# homework3 - 8 A system with Hamiltonian H(0 is sented by...

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8. A system with Hamiltonian H (0) is subject to a perturbation H (1) , which in a certain ONB can be repre- sented by the following matrices h H (0) i = ε 0 00 0 0 0 ε 0 00 0 ε 0 0002 ε 0 0 000 02 ε 0 h H (1) i = 0 0 0 0 0 000 0 i 000 i 0 Find the new energies of this system, correct to f rst order in the perturbation. 9. An otherwise free particle of charge e is constrained to move on the circumference of a circle of radius R .I f θ denotes the angular position of the particle, its (rotational) kinetic energy is associated with the operator H 0 = L 2 z 2 mR 2 where L z is 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 function ψ ( θ )= h θ | ψ i the operator L z has the action h θ | L z | ψ i = i ~ ∂ψ ( θ ) ∂θ .
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