Bound state of a spin 3/2 particle in a potential

[yarospo]

Hello,
Up until now I was certain that a bound state is a state with energy below the minimum of the potential at infinity. However, in this question I don't know at all how to proceed.

Homework Statement

A spin 3/2 particle moves in a potential
$$V=V_0(r)+\frac{V_1}{r^3}L\cdot S$$
and V₀ > 0.
We define the eigenvalue of the operator $$L\cdot S$$ in the basis |J,L,S> as a_J,L,S while J=L+S.
What is the necessary condition on V₁a to get a bound state. Should it
be positive or negative?

Homework Equations

The Attempt at a Solution

Right from the start what puzzles me is that I have no idea how V₀ looks aside from the fact that it's positive. I also note that the potential seems to depend on angular momentum and spin, but their effects cancel out when r goes to infinity it goes to 0, and since V₀>0 the energy threshold for bound states depends solely on it. But if I know nothing about it, then for example, taking $$V_0(r)=r^2$$ there is absolutely no restriction on the factor of $$\frac{1}{r_3}$$ - all the states will be bound.

Well, here's how I intended to continue, once I knew what I'm actually looking for:
Applying the Hamiltonian on the basis vectors (Using central potential formulas), I get:
$$E_{J,L,S}=<J,L,S|H|J,L,S> = \left.<J,L,S|V_0(r)\right|J,L,S>+a_{J,L,S}V_1<J,L,S\left|\frac{1}{r^3}\right|J,L,S>$$
$$=\int _0^{\infty }dr r^2V_0(r)\left[R_{\text{nl}}(r)\right]{}^2+\frac{a_{J,L,S}V_1}{a_0^3n^3l\left(l+\frac{1}{2}\right)(l+1)} < ??$$

 

[NateD]

>Where n, l, a0 are the usual quantum numbers related to the radial wave function.At this point I'm stuck, since I don't know what the integral in the first term should be (since I don't know what V_0(r) is), and I don't know what the matrix element in the second term is.Any help would be much appreciated! Thank you!