9780030839931, Chapter 17, Problem 1P

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Chapter 17, Problem 1P Problem Derivation of the Hartree Equations from the Variational Principle (a) Show that the expectations value of the Hamiltonian in a state of the form 10) is (17.68) provided that all the wi satisfy the normalization condition dr|wi|2 =1. (b) Expressing the constraint of normalization for each with Lagrange multiplier and taking and as independent show that the stationary condition 0 (17.69) leads directly to the Hartree equations Step by step solution Step Step The expectation value of any function is given position vector and dV is small volume element and f (r) is conjugate of f(r) Step of 6 (a) The expectation value of the one body part of the Hamiltonian operator is given as, As the wave functions are normalized, So, the above equation reduces The body operator given as, The expectation value of the Hamiltonian is, Step Substitute 2m and for in the above expression, Thus, the expectation value of the Hamiltonian IS, Hence proved. Step of 6 (b) The expected value with the normalization condition as Lagrange multiplier can be written as follows: the term is given as, Take the variation with respect to of the one body operator. Thus, 2m For the two-body problem, Sw, 2 Now. =0 Substitute for and 2 for П. Divide the above equation by on both sides gives, Consider and substitute 2 for in the above equation which gives, Step of 6 The above equation is the Hartree equation given in equation 17.7 which is given in the text