Statistical thermodynamics: number of states of particle in central potential

[Derivator]

Homework Statement

Give the number of states (energy of the state smaller than E<0) $$\Phi(E)$$ of a spinless particle with mass $$m$$ in the central potential $$V(\vec{r})=-\frac{a}{\left|\vec{r}\right|}$$.

Homework Equations

The Attempt at a Solution

Hi,

the hamiltonian of this problem is given by

$$\mathcal{H}=\frac{p^2}{2m}-\frac{a}{r}$$
with $$|\vec{r}|=r$$

I know, that the energy eigenvalues of such an potential can be expressed by:

$$E_n = -\frac{E_0}{n^2}$$

where E_0 is the ground state energy.

But how do I count the number of states? Isn't the number of states that are smaller than a specific E<0 infinite?


derivator

 

[nickjer]

It is asking for the number of states $$\Phi(E)$$, so it seems to be energy dependent. I believe they want the number of states below the energy $$E$$.