852HW5_Solutions

# 852HW5_Solutions - PHYS852 Quantum Mechanics II Spring 2009...

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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 ℓm (0) ℓ | Z | 3 ℓ ′ m ′ ℓ (0) ) , which vanish unless ℓ ′ = ℓ ± 1 and m ′ ℓ = m ℓ . a.) Write the matrix element ( 3 ℓm (0) ℓ | Z | 3 ℓ ′ m ′ ℓ (0) ) out as an integral over r,θ,φ . Evaluate the inte- gral for all transitions which obey the selection rules. Answer: Inserting the projector I = integraltext ∞ rdr integraltext π sin( θ ) dθ integraltext 2 π dφ | rθφ )( rθφ ) gives ( 3 ℓm ( b ) ℓ | Z | 3 ℓ ′ m ′ ℓ ( b ) ) = integraldisplay ∞ rdr integraldisplay π sin( θ ) dθ integraldisplay 2 π dφR 3 ℓ ( r ) bracketleftbig Y m ℓ ℓ ( θ,φ ) bracketrightbig ∗ r cos θR 3 ℓ ′ ( r ) Y m ′ ℓ ℓ ′ ( θ,φ ) , where we used | n,ℓ,m (0) ℓ ) → | n,ℓ,m ( b ) ℓ ) in anticipation of these forming a ‘bad’ basis for pertur- bation theory. The radial wavefunction (from lecture) is: R nℓ ( r ) = bracketleftBigg parenleftbigg 2 na parenrightbigg 3 ( n − ℓ − 1)! 2 n ( n + ℓ )! bracketrightBigg 1 / 2 exp parenleftbigg − r na parenrightbiggparenleftbigg 2 r na parenrightbigg ℓ L 2 ℓ +1 n − ℓ − 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 ℓ ,ℓ,ℓ ′ ): 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 | nℓm (0) ℓ ) 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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## This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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852HW5_Solutions - PHYS852 Quantum Mechanics II Spring 2009...

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