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hw4 - HOMEWORK ASSIGNMENT 4 PHYS852 Quantum Mechanics I...

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HOMEWORK ASSIGNMENT 4 PHYS852 Quantum Mechanics I, Spring 2008 Topics covered: Atomic Physics applications: STARK EFFECT, Zeeman effect, spin-orbit coupling 1. [15 pts] Compute the Stark effect to lowest non-vanishing order for the n = 3 level of the hydrogen atoms. Fully evaluate whatever matrix elements, ( nℓm | Z | nℓ m ) , appear. Also remember to identify the ‘good’ states before applying perturbation theory. Include a sketch of the energy levels versus E 0 , with each level labeled by its state(s). 2. [10 pts] Do the same as the previous problem, but for the Zeeman effect. 3. [5 pts] Prove that [ J z , vector L · vector S ] = 0. Is m j is conserved in the spin-orbit interaction? 4. [10 pts] Let the hamiltonian H for some system depend on the parameter λ , so that H = H ( λ ). The eigenstates and eigenvalues are then also functions of λ , i.e. E n = E n ( λ ) and | n ) = | n ( λ ) ) . Expand H , the eigenvalues, and the eigenvectors around λ λ 0 and use first-order perturbation theory (with respect to the small parameter ǫ = λ λ 0
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