Question 03/97

1. It is known that a real positive function

(x) is a groundstate of a one-dimensional quantum Hamiltonian H. Find the potential V(x) of that Hamiltonian.

2. Suggest a way for construction of potential V(x) and two functions

(x) and

(x) (i.e. all three functions can be chosen by you) in such a way that the functions correspond to the ground state and the first excited state of the quantum Hamiltonian.

The first part of this question appeared as a qualifying exam question at MIT, while the second part was suggested by M. Schwartz

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