Question

3. The Pauli Hamiltonian The Hamiltonian of an electron of mass m, charge q, spinn σ(ox, σ" σ Pauli matrices), placed in an electromagnetic field described by the vector poten tial A(r, /) and the scalar potential U(r, /). is written: qh 2m The last term represents the interaction between the spin magnetic moment _ơ and the magnetic field B(R, ) - Vx A(R. 1). also be written in the following form ("the Pauli Hamiltonian"): Show, using the properties of the Pauli matrices, that this Hamiltonian can

Answer 1

The Pauli vector identity (sigma^bar middot A^bar) = A^bar middot B^bar + I sigma^bar middot A^bar times B^bar if A^bar = B^bar = pi^bar (sigma^bar middot pi^bar) (sigma^bar middot pi^bar) = pi^bar middot pi^bar + I sigma^bar middot (pi^bar times pi^bar) (sigma^bar middot pi^bar)^2 = pi^2 + I sigma^bar middot (pi^bar times pi^bar) h = 1/2 m {sigma^bar middot [p^bar - q A^bar (R^bar, t)]}^2 + q u (R, t) Applying H for any Orbitrary wave function psi H psi = {1/2 m{[p^bar - q A^bar (R^bar, t)]^2 + I sigma middot {[p^bar - q A^bar (R^bar - q A^bar)} + q u (R, t)} psi [p^bar - q A^bar (R^bar, t)] times [p^bar - q A^bar (R^bar, t)] psi = [p^bar times p^bar - q p^rightarrow times A rightarrow (R^rightarrow, t) - q A^bar (R, t) times p^bar + q^2 A^bar times A^bar] psi = 0 - q p^bar times A^bar psi - q A^bar times p^bar psi + 0 = -[q (p^bar times A^bar) psi + q (p^bar psi) times A^bar] + q A^bar times p^bar psi