Answer to the Question 03/97
The question was: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.
(8/97) The problem was solved correctly by Adi Armoni from TAU (e-mail: armoni@post.tau.ac.il).
The first part of the problem is simple: simply write down the Schrodinger equation and express V(x) in terms of everything else.
The second part is slightly more complicated: One assumes that the ratio between the eigenfunction of the first excited state and the ground state is some known function f(x). (E.g., f(x)=xn.) By writing down Schrodinger equation for
(x) and
=f(x)(x)
we can get a first order differential equation for
which can be readily integrated.
For a detailed solution see the following postscript file.
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