12_variation_perturbation_matrixhq

2 105 7 5 m 7 6 5 4 3 m 105 2 10 15 6 2

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Unformatted text preview: 2 2 6 105 30 2 30 140 6 630 105 30 1 30 140 6 630 105 30 Etrial 4 2 2 2 m 2 1 1 1 1 1 1 1 1 1 2 30 630 140 30 630 140 30 105 140 2 56 2128 2 51.065125, 4.934875 m m 2 Etrial 2 h 2 4.934875 h 2 4.934875 0.125002 2 m m m2 Etrue h2 h2 1 h2 0.125000 8ma 2 m 8 m pretty good! doesn’t do odd levels 19 10/7/2013 5x5 Matrix Example - Particle-in-a-Box with a Trial Wavefunction i 0 5 j 0 5 a 1 hbar 1 i 1 basis functions x ( a x) 2nd der. with res. to x i 1 m 1 i 1 i ( i 1) ( a x) x on the computer i x ( i 1) ( a x) der. with res. to x i 1 i 2 i 1 i 1 x i 1 i 2 x ( i 1) ( a x) x i ( a x) ( i 1) ( a x) i 1 i ( i 1) a 2 hbar i 1 i 1 j 1 j 1 j 2 j j 1 j 1 H x ( a x) x j ( j 1) ( a x) 2 x ( j 1) ( a x) x ( a x) j ( j 1) dx i j 2 m 0 a i 1 i 1 j 1 j 1 S x ( a x) x ( a x) dx i j 0 E genvals ( H S) H 1.739 103 484.675 244.030 E 123.372 44.413 4.935 c genvecs ( H S) S 4.059 10 5 2.171 10 6 1.509 10 5 1.168 10 4 3 4 4 4 5.301 10 2.691 10 2.755 10 1.752 10 3 3 0.027 0.015 6.760 10 9.015 10 c 0.144 0.093 0.039 0.187 0.614 0.570 0.493 0.292 0.809 0.865 0.956 0.766 E 5 E 4.935 5 ( x) 2 2 ci 5x i 1 ( a x) 0.1250000000 i 1 ( 1) 0.011 0.137 0.440 0.481 0.745 exact 0.1250000000 a 2 N ( x) dx ( x) 1 2 ( x) ( x) a N 0 i 0.685 0.243 0.045 3 5.022 10 4 3.980 10 0.685 0.011 x a sin N 0.024 1.5 ( x) 1 ( x) 0.5 0 0 0.2 0.4 0.6 0.8 x E E 44.413 4 4 2 2 1.1250000002 exact 1.1250000000 3( x) 3( x) c xi 1 ( a x) i 1 ( 1) i 4 a 2 N3 3( x) dx 0 i 3( x) 1 2 a 3 x a sin 3( x) N3 2 1 3( x) 3( x) 0 1 2 0 0.2 0.4 0....
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This document was uploaded on 01/19/2014.

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