Hellmann–Feynman theorem
The Hellmann - Feynman theorem is a theorem in quantum mechanics, which, converts the energy eigenvalues of a time-independent Hamiltonian with respect to the parameters it contains. It is named after its discoverers Hans Hellmann (1936) and Richard Feynman (1939 ) named.
In general, the theorem states:
Is the parameterized Hamiltonian,
Is the nth eigenvalue of the Hamiltonian,
Is the nth eigenvector of the Hamiltonian,
Is the parameter of interest
And represents a complete integration over the entire domain of the eigenvectors.
The proof
The proof is, if done purely formal, pretty easy. In the Dirac Bra- Ket notation can be written:
There holds:
For a critical, mathematical consideration of this evidence, see.