NRQM10_2

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

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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 = ( vectorσ · 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 φ ( 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 “g”–factor and vector S ≡ 1 2 planckover2pi1 vectorσ 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 ρ = | ψ ( vector r ) | 2 . Both quantities are related through the continuity equation....
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NRQM10_2 - Physics 216 Problem Set 2 Spring 2010 DUE...

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