C the coulomb potential is an example of a two body

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Unformatted text preview: 1 sÆ \3 ÅÅÅ ƒ ƒ r1 ƒ ƒ ƒ ƒ ƒ 1 =0 because of the orthogonality of single-particle states. Therefore, the expectation value of H0 is given by the diagonal pieces only, and we find X1 s2 2 s » H0 » 1 s2 2 s\ 1 = ÅÅÅÅ HX1 sÆ 1 s 2 s » H0 » 1 sÆ 1 s 2 s\ + X1 s 2 s 1 sÆ » H0 » 1 s 2 s 1 sÆ \ + X2 s 1 sÆ 1 s » H0 » 2 s 1 sÆ 1 s \ + 6 X1 sÆ 2 s 1 s » H0 » 1 sÆ 2 s 1 s \ + X2 s 1 s 1 sÆ » H0 » 2 s 1 s 1 sÆ \ + X1 s 1 sÆ 2 s » H0 » 1 s 1 sÆ 2 s \L and each term is simply E1 s + E1 s + E2 s . Therefore, X1 s2 2 s » H0 » 1 s2 2 s\ = E1 s + E1 s + E2 s , just the sum of single-particle energies without any prefactor. This is a general result for any Slater determinants if the operator is a "single-body operator". because of the orthogonality of single-particle states. Therefore, the expectation value of H0 is given by the diagonal pieces only, and we find X1 s2 2 s » H0 » 1 s2 2 s\ 1 HW6.nb sÆ 1 s 2 s » H0 » 1 sÆ 1 s 2 s\ + X1 s 2 s 1 sÆ » H0 » 1 s 2 s 1 sÆ \ + X2 s 1 sÆ 1 s » H0 » 2 s 1 sÆ 1 s \ + 3 = ÅÅÅÅ HX1 6 X1 sÆ 2 s 1 s » H0 » 1 sÆ 2 s 1 s \ + X2 s 1 s 1 sÆ » H0 » 2 s 1 s 1 sÆ \ + X1 s 1 sÆ 2 s » H0 » 1 s 1 sÆ 2 s \L and each term is simply E1 s + E1 s + E2 s . Therefore, X1 s2 2 s » H0 » 1 s2 2 s\ = E1 s + E1 s + E2 s , just the sum of single-particle energies without any prefactor. This is a general result for any Slater determinants if the operator is a "single-body operator". (c) The Coulomb potential is an example of a "two-body operator." The expectation value of D H = ‚ 3 e2 ÅÅÅÅÅÅÅ i< j ri j 2 2 2 e e e = ÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅ for the 1 s2 2 s configuration is r12 r13 r23 X1 s2 2 s » D H » 1 s2 2 s\ 1 = ÅÅÅÅ HX1 sÆ 1 s 2 s » + X1 s 2 s 1 sÆ » + X2 s 1 sÆ 1 s » - X1 sÆ 2 s 1 s » - X2 s 1 s 1 sÆ » - X1 s 1 sÆ 2 s »L 6 D H H » 1 s Æ 1 s 2 s\ + » 1 s 2 s 1 s Æ \ + » 2 s 1 s Æ 1 s \ Æ - » 1 s 2 s 1 s \ - » 2 s 1 s 1 sÆ \ - » 1 s 1 sÆ 2 s \L For instance, if I calculate the term e2 e2 e2 X1 sÆ 1 s 2 s » D H » 1 sÆ 1 s 2 s\ = Y1 sÆ 1 s 2 s … ÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅ … 1 sÆ 1 s 2 s] r12 r13 r23 e e e = Y1 s...
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