Exercise 2: Perturbation Theory vs. Variational Principle [8 points] A quantum system is ruled by the Hamiltonian H, whi

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Exercise 2 Perturbation Theory Vs Variational Principle 8 Points A Quantum System Is Ruled By The Hamiltonian H Whi 1 (50.47 KiB) Viewed 15 times

Exercise 2: Perturbation Theory vs. Variational Principle [8 points] A quantum system is ruled by the Hamiltonian H, which has two orthonormal solutions: a with energy Ea and with energy Es, where Ea < E. A weak perturbation acting on the system has the following matrix elements (vaĤ¹|va) = (vrĤ¹|v) = 0 (vaĤ' vb) = (v¹|va) = h, where h is a constant. (a) Find the eigenvalues of the total Hamiltonian, H = Ho + H'. [2 points] Quantum Structure of Matter Exercise Sheet 10-29.06.2022 (b) Find the second-order corrections to the energies. (It is straightforward to show that the first-order corrections vanish.) [2 points] = (c) Estimate the ground state energy of the perturbed system with Hamiltonian H Ho+H' using the variational principle with a trial function of the form = (coso) + (sino). where is an adjustable parameter. [2 points] (d) Compare the results obtained from perturbation theory and from the variational principle. Which method is more accurare and why? [2 points]