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Unformatted text preview: Chapter 7: Slide 1 Chapter 7 The Helium Atom Chapter 7: Slide 2 Outline • The Variational Method • Applications of the Variational Method • Better Variational Wavefunctions • The Helium Atom • Perturbation Theory Treatment of Helium • Variational Method Treatment of Helium Chapter 7: Slide 3 The Variational Method In quantum mechanics, one often encounters systems for which the Schrödinger Equation cannot be solved exactly. There are several methods by which the Equation can be solved approximately, to whatever degree of accuracy desired. One of these methods is Perturbation Theory, which was introduced earlier. A second method is the Variational Method , which is developed here, and will be applied to the Helium atom Schrödinger Equation. Chapter 7: Slide 4 The Variational Theorem This theorem states that if one chooses an approximate wavefunction, φ , then the Expectation Value for the energy is greater than or equal to the exact ground state energy, E . φ φ φ φ H E E trial = = < Proof: * * E d d H ≥ = ∫ ∫ τ φ φ τ φ φ * * * * ≥ = < ∫ ∫ ∫ ∫ τ φ φ τ φ φ τ φ φ τ φ φ d d E d d H E E ( 29 ∫ ∫ = < τ φ φ τ φ φ d d E H E E * * Assume that we know the exact solutions, ψ n : n n n E H ψ ψ = ≥ ? Chapter 7: Slide 5 In Chapter 2, it was discussed that the set of eigenfunctions, ψ n , of the Hamiltonian form a complete set . of orthonormal functions. That is, any arbitrary function with the same boundary conditions can be expanded as a linear combination (an infinite number of terms) of eigenfunctions. ∑ ∑ ∞ = = = n n n n n n c c ψ ψ φ This can be substituted into the expression for <E> to get: ( 29 ∫ ∫ = < τ φ φ τ φ φ d d E H E E * * ( 29 ∫ ∑ ∑ ∫ ∑ ∑  = τ ψ ψ τ ψ ψ d c c d c E H c n n n m m m n n n m m m * * ( 29 ∫ ∑ ∑ ∫ ∑ ∑  = < τ ψ ψ τ ψ ψ d c c d E H c c E E n n n m m m n n n m m m * * Chapter 7: Slide 6 ( 29 ∫ ∑ ∑ ∫ ∑ ∑  = < τ ψ ψ τ ψ ψ d c c d E E c c E E n n n m m m n n n n m m m * * ( 29 ∫ ∑∑ ∫ ∑∑ = τ ψ ψ τ ψ ψ d c c d E E c c n m m n n m n m n m n n m * * * * ( 29 mn m n n m mn n m n n m c c E E c c E E δ δ ∑∑ ∑∑ = < * * because mn n m d δ τ ψ ψ = ∫ * ψ orthonormality ∑ ∑ = < n n n n n n n c c E E c c E E * * ) ( ≥ * ≥ ≥ E E c c n n n because Therefore: E H E E trial ≥ = = < φ φ φ φ Chapter 7: Slide 7 Outline • The Variational Method • Applications of the Variational Method • Better Variational Wavefunctions • The Helium Atom • Perturbation Theory Treatment of Helium • Variational Method Treatment of Helium Chapter 7: Slide 8 Applications of the Variational Method The Particle in a Box In Chapter 3, we learned that, for a PIB:...
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This note was uploaded on 04/11/2011 for the course CHEM 5210 taught by Professor Staff during the Spring '08 term at North Texas.
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