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Physics LibreTexts

11.1: Exercises

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  1. Consider the two-state system investigated in Section 1.3. Show that the most general expressions for the perturbed energy eigenvalues and eigenstates are E1=E1+e11+|e12|2E1E2+O(ϵ3),E2=E2+e22|e12|2E1E2+O(ϵ3), and ψ1=ψ1+e12E1E2ψ2+O(ϵ2),ψ2=ψ2e12E1E2ψ1+O(ϵ2), respectively. Here, ϵ=|e12|/(E1E2)1. You may assume that |e11|/(E1E2), |e22|/(E1E2)O(ϵ).
  2. Consider the two-state system investigated in Section 1.3. Show that if the unperturbed energy eigenstates are also eigenstates of the perturbing Hamiltonian then E1=E1+e11,E2=E2+e22, and ψ1=ψ1ψ2=ψ2 to all orders in the perturbation expansion.
  3. Consider the two-state system investigated in Section 1.3. Show that if the unperturbed energy eigenstates are degenerate, so that E1=E2=E12, then the most general expressions for the perturbed energy eigenvalues and eigenstates are E±=E12+e±, and ψ±=1|ψ±ψ1+2|ψ±ψ2, respectively, where e±=12(e11+e22)±12[(e11e22)2+4|e12|2]1/2, and 1|ψ±2|ψ±=(e12e11e±)=(e22e±e12). Demonstrate that the ψ± are the simultaneous eigenstates of the unperturbed Hamiltonian, H0, and the perturbed Hamiltonian, H1, and that the e± are the corresponding eigenvalues of H1.
  4. Calculate the lowest-order energy-shift in the ground state of the one-dimensional harmonic oscillator when the perturbation V=λx4 is added to H=p2x2m+12mω2x2.

Contributors and Attributions

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 11.1: Exercises is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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