8. A system with HamiltonianH(0)is subject to a perturbationH(1),which in a certain ONB can be repre-sented by the following matriceshH(0)i=⎛⎜⎜⎜⎜⎝ε000 0 00ε0000ε00002ε00000 02ε0⎞⎟⎟⎟⎟⎠hH(1)i=⎛⎜⎜⎜⎜⎝0∆0∆0∆0∆0000 0i∆000−i∆0⎞⎟⎟⎟⎟⎠Find the new energies of this system, correct tofrst order in the perturbation.9. An otherwise free particle of chargeeis constrained to move on the circumference of a circle of radiusR.Ifθdenotes the angular position of the particle, its (rotational) kinetic energy is associated with the operatorH0=L2z2mR2whereLzis the component of angular momentum perpendicular to the plane in which the particle moves.In the (angular) position representation, in which the particle’s state is represented by the wave functionψ(θ)=hθ|ψithe operatorLzhas the actionhθ|Lz|ψi=−i~∂ψ(θ)∂θ.
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