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supp08 - Supplement 8-A A Useful Theorem The following...

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Supplement 8-A A Useful Theorem The following useful result appears in Pauli’s 1930 “Handbuch Article on Quantum Theory”: Consider eigenvalues and eigenfunctions of a Hamiltonian that depends on some pa- rameter—for example, the mass of the electron, or the charge of the electron, or any other parameter that may appear in more complicated problems. The Schrödinger eigenvalue equation may then be written with the parameter explicitly indicated as, (8A-1) It follows that with the eigenfunctions normalized to unity, (8A-2) that (8A-3) Let us now differentiate both sides with respect to . We get Consider now the first two terms on the right-hand side. Using the eigenvalue equation and its complex conjugate (with hermiticity of H ), we see that they add up to We are therefore left with (8A-4) The utility of this result is somewhat limited, because it requires knowing the exact eigen- values and, for the calculation on the right-hand side, the exact eigenfunctions.
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