBjxk is a s...
3. A particle with mass m and charge q moves in a uniform magnetic filed of magnitude B that is oriented along the z axis. (a) Neglecting the effects of spin and using the so-called Landau gauge with the vector po- tential given by A = (-By,0,0), show that the Hamiltonian may be written as À = 2m 2 ++øp +29BD2y +(, 2] (1) с (b) Because Pa and Êz commute with Ĥ, the time-independent Schrödinger equation for (x, y,...
A spin-1 particle interacts with an external magnetic field B = B. The interaction Hamiltonian for the system is H = gB-S, where S-Si + Sỳ + SE is the spin operator. (Ignore all degrees of freedom other than spin.) (a) Find the spin matrices in the basis of the S. S eigenstates, |s, m)) . (Hint: Use the ladder operators, S -S, iS, and S_-S-iS,, and show first that s_ | 1,0-ћ /2 | 1.-1)) . Then use these...
3. (10 points) The Hamiltonian for a particle moving in a vertical uniform gravitational field g is 2 Hmgq where q is the altitude above the ground. We want to find any canonical transformation from old variables (q,p) to new variables (Q,P) which provides a cyclic coordinate. To do this, define new variables as where a, b are constants. a) Determine any combination of constants a and b, which provides a canonical transformation b) Find the type 1 generating function,...