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Unformatted text preview: Physics 216 Solutions to Problem Set 2 Spring 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. The Pauli matrices, σ i satisfy the identity, σ i σ j = δ ij I + iǫ ijk σ k , (1) where there is an implicit sum over the repeated index k . Note that if we multiply eq. (1) by A i B j and sum over i and j , we recover eq. (14.3.39) on p. 382 of Shankar. Moreover, if A = B = p , then the resulting identity is ( vectorσ · vector p ) 2 = vector p 2 I , since ǫ ijk p i p j = 0 follows from the fact that ǫ ijk is antisymmetric under the interchange of i and j whereas p i p j is symmetric under the interchange of i and j . Hence, if follows that the two candidate Hamiltonians H 1 and H 2 are 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? Applying the principle of minimal substitution to H 1 , we obtain a spin-independent Hamiltonian, H = 1 2 m parenleftBigg vector p − q vector A c parenrightBigg 2 I + qφ I , (2) 1 for a particle in an external electromagnetic field that derives from a scalar potential φ and a vector potential vector A . In contrast, if we apply the principle of minimal substitution to H 2 , we obtain H = 1 2 m vectorσ · parenleftBigg vector p − q vector A c parenrightBigg vectorσ · parenleftBigg vector p − q vector A c parenrightBigg + qφ I . We can simplify the first term above by writing: vectorσ · parenleftBigg vector p − q vector A c parenrightBigg vectorσ · parenleftBigg vector p − q vector A c parenrightBigg = summationdisplay ij σ i σ j parenleftbigg p i − qA i c parenrightbiggparenleftbigg p j − qA j c parenrightbigg = summationdisplay ijk ( δ ij I + iǫ ijk σ k ) parenleftbigg p i −...
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This document was uploaded on 10/31/2011.
- Spring '09
- Schrodinger Equation