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Problem 4.Consider one space dimension (1D). Define an operator O and its Hermitian conjugateOf0 =i=p + W(x), Of =-iV2mV2mp + W(x) ,(5)where p is the momentum operator and W(a) is a real function of coordinate x.Consider two physics systems with HamiltonianH1 = Oto =p22m+ Vi(I) ,p-(6)H2 = 00t = .2m+ V2(I) ,respectively.Express potential energy Vi(x) and V2(x) in terms of W(x).Show that if 71(a) is the eigenstate of the Hamiltonian H1 with energy eigen-value E1, then Or(x) is the eigenstate of H2 with the same energy, E1.Similarly, if w/2(x) is the eigenstate of the Hamiltonian H2 with energy eigenvalueE2, then Of12(x) is the eigenstate of H, with the same energy, E2.
