12_variation_perturbation_matrixhq

To 2 2 2 2 2 2 2 2 derivative

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Unformatted text preview: 2 H1,1S 2, 2 H 2, 2 S1,1 H1,1 H 2, 2 H12, 2 4 2 S1,1S 2, 2 S1, 2 S1,1S 2, 2 S12, 2 2 boils down to values for matrix elements , and , matrix elements ˆ H i , j f i Hf j d Si , j f i f j d When S is not the Identity Matrix (Basis Functions are Not Orthonormal) E has eigenvalues on diagonal, zero off diagonal C is the matrix of each column of eigenvectors H is the Hamiltonian matrix S is the overlap matrix for the basis functions (H-ES)C=0 right multiply each term by C, rearrange HC=ESC left multiply each side by C-1S-1, rearrange C-1S-1HC=C-1S-1ESC E is diagonal C-1S-1HC=EC-1S-1SC matrix times its inverse is identity matrix C-1S-1HC=E It takes both the eigenvector coefficients and the basis overlap elements to diagonalize the Hamiltonian If Basis Functions are Orthogonal (S=I): it takes only c-1Hc=E to diagonalize the Hamiltonian 17 10/7/2013 2x2 Matrix Example - Particle-in-a-Box with a Trial Wavefunction Symmetric about center at /2 and goes to zero at walls ( 0, ). Can only model the symmetric states, i.e. all of the odd quantum number solutions for particle in a box f n ( x) x n ( a x) n 1 use only two terms and f1 ( x) x(1 x) x x boils down to values for matrix elements 1 2 d 2 ˆ H1,1 f1* Hf1d x x 2 x x2 2 0 2m dx 1 2 dx x x2 0 2m m 2 2 2 31 2 0 2 0 2 1 1 m 2 3 2 x 1 0 2 dx x 1 0 2 2 d 1 2 x dx 2x3 x 4 2m dx 2 x3 2 x 4 x5 2 1 1 1 3 m 4 5 0 m 3 2 5 1 2 x 3 x 4 dx 1 2 d 2 2 ˆ H 2, 2 f 2* Hf1d x 2 2 x 3 x 4 x 2 x3 x 4 2 0 2m dx 2 2 2 12 x 12 x 2 dx x 2 2...
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