Z for classical particles in B-field

[cscott]

Homework Statement

Show that the free energy of classical particles with no internal magnetic moment is always independent of magnetic field. Hint: Write down Z for N classical particles. Let the particles interact by U which depends only on the positions of the interacting particles. Show that the magnetic field dependence can be eliminated from the partition function by a change of variable.

Homework Equations

Partition function
$$Z = \frac{1}{N!} = \int \exp \left (-\beta H \right) d^3p_1..d^3p_Nd^3x_1..d^3x_N$$

Free energy:
$$\Omega = -\frac{\ln(Z)}{\beta}$$

The Attempt at a Solution

I would start by trying to write down H
$$H = \sum_n \left [ \frac{\vec{p_n}^2}{2m_n} + U(\vec{r_n}) \right]$$

but I don't know how to make any more use of that hint.

 

[fzero]

If there's no magnetic moment, the particles couple to the magnetic field through the momentum term via

$$H = \sum_n \left [ \frac{(\vec{p_n}-q\vec{A})^2}{2m_n} + U(\vec{r_n}) \right]$$

where $$\vec{A}$$ is the scalar potential and $$q$$ is the charge of a single particle.

 

[cscott]

So because U has nothing to do with the magnetic field, I need to eliminate the vector potential from,

$$\Pi_n \int \exp \left [-\frac{\beta}{2m_n} (\vec{p}_n - q \vec{A})^2 \right] d^3 p_n$$

and change variable,

$$\vec{u}_n = \vec{p_n}-q\vec{A}$$

thus eliminating vector potential?

Are the end points of the integral +/-infinity?