- #1

MathematicalPhysicist

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## Homework Statement

1.Consider a one dimensional attraction potential V(x) s.t V(x)<0 for each x.

Using the variational principle, show that such a potential has at least one bound state.

Hint: use a gaussian in x as a trial functio.

2. A particle with charge e and mass m is confined to move on the circumference of a circle of radius r. Find the eigenstates and eigenvalues of the Hamiltonian.

## The Attempt at a Solution

1. Now sure where to start, I have calculated [tex]<H(a)>=<\psi|H|\psi>/<\psi|\psi>[/tex] where [tex]\psi(x)=exp(-ax^2)[/tex], and I know that it's bigger than E0, usually in order for us to find a strict upper bound for E0, we need to differentiate <H(a)> wrt to a and find its minimum, but here V(x) is not given explicitly so I guess I need to show somehow that <H(a)> is bounded, I've showed that it's bounded by [tex]\frac{a\hbar^2}{m}[/tex], is that enough to show that for this potential it has at least one bounded eigenstate.

2. I have no idea, I mean the potential is: [tex]V(r)=-mw^2r^2/2[/tex], how do I proceed from here?

Thanks in advance.

QM rulessssssssssssssss! :-)