hw4solutions - HOMEWORK ASSIGNMENT 4 PHYS852 Quantum...

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Unformatted text preview: 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, ( nm | Z | n m ) , appear. Also remember to identify the good states before applying perturbation theory. Include a sketch of the energy levels versus E , with each level labeled by its state(s). Answer : The n = 3 energy level is nine-fold degenerate, containing the states | 300 ) , | 310 ) , | 31 1 ) , | 320 ) , | 32 1 ) , and | 32 2 ) . The selection rules for the Stark effect are = 1 and m = m . The states which are mixed are therefore a closed three-level manifold ( m = 0), | 300 ) | 310 ) | 320 ) , and two closed two-level manifolds ( m = 1), | 31 1 ) | 32 1 ) . The good eigenstates of the m = 0 manifold are found by diagonalizing the matrix V = z 01 z 01 z 12 z 12 with z 01 = eE ( 300 | Z | 310 ) and z 12 = eE ( 310 | Z | 320 ) We can determine z 01 and z 12 by integration z 01 = eE integraldisplay r 2 dr integraldisplay sin d integraldisplay 2 d ( R 30 ( r ) Y ( , ) ) r cos R 31 ( r ) y 1 ( , ) = 3 6 ea E z 12 = eE integraldisplay r 2 dr integraldisplay sin d integraldisplay 2 d ( R 30 ( r ) Y 1 ( , ) ) r cos R 31 ( r ) y 2 ( , ) = 3 3 ea E This leads to the eigenvalues of V being v 01 = 9 ea E , v 02 = 0 and v 03 = 9 ea E , with correspond- ing eigenvectors | v 01 ) = 1 6 parenleftBig 2 | 300 ) 3 | 310 ) + | 320 ) parenrightBig | v 02 ) = 1 3 parenleftBig | 300 ) 2 | 310 ) parenrightBig | v 03 ) = 1 6 parenleftBig 2 | 300 ) + 3 | 310 ) + | 320 ) parenrightBig The good eigenstates of the m = 1 manifolds are eigenstates of the matrix V = parenleftbigg z 12 z 12 parenrightbigg where z 12 = eE ( 31 1 | Z | 32 1 ) = 9 2 ea E 1 Here the eigenvalues are v 11 = 9 2 ea E and v 12 = 9 2 ea E , with corresponding eigenvectors | v 11 ) = 1 2 ( | 31 1 ) | 32 1 ) ) | v 12 ) = 1 2 ( | 31 1 ) + | 32 1 ) ) To first-order in perturbation theory we then find that the E 3 level splits into 5 levels, with energy shifts of 9 ea E , (9 / 2) ea E , 0, (9 / 2) ea E , and 9 ea E . The corresponding level manifolds as are | v 01 ) , {| v 11+ ) , | v 11 )} , {| v 02 ) , | 322 ) , | 32 2 )} , {| v 12+ ) , | v 12 )} , and | v 03 ) , respectively....
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This note was uploaded on 11/26/2010 for the course PHYSICS PHYS 852 taught by Professor Michaelmoore during the Spring '10 term at Michigan State University.

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hw4solutions - HOMEWORK ASSIGNMENT 4 PHYS852 Quantum...

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