Download Spins in Magnetic Fields: Precession and Resonance - PHY662, Spring 2004 and more Assignments Quantum Mechanics in PDF only on Docsity! PHY662, Spring 2004 Outline for Tues. Jan. 27, 2004 Spins in static and dynamic magnetic fields: precession and resonance 26th January 2004 1 Administration Today 1. Homework review: also a chance to review precession. Note µ-decay: using g − 2 to look for supersymmetry. 2. Conservation and symmetry. 3. Magnetic resonance. 2 Homework review [See KEY for HWK #2 - we will spend more time on problem #3]. One of the more interesting scientific uses for precession is to look at g− 2. This is not zero, as predicted by, say, Dirac theory, due to corrections from, if you like, from virtual particles. The brief existence of extra particles affects the response of the particle to external magnetic fields. For the electron, g = 2.002 319 304 374 ± 0.000 000 000 008. This is extremely accu- rate and in good agreement with theoretical calculations. [Source: Particle Data Group, 2003.] For the muon, g = 2.002 331 841 6 ± 0.000 000 001 2. [From the BNL experiment, January, 2004; see arxiv.org/hep-ex/0401008.] The experimental result for the muon is in mild disagreement with the theoretical pre- dictions (1 or 2 standard deviations). Does this indicate the existence of unknown particles, such as supersymmetry? We will have to wait and see. The theoretical pre- dictions are somewhat unsettled, actually. Also, disagreement at the several standard 1 deviation level often vanishes as more data is taken. But no one is taking data right now. 3 Conservation and Symmetry We didn’t have time last meeting to cover this properly. Here are the details, which I will review in class: • A symmetry is a (time-independent) unitary operator that does not change the equations of motion. • Let U be a symmetry operator. Then if |ψ〉 obeys the equation of motion, so does U |ψ〉: HU |ψ〉 = ih̄ ∂ ∂t U |ψ〉 . This implies that HU |ψ〉 = ih̄U ∂ ∂t |ψ〉 = UH|ψ〉 , as |ψ(t)〉 obeys the Schrodinger equation. Thus HU = UH . As U†U = 1 for unitary operators, this can be written as U†HU = I = UHU† . • Suppose there is a set of symmetries that are parametrized by a single contin- uous parameter: U(a). By composition, U(a)U(a) = U(2a). Examples are translations by a distance a or rotations by an angle a about a given axis. Write U(a) = exp(aG/ih̄). The operator G is said to be a generator (or sometimes charge) for the family of symmetries U(a). Note that ∂∂aU(a) = G ih̄U and ∂ ∂aU † = ∂∂ae aG†/(−ih̄) = ∂∂ae −aG/ih̄ = −Gih̄ U . Since U(a) is unitary, G must be Hermitian. One way to see this is (using [G,U ] = 0) by noting 0 = ∂ ∂a U†U = G† −ih̄ U†U + U† G ih̄ U = −G† +G . We can use the same trick, this time taking the derivative of H , rather than 1, 0 = ∂ ∂a H = ∂ ∂a (U†HU) = ( G† −ih̄ )U†HU + U†H( G ih̄ )U . This gives GH −HG = [G,H] = 0 . • Applying Ehrenfest’s theorem to G then gives ddt 〈G〉 = 〈[G,H]〉 = 0 whenever G is a generator of a symmetry U(a) for a given Hamiltonian. 2
