852_2010hw8 - PHYS852 Quantum Mechanics II Spring 2010...

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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 8 Topics covered: hydrogen fine structure 1. [10 pts] Let the hamiltonian H depend on the parameter λ , so that H = H ( λ ). The eigenstates and eigenvalues of H are then also functions of λ , i.e. E n = E n ( λ ) and | n i = | n ( λ ) i . Use the property H | n i = E n | n i to prove the Feynman-Hellmann theorem: ∂E n ( λ ) ∂λ = h n ( λ ) | ∂H ( λ ) ∂λ | n ( λ ) i 2. [15 pts] The effective hamlitonian which governs the radial wave equation is H =- ~ 2 2 M ∂ 2 ∂r 2 + ~ 2 ‘ ( ‘ + 1) 2 Mr 2- e 2 4 π r . The exact eigenvalues in terms of e and ‘ are E n =- Me 4 32 π 2 2 ~ 2 n ( ‘ ) 2 , where n ( ‘ ) = n r + ‘ + 1, with n r being the highest power in the series expansion or R n‘ ( r ). Apply the Feynman-Hellman theorem with λ = e to derive: h n‘ (0) | R- 1 | n‘ (0) i = 1 n 2 a . Then use λ = ‘ , with ‘ treated as a continuous parameter, to derive: h n‘ (0) | R- 2 | n‘ (0) i = 1 ( ‘ + 1 / 2) n 3 a 2 ....
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This note was uploaded on 12/21/2011 for the course PHYS 852 taught by Professor Moore during the Spring '11 term at Michigan State University.

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852_2010hw8 - PHYS852 Quantum Mechanics II Spring 2010...

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