12_variation_perturbation_matrixhq

246 1 22 1 1 x 4 2 x2 1 x 2 2 dx

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Unformatted text preview: 2 dx 2 1/ 2 6 10/7/2013 Find Optimal Value of Parameter 4 parameterized energy expression 2 Find Best Value of Parameter 4 2 4 2 0 2 4 2 plug this back into energy expression 2 , 4 2 2 4 2 1 2 / 0.707 2 0.500 Compare to True Result 1 Get Trial Wavefunction Normalize 1 1 ∗ 1 1 1 2 2 1 1 1 1 1 x 22 dx 2 1/ 2 2 Compare to True Wavefunction / True where is the same Trial 7 10/7/2013 Textbook Harmonic Oscillator / / and , where ! Dr. Coe’s Lecture Harmonic Oscillator / / where and ! 1/ Therefore Variational Method: Example 3 – Ground State Energy of He r1 Internal Hamiltonian 2 1 2 1 2 4 1 4 1 1 4 -e r12 Z=+2e helium r2 -e Hard to solve exactly due to electron-electron interactions If we ignore this term, then the operator is separable 1 2 and ( 4.000 ) and since Trial Wavefunction / / Find Expression for Trial Energy Treat Z like a variational parameter, then after a great deal of algebraic work (Ch. 8) 27 in atomic units 8 8 10/7/2013 Variational Method: Example 3 – Ground State Energy of He Find Optimal Value of Parameter 27 8 parameterized energy expression Find Best...
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