In the hamiltonian the factor e2 4 appears the

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Unformatted text preview: ltonian, the factor e2 4 c r 0r appears. The eigenvalue has the form 1 mc2 2 n2 Enl 2 If we take as our parameter to be , then we get c1 r n,l 1 r nl mc2 n2 Enl (8A-8) so that mc n2 1 a0n2 (8A-9) In the radial Hamiltonian, there is a term 2 l(l 2m r If we treat l as the parameter and recall that n 2 2l 2m 1 r2 1) 2 nr l 12 mc 2 1, we get 2 2 n3 (8A-10) which is equivalent to 1 r2 1 nl a2 n3(l 0 1 2 ) (8A-11) A Useful Theorem W-37 Using an observation of J. Schwinger that the average force in a stationary state must vanish, we can proceed from F dV(r) dr d dr to F(r) 0r 4 0r l(l 1) 2mr 2 (8A-12) 2 e2 4 2 e2 2 l(l 1) mr 3 0 and thus obtain 1 r3 nl e2 m 1 2 4 0 r2 l(l 1) 1 nl a3n3l(l 0 1 2 )(l 1) (8A-13)...
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