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.Science