Energy and states of a particle in a box.

[Higgsono]

Suppose we have a classical particle in box. The number of degrees of freedom is 6. The position of the particle and its momenta.

Now if we want to calculate the entropy of the system as a function of the energy we only need to find a relation between all the possible states the particle can be in and its kinetic energy.

The possible states in momentum space is proportional to the surface area of a sphere with radius p, where E=p^2/(2m).

But what I don't understand is this: Why do the number of possible states in momentum space increase with an increase in the kinetic energy of the particle? Sure, the area of the sphere increses with an increase in radius. But why can't we just say that the number of possible states is proportional to the direction of the vector p from the origin? In this case the number of directions which would be the same as the number of possible states of p does not increase with an increase in energy. The number of possible directions is the same independent of the magnitude of p.

So why do we say that the number of possible states increases when the particles momentum increases?

 

[Orodruin]

Essentially because the phase space volume element is $dp_1dp_2dp_3dV$. Going to spherical coordinates in the momentum variables gives you a factor $p^2$ from the Jacobian determinant.

In essence, the reason is therefore the same as why a sphere with larger radius has a larger surface area.

 

[Higgsono]

Essentially because the phase space volume element is $dp_1dp_2dp_3dV$. Going to spherical coordinates in the momentum variables gives you a factor $p^2$ from the Jacobian determinant.

In essence, the reason is therefore the same as why a sphere with larger radius has a larger surface area.

But the number of states should be equal to the number of positions times the number of directions in which the particle can move through this point. And since an increase in momentum doesn't increase the number of possible directions, the number of possibles states should not increase either.

 

[Orodruin]

But the number of states should be equal to the number of positions times the number of directions in which the particle can move through this point. And since an increase in momentum doesn't increase the number of possible directions, the number of possibles states should not increase either.

This is just not true. If you look at what the phase space volume actually is, you get the factor of $p^2$. The "number of directions" is a handwaving argument with no basis in the mathematics. If you do the math right using the actual phase space volume you get the right result.