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8_Variational_Principle

# 8_Variational_Principle - VIII The Variational Principle...

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VIII. The Variational Principle Griffiths Chapter 7 1

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A. The Variational Principle The variational principle is another approximation method for quantum mechanics. Perturbation theory allowed us to find approximate wave functions and energies for systems that we couldn’t solve the Schrödinger equation for exactly. 2
A. The Variational Principle The variational principle only tells us about the energy of a system, and only of the ground state. Furthermore, we can only find an upper bound on the ground state energy. On the other hand, the variational principle is easy to understand and use, and the ground state energy of a system is often the most important thing we want to know. 3

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The ground state energy of a system is, according to the variational principle, A. The Variational Principle E gs ψ | H | ψ ground state expectation value of H, H See board VIII.A.1 1. Statement of the Principle 4
Griffiths Example 7.1: Find an upper bound on the ground state energy for the one dimensional harmonic oscillator. A. The Variational Principle 2. An Example H = 2 2 m d 2 dx 2 + 1 2 m ω 2 x 2 Solution: See board VIII.A.2 E gs = 1 2 ω What luck! Equal to the exact result. Used a Gaussian trial function with free parameter and then MINIMIZED with respect to the free parameter. 5

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Griffiths Example 7.2: Find an upper bound on the ground state energy for the delta function potential. A. The Variational Principle 3. Another Example Solution: See board VIII.A.3 For exact result, replace π with 2. H = 2 2 m d 2 dx 2 αδ ( x ) E gs ≤ − m α 2 π 2 Negative for bound state Used a Gaussian trial function again with a free parameter and then MINIMIZED with respect to the free parameter. 6
Griffiths Example 7.3: Find an upper bound on the ground state energy for the one dimensional square well. A. The Variational Principle 4. A Third Example Solution: See board VIII.A.4 Could NOT use a Gaussian trial function here. H = 2 2 m d 2 dx 2 + V ( x ) where V ( x ) = 0 , if 0 x a , otherwise E gs 12 2 2 ma 2 It “leaks” outside the potential – it is non-zero in the infinity region of the potential For exact result, replace 12 with π 2 .

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8_Variational_Principle - VIII The Variational Principle...

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