Ground state energy of a two particle gravitational atom

[quasar_4]

Homework Statement

Two neutral spinless particles of mass m are gravitationally bound to one another. What is the ground state energy of this two-particle gravitational atom?

Homework Equations

The Attempt at a Solution

So, it's a two particle system, but

$$H_{total} = H_1 + H_2$$ and $$\psi_{12} = \psi_1 \psi_2$$ implies

$$H_{total} \psi_{12} = (H_1 + H_2) \psi_1 \psi_2 = (E_1 + E_2) \psi_1 \psi_2$$.

So I should be able to find the energy eigenvalues for one of the particles since they're identical, then (assuming spinless = zero spin = bosons), I should just be able to add the two ground state energies together.

My Hamiltonian for a single particle is

$$\frac{p^2}{2m} - \frac{Gm^2}{x^2} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x ^2} - \frac{Gm^2}{x^2}$$

so I get an ODE of the form

$$\frac{\partial^2\psi}{\partial x^2} - \frac{2 G m^3}{\hbar^2 x^2}\psi + \frac{2 mE}{\hbar^2} \psi = 0$$

What I don't see is how on Earth energy is quantized. I guess I am under the impression that to quantize any observable, we need boundary conditions (i.e., isn't it usually the presence of a boundary condition that causes quantization)? In any case, for a bound state, the total energy must be less than the potential for the system... but how do I fit that into this problem?

Or, option 2: have I set it up totally wrong? I could try to use the center of mass frame, in which things might start to look more like the hydrogen atom. Maybe I'll work on that until this gets some replies.

 

[kuruman]

What makes you think this is a one-dimensional problem? I would treat it in 3-d, just like the hydrogen atom, except that the constants are different.