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ProblemSet5 - Chem 120A Problem Set 5 Due 1 When a hydrogen...

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Chem 120A Problem Set 5 Due 04/24/06 1. When a hydrogen atom is in a uniform electric field E = ε ˆ z , the atom experiences a perturbation V ( r ) = - μ · E = + e ε rcos θ (1) where the dipole moment operator of the electron is given by μ = - er . a) Evaluate the first-order correction to the ground state energy of the H atom as a function of the electric field strength ε . Discuss your answer b) For the n=2 state there are 4 degenerate states in the absence of the electric field. Construct the secular equation which needs to be solved to get the first order correction to the energy and show that it reduces to one 2 × 2 system and two 1 × 1 systems. Which states are unaffected? c) Solve the 2 × 2 system to obtain the levels and hence the splitting of the degenerate states as a function of the field strength ε . Show that the two new wavefunctions resulting from application of the electric field are Ψ + = 1 2 ( Ψ 2 s - Ψ 2 p z ) and Ψ - = 1 2 ( Ψ 2 s + Ψ 2 p z ) (2) and identify which solution has which energy. 2. Using a Gaussian trial function e - α r 2 for the ground state of the hydrogen atom, show that E ( α ) = h 2 α 2 μ - e 2 α 1 / 2 2 1 / 2 ε 0 π 3 / 2 (3) and hence derive the minimum energy for this gaussian variational function (note that the trial wave- function is not normalized). Convert your answer to a.u. and evaluate the percentage error in your variational result.
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