852HW5_Solutions - PHYS852 Quantum Mechanics II, Spring...

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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2009 HOMEWORK ASSIGNMENT 5: Solutions 1. [20 pts] The goal of this problem is to compute the Stark effect to first-order for the n = 3 level of the hydrogen atoms. The stark shift is governed by the potential: V E = eE Z, so that you will be computing the matrix elements ( 3 m (0) | Z | 3 m (0) ) , which vanish unless = 1 and m = m . a.) Write the matrix element ( 3 m (0) | Z | 3 m (0) ) out as an integral over r,, . Evaluate the inte- gral for all transitions which obey the selection rules. Answer: Inserting the projector I = integraltext rdr integraltext sin( ) d integraltext 2 d | r )( r ) gives ( 3 m ( b ) | Z | 3 m ( b ) ) = integraldisplay rdr integraldisplay sin( ) d integraldisplay 2 dR 3 ( r ) bracketleftbig Y m ( , ) bracketrightbig r cos R 3 ( r ) Y m ( , ) , where we used | n,,m (0) ) | n,,m ( b ) ) in anticipation of these forming a bad basis for pertur- bation theory. The radial wavefunction (from lecture) is: R n ( r ) = bracketleftBigg parenleftbigg 2 na parenrightbigg 3 ( n 1)! 2 n ( n + )! bracketrightBigg 1 / 2 exp parenleftbigg r na parenrightbiggparenleftbigg 2 r na parenrightbigg L 2 +1 n 1 parenleftbigg 2 r na parenrightbigg Letting Mathematica handle the integrals, we find the matrix-elements which satisfy these se- lection rules are (labelled as z m ,, ): z 001 = ( 3 , , ( b ) | Z | 3 , 1 , ( b ) ) = 3 6 a z 012 = ( 3 , 1 , ( b ) | Z | 3 , 2 , ( b ) ) = 3 3 a z 112 = ( 3 , 1 , 1 ( b ) | Z | 3 , 2 , 1 ( b ) ) = (9 / 2) a z 112 = ( 3 , 1 , 1 ( b ) | Z | 3 , 2 , 1 ( b ) ) = (9 / 2) a from Z = Z we then know that: z 010 = z 001 = 3 6 a z 021 = z 012 = 3 3 a z 121 = z 112 = (9 / 2) a z 121 = z 112 = (9 / 2) a 1 b.) List all of the degenerate | nm (0) ) states in the n = 3 subspace. Then use the selection rules to group the levels into closed sets of coupled states. Answer: The n = 3 manifold consists of nine states: | 3 , , ( b ) ) , | 3 , 1 , 1 ( b ) ) , | 3 , 1 , ( b ) ) , | 3 , 1 , 1 ( b ) ) , | 3 , 2 , 2 ( b ) ) , | 3 , 2 , 1 ( b ) ) , | 3 , 2 , ( b ) ) , | 3 , 2 , 1 ( b ) ) , and | 3 , 2 , 2 ( b ) ) . From this we see that the n = 3 subspace can divided into a closed thee-state manifold, and two closed two-state manifolds. They are: | 3 , , ( b ) ) | 3 , 1 , ( b ) ) | 3 , 2 , ( b ) ) | 3 , 1 , 1 ( b ) ) | 3 , 2 , 1 ( b ) ) | 3 , 1 , 1 ( b ) ) | 3 , 2 , 1 ( b ) ) The remaining two states do not couple with any states in the degenerate subspace: | 2 , 1 , 2 ( b ) ) | 2 , 1 , 2 ( b ) ) c.) For each group in part b.) with more than one element, find the good eigenstates, by diago-c....
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852HW5_Solutions - PHYS852 Quantum Mechanics II, Spring...

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