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 , so that H = H ( ). The eigenstates and eigenvalues of H are then also functions of , i.e. E n = E n ( ) and | n i = | n ( ) i . Use the property H | n i = E n | n i to prove the Feynman-Hellmann theorem: E n ( ) = h n ( ) | H ( ) | n ( ) 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 r . The exact eigenvalues in terms of e and are E n =- Me 4 32 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 = e to derive: h n (0) | R- 1 | n (0) i = 1 n 2 a . Then use = , 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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852_2010hw8 - PHYS852 Quantum Mechanics II, Spring 2010...

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