Consider an ideal gas of highly relativistic particles

Chapter 6, Problem 52P

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QUESTION:

Consider an ideal gas of highly relativistic particles (such as photons or fast-moving electrons), whose energy-momentum relation is E = pc instead of \(E=p^{2} / 2 m\). Assume that these particles live in a one-dimensional universe. By following the same logic as above, derive a formula for the single-particle partition function, \(Z_1\), for one particle in this gas.

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QUESTION:

Consider an ideal gas of highly relativistic particles (such as photons or fast-moving electrons), whose energy-momentum relation is E = pc instead of \(E=p^{2} / 2 m\). Assume that these particles live in a one-dimensional universe. By following the same logic as above, derive a formula for the single-particle partition function, \(Z_1\), for one particle in this gas.

ANSWER:

Step 1 of 3

Given energy-momentum relation of the ideal gas

For a one dimensional universe, the allowed momenta are

We have the relation between the energy and momentum

So the allowed energies are

 

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