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. Questions are associated to Introductory Quantum Mechanics.

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4.png

4.png
Problem 4.
Consider one space dimension (1D). Define an operator O and its Hermitian conjugate
Of
0 =
i
=p + W(x), Of =-
i
V2m
V2m
p + W(x) ,
(5)
where p is the momentum operator and W(a) is a real function of coordinate x.
Consider two physics systems with Hamiltonian
H1 = Oto =
p2
2m
+ 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 eigenvalue
E2, then Of12(x) is the eigenstate of H, with the same energy, E2.

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