A hydrogen atom is placed in a (time-dependent) electric field E = E(t)k. Calculate all four matrix elements H′ij of the perturbation H' = eEz between the ground state (n = 1) and the (quadruply degenerate) first excited states (n = 2). Also show that H'ij = 0 for all five states. Note: There is only one integral to be done here, if you exploit oddness with respect to z; only one of the n = 2 states is "accessible" from the ground state by a perturbation of this form, and therefore the system functions as a two-state configuration—assuming transitions to higher excited states can be ignored.
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