The Hamiltonian of a system has the form 1 d² 1 · + ²⁄3 x² + √4x² = Ĥo + Y₁X² 2 dx2 2 Ĥ = == Let un(x) = |n) be the eigenstates of Fo, with Fo|n) = (n + ) [n), n = 0, 1, 2, ... . In this problem, we will first utilize the linear variational method to set up the secular determinant for finding the lowest energy state for the trial function, lp) = col0) + c₂|2). (a) In setting up the secular determinant for this problem, we will need to evaluate the Hamiltonian and overlap matrix elements, Hmn = (m|F|n) and Smn = (m/n), respectively, for m, n = 0, 2. Find the overlap (S) matrix, assuming [0) and [2) are both
The Hamiltonian of a system has the form 1 d² 1 · + ²⁄3 x² + √4x² = Ĥo + Y₁X² 2 dx2 2 Ĥ = == Let un(x) = |n) be the eigenstates of Fo, with Fo|n) = (n + ) [n), n = 0, 1, 2, ... . In this problem, we will first utilize the linear variational method to set up the secular determinant for finding the lowest energy state for the trial function, lp) = col0) + c₂|2). (a) In setting up the secular determinant for this problem, we will need to evaluate the Hamiltonian and overlap matrix elements, Hmn = (m|F|n) and Smn = (m/n), respectively, for m, n = 0, 2. Find the overlap (S) matrix, assuming [0) and [2) are both
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Transcribed Image Text:The Hamiltonian of a system has the form
1
A = -1¹ ²³² + ²x² + Y₁x² = H₁ + V₁x¹
2 dx2
2
Let un(x) = |n) be the eigenstates of Fo, with Fo|n) = (n + ½) |n), n = 0, 1, 2, ... . In this
problem, we will first utilize the linear variational method to set up the secular determinant
for finding the lowest energy state for the trial function, lp) = co10) + C₂12).
(a) In setting up the secular determinant for this problem, we will need to evaluate the
Hamiltonian and overlap matrix elements, Hmn = (m|Â|n) and Smn = (m[n),
respectively, for m, n = 0, 2. Find the overlap (S) matrix, assuming [0) and 12) are both
normalized.
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