NRQM10_2

NRQM10_2 - Physics 216 Problem Set 2 Spring 2010 DUE:...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Physics 216 Problem Set 2 Spring 2010 DUE: TUESDAY, APRIL 27, 2010 1. We wish to determine the correct form of the Schrodinger equation for a spin- 1 2 particle in an external electromagnetic field. The wavefunction for a spin- 1 2 particle is a two-component spinor wave function, where each component is an independent function of vectorx and t . Thus, the Hamiltonian must be a 2 2 matrix, whose elements are also operators on the Hilbert space of square integrable functions. (a) First consider a free spin- 1 2 particle. We demand that H is rotationally invari- ant. Two candidate Hamiltonians are: ( i ) H 1 = vector p 2 2 m I , ( ii ) H 2 = ( vector vector p ) 2 2 m . where I is the 2 2 identity matrix. Show that H 1 and H 2 are in fact identical. (b) Consider now a spin- 1 2 particle with charge q in an external electromagnetic field. Using the principle of minimal coupling, deduce the Hamiltonians corresponding to (i) and (ii) above, which include the dependence on the scalar potential ( vectorx , t ) and the vector potential vector A ( vectorx , t ) [do not choose a specific gauge]. Show that the two resulting Hamiltonians differ. In particular, in case (ii) above, a term in the Hamiltonian arises of the form ge 2 mc vector S vector B , where g is called the gfactor and vector S 1 2 planckover2pi1 vector is the spin operator. Using the fact that the electron has q = e , what value of g is predicted by this approach? 2. The current density vector j (in the absence of an external electromagnetic field) was introduced by Shankar in eq. (5.3.8) on p. 166. The corresponding number density is given by = | ( vector r ) | 2 . Both quantities are related through the continuity equation....
View Full Document

Page1 / 3

NRQM10_2 - Physics 216 Problem Set 2 Spring 2010 DUE:...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online