The Hamiltonian of a system has the form 1 d² 2 dx² अ = • 1⁄2 x² + √4x¹ = Ĥ0 + V4Xª Let ½(x) = |n) be the eigenstates of Ño, with Ño|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, lø) = c₂|0} + 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|Â|n) and Smn = (m|n), respectively, for m, n = 0, 2. Find the overlap (S) matrix, assuming 10) and 12) are both normalized. 3 3 (b) If (0|x4|0) = ³, (0|x4|2) = (2|x4|0) = - 39 and (2|x¹|2) = = find expressions for all
The Hamiltonian of a system has the form 1 d² 2 dx² अ = • 1⁄2 x² + √4x¹ = Ĥ0 + V4Xª Let ½(x) = |n) be the eigenstates of Ño, with Ño|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, lø) = c₂|0} + 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|Â|n) and Smn = (m|n), respectively, for m, n = 0, 2. Find the overlap (S) matrix, assuming 10) and 12) are both normalized. 3 3 (b) If (0|x4|0) = ³, (0|x4|2) = (2|x4|0) = - 39 and (2|x¹|2) = = find expressions for all
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Transcribed Image Text:The Hamiltonian of a system has the form
1
Ĥ
π =
-1
1¹2²2² + ²x² + 4x¹ = Fo+ Y4x¹
2
dx2
Let un(x) = |n) be the eigenstates of Ho, with Ĥo|n) = (n + ¹) |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) = co|0) + 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
normalized.
3
39
(b) If (0|x4|0) =, (0|x4|2) = (2|x4|0) =,and (2|x4|2) = 32, find expressions for all
the Hamiltonian elements, Hoo, H22, Ho2, and H₂0 (you will be asked to identify one of
them from a list of possibilities on the quiz).
(c) Use your results for Hmn and Smn to construct the secular equation for this system and
evaluate the determinant to obtain a polynomial equation for the system energy, E. On
the submission quiz, you will be asked to identify the coefficients in this equation.
(d) Suppose 14 = 0.1. Use this value and solve for the variational energies of the system
using the result from part (c). You can solve either by using the quadratic formula or
WolframAlpha (or equivalent software).
(e) Using the information provided above, evaluate the first-order perturbation correction
to the ground state energy of the system. How does the corrected energy compare with
the variational result?
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