Show that for any normalized |u> , the inequality E0 <= <u|H|u> holds, where E0 is teh lowest energy eigenvalue.
Prove the following theorem: Every attractive potential in one dimension has at least one bound state.
Consider the potential V = -aV0 * delta(x)
Show that it admits a bound state of energy E = -ma^2 * V0^2 / 2h^2
Please see the attachment.
Let the eigenbasis of the Hamiltonian H be denoted by such that:
Also we can order the energy levels in such a way that:
Now, we can write the normalized wave function as a linear combination of the eigenstates:
And since is normalized we have:
Due to (1.2) and the fact that we can write equation (1.5) as an inequality:
Using (1.4) ...
The 8 pages solution shows in detail how to approach and solve the problems.