Distance between atoms - Minimum energy

[mgal95]

Homework Statement

The effective potential between two atoms of same mass $m$ is:
$$V(x)=-a\frac{1}{x}+b\frac{1}{x^2}$$
where $a,b>0$ and $x$ is the distance between them.
(a) Calculate (order of magnitude) the distance between the two atoms in the molecule and its minimum possible energy.
(b) Compare the minimum energy with the minimum potential energy. Could we have $a=0$ and $b<0$?
(c) Can a molecule with $$V(x)=-\frac{|b|}{x^{2n}} , 1\leq n$$ exist?

Homework Equations

This is a problem given in the finals of a third year Quantum Mechanics class at my University. I suppose that all is needed are the uncertainty inequalities, since I do not even know if the corresponding Schroedinger Equation has an analytic solution (let alone find it).

The Attempt at a Solution

I do not even know where to start from. Some ideas:
$$\left< E\right>=\left< H\right>=\frac{\left<p_1^2 \right>}{2m} + \frac{\left<p_2^2 \right>}{2m} +\left< V\right>$$
$$\Delta p\Delta x\geq \hbar/2$$
$$\left(\Delta p\right)^2=\left< p^2\right>-\left< p\right>^2$$
Thus:
$$\left<p^2 \right>\geq \left(\Delta p\right)^2\geq \frac{\hbar}{2\left(\Delta x\right)^2}$$
and then
$$\left< E\right>\geq \frac{\hbar}{\left(\Delta x\right)^2} -a\left< \frac{1}{x}\right>+b\left<\frac{1}{x^2} \right>$$
My idea now is to form the right part of the last inequality in a function $f=f(\Delta x)$. Then I can find the minimum of this function equating the derivative with zero. Then the equality would give me the minimum energy. The value of $\Delta x$ that I will get will be in the order of magnitude of the distance between the atoms. Since nature prefers minimum energy states, the minimum energy would be the ground state and thus I can assume $\Delta x\simeq x$ The problem is that I cannot form such a function for this potential. On the other hand, I can do this for the potential given in (c), since $-b\left< x^{-2n}\right> \geq -b\left(\Delta x\right)^{-2n}$ (proven using the definition of the uncertainty).
Another idea for the distance is to calculate $\left< x\right>$, but I do not have a clue how to do that.

Please excuse my English
Thank you

 

[mfb]

I think you are overthinking this. To get an order of magnitude estimate, you can approximate the potential around the minimum by a parabola, and you should know how the energy levels look there.