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Consider a four-state system whose Hamiltonian is given by Vo +A 0 8Vo 3V -21 0 -21 7V where Vo is a real-valued constant and A is a real-valued parameter. (c) Now decompose this Hamiltonian into an unperturbed Hamiltonian, Ho, and a perturbing piece, H'. Solve the unperturbed system, that is, find the (four) eigenvalues and (four) eigenvectors of your unperturbed Hamiltonion, Ho. (d) Using nondegenerate perturbation theory, find the first- and second-order corrections to the unperturbed energies in the presence of the perturbing Hamiltonian, H'. (e) Solve the system exactly, by diagonalising the full Hamiltonian, H, to find the exact eigen- values of the full Hamiltonian, H. (f) Expand your solutions to part (e) to the appropriate order in A and compare these results to your results from nondegenerate perturbation theory in part (d). What condition must you impose on A to ensure that the leading energy correction from perturbation theory is a good approximation to the exact result?

Consider a four-state system whose Hamiltonian is given by Vo +A 0 8Vo 3V -21 0 -21 7V where Vo is a real-valued constant and A is a real-valued parameter. (c) Now decompose this Hamiltonian into an unperturbed Hamiltonian, Ho, and a perturbing piece, H'. Solve the unperturbed system, that is, find the (four) eigenvalues and (four) eigenvectors of your unperturbed Hamiltonion, Ho. (d) Using nondegenerate perturbation theory, find the first- and second-order corrections to the unperturbed energies in the presence of the perturbing Hamiltonian, H'. (e) Solve the system exactly, by diagonalising the full Hamiltonian, H, to find the exact eigen- values of the full Hamiltonian, H. (f) Expand your solutions to part (e) to the appropriate order in A and compare these results to your results from nondegenerate perturbation theory in part (d). What condition must you impose on A to ensure that the leading energy correction from perturbation theory is a good approximation to the exact result?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.1: Introduction To Eigenvalues And Eigenvectors
Problem 16EQ
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Consider a four-state system whose Hamiltonian is given by Vo +A 8Vo H 3Vo -2) 7Vo -2) where Vo is a real-valued constant and A is a real-valued parameter. (c) Now decompose this Hamiltonian into an unperturbed Hamiltonian, Ho, and a perturbing piece, H'. Solve the unperturbed system, that is, find the (four) eigenvalues and (four) eigenvectors of your unperturbed Hamiltonion, Ho. (d) Using nondegenerate perturbation theory, find the first- and second-order corrections to the unperturbed energies in the presence of the perturbing Hamiltonian, H'. (e) Solve the system exactly, by diagonalising the full Hamiltonian, H, to find the exact eigen- values of the full Hamiltonian, H. (f) Expand your solutions to part (e) to the appropriate order in A and compare these results to your results from nondegenerate perturbation theory in part (d). What condition must you impose on A to ensure that the leading energy correction from perturbation theory is a good approximation to the exact result?
Transcribed Image Text:Consider a four-state system whose Hamiltonian is given by Vo +A 8Vo H 3Vo -2) 7Vo -2) where Vo is a real-valued constant and A is a real-valued parameter. (c) Now decompose this Hamiltonian into an unperturbed Hamiltonian, Ho, and a perturbing piece, H'. Solve the unperturbed system, that is, find the (four) eigenvalues and (four) eigenvectors of your unperturbed Hamiltonion, Ho. (d) Using nondegenerate perturbation theory, find the first- and second-order corrections to the unperturbed energies in the presence of the perturbing Hamiltonian, H'. (e) Solve the system exactly, by diagonalising the full Hamiltonian, H, to find the exact eigen- values of the full Hamiltonian, H. (f) Expand your solutions to part (e) to the appropriate order in A and compare these results to your results from nondegenerate perturbation theory in part (d). What condition must you impose on A to ensure that the leading energy correction from perturbation theory is a good approximation to the exact result?
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