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# hw6 - PHYS 507 Homework VI(Fall'05 Assigned Monday Due...

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PHYS 507 Homework VI (Fall ’05) Assigned: November 23, 2005, Monday. Due: November 30, 2005, Wednesday, at 5:00 pm. 1. In the Hydrogen atom problem H = ~ p 2 / 2 m - Ze 2 /r , the particular gauge chosen is ~ A = 0 and φ = Ze/r . In another possible gauge ( ~ ˜ A, ˜ φ ), we might choose ˜ φ = 0. In that case the Hamiltonian becomes ˜ H = 1 2 m ~ p - ( - e ) c ~ ˜ A 2 , i.e., consisting of kinetic energy only. One price paid is ~ ˜ A = ~ ˜ A ( ~ r, t ) and hence ˜ H = ˜ H ( t ) is now time dependent. But this Hamiltonian still provides a valid description of the Hydrogen atom problem. (a) What is ~ ˜ A ( ~ r, t )? (b) What is the scalar function Λ used in the gauge transformation from ( ~ A, φ ) to ( ~ ˜ A, ˜ φ )? (c) If ψ is a position-space wavefunction for a particular state of the electron in the gauge ( ~ A, φ ), what is the position-space wavefunction ˜ ψ that needs to be used to describe the same state in the gauge ( ~ ˜ A, ˜ φ )? 2. Consider a particle in 1D having charge Q moving under a time and position inde- pendent electric field E . There are two frequently used Hamiltonians to describe the

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