Question

Problem 9.2 Consider a system whose Hamiltonian is given by $\hat{H} = E_0 \begin{pmatrix} 1+\lambda & 0 & 0 & 0\\ 0 & 8 & 0 & 0\\ 0 & 0 & 3 & -2\lambda\\ 0 & 0 & -2\lambda & 7 \end{pmatrix}$, where $\lambda << 1$. (a) By decomposing this Hamiltonian into $\hat{H} = \hat{H}_0 + \hat{H}_p$, find the eigenvalues and eigen- states of the unperturbed Hamiltonian $\hat{H}_0$. (b) Diagonalize $\hat{H}$ to find the exact eigenvalues of $\hat{H}$; expand each eigenvalue to the second power of $\lambda$. (c) Using first- and second-order nondegenerate perturbation theory, find the approximate eigenenergies of $\hat{H}$ and the eigenstates to first order. Compare these with the exact values obtained in (b).

          Problem 9.2
Consider a system whose Hamiltonian is given by $\hat{H} = E_0 \begin{pmatrix} 1+\lambda & 0 & 0 & 0\\ 0 & 8 & 0 & 0\\ 0 & 0 & 3 & -2\lambda\\ 0 & 0 & -2\lambda & 7 \end{pmatrix}$,
where $\lambda << 1$.
(a) By decomposing this Hamiltonian into $\hat{H} = \hat{H}_0 + \hat{H}_p$, find the eigenvalues and eigen-
states of the unperturbed Hamiltonian $\hat{H}_0$.
(b) Diagonalize $\hat{H}$ to find the exact eigenvalues of $\hat{H}$; expand each eigenvalue to the second
power of $\lambda$.
(c) Using first- and second-order nondegenerate perturbation theory, find the approximate
eigenenergies of $\hat{H}$ and the eigenstates to first order. Compare these with the exact values obtained
in (b).

Problem 9.2
Consider a system whose Hamiltonian is given by Ĥ = E0
< p m a t r i x >,
where λ << 1.
(a) By decomposing this Hamiltonian into Ĥ = Ĥ0 + Ĥp, find the eigenvalues and eigen-
states of the unperturbed Hamiltonian Ĥ0.
(b) Diagonalize Ĥ to find the exact eigenvalues of Ĥ; expand each eigenvalue to the second
power of λ.
(c) Using first- and second-order nondegenerate perturbation theory, find the approximate
eigenenergies of Ĥ and the eigenstates to first order. Compare these with the exact values obtained
in (b).

Added by Natalie C.

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