D19c MATHEMATICAL TRIPOS PART II Lent 2016 Prof. N.Dorey n.dorey@damtp.cam.ac.uk Applications of Quantum Mechanics - Examples III (Starred questions are optional) 1. The Schrödinger equation for a particle in a background electromagnetic field is ih at 2m ?2 (+ + 2 A ) 4 - cov - MB . sv . Under a gauge transformation, the potentials A and ø transform to A = A + Vf and ø = 0 - . Show that with a suitable transformation of v, the Schrödinger equation transforms into itself, i.e. is gauge invariant. Show that the probability density |v|2 and the expectation value of -ihV + eA are gauge invariant. 2. Consider a particle of charge -e and mass m moving in the x-y plane under the influence of a constant, uniform magnetic field B = (0, 0, B). Write down the classical Hamiltonian H in the gauge A = B(-y/2, x/2, 0) and use Hamilton's equations, *= >p' OH OH . to show that that classical orbits are circles. Show that the coordinates (x0, y0, 0) of the center of any circlular orbit are related to the position and momentum of the particle as, yo = y Px 2 TO = 2 - CB' - eB Copyright @ 2016 University of Cambridge. Not to be quoted or reproduced without permission. Show that the quantum operators corresponding to the classical quantities x0 and y0 are individually conserved but cannot be measured simultaneously. 3. Show that by a suitable rescaling, the Hamiltonian for an electron moving essentially in the x, y plane in a time-independent magnetic field of strength B = (0, 0, B(y)) (i.e. depending on y but not x) can be reduced to ? 2 ? Jy 2 + b(y)03, where da = b(y). Use the Pauli matrix algebra to show that H can be written as Q2 where Q = i (or - ia (y) 02 - 1 001 . Check that Q is hermitian and use this to deduce that any zero energy state is annihilated by Q. Show that if there are spin down, zero energy states, they are of the form 0 (e-(y) f(x - ig) where dc = a(y). Show that if b(y) > bo > 0, then c(y) grows at least quadratically as [y] > >. 4. A quantized particle of mass m moves in one dimension under the influence of a potential V(x) = ~ 8(x). Show that there is a bound state with energy -h2>2/2m. Now suppose the potential becomes 00 V(x) = - m ~ 1 =- 00 8(x - la) . 1
?2 Show that in this case there is (for A not too small) a negative energy band - 2m KESES - ?2 .2 2m Ký where KL is determined by 1 = cosh KLa - À sinh KLa and Ku satisfies a similar equation. Show that when a -> o the band narrows down to the bound state energy. [Hint: Use the obvious basis of negative energy solutions in