RayleighRitz - PHY4221 Quantum Mechanics I Fall 2004...

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PHY4221 Quantum Mechanics I Fall 2004 Rayleigh-Ritz linear variational method Given the eigenvalue problem of the Hamiltonian operator H : H | Φ i = E | Φ i , (1) the Variational Principle states that any normalized state vector | Ψ i for which the expectation value E ([Ψ]) ≡ h Ψ | H | Ψ i of the Hamiltonian operator H , i.e. the average energy, is stationary is an eigenvector of H , and the corresponding energy eigenvalue is the stationary value of E ([Ψ]). That is, if we could actually carry out a variation of | Ψ i , starting with some initial | Ψ Initial i and varying | Ψ i by small steps, namely by rotating | Ψ i by small amounts in the Hilbert space, such that we could end by finding the true minimum of the functional E ([Ψ]), we would attain the ground state eigenvector and eigenenergy exactly. Similarly, a local minimum would give us an excited state eigenvector and eigenenergy. Obviously, this is not feasible even with modern computers.
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