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Chapter 7: Q 7.40 (page 293)

Starting from equation 7.83, derive a formula for the density of states of a photon gas (or any other gas of ultra relativistic particles having two polarisation states). Sketch this function.

Short Answer

Expert verified

Hence, the formula for density of states of a photon gas isg(ϵ)=8πVϵ2(hc)3

Step by step solution

01

Given information

The equation 7.83 is

UV=8π(hc)3∫0∞ϵ3eϵ/kT-1dϵ

02

Explanation

The equation 7.83 is:

UV=8π(hc)3∫0∞ϵ3eϵ/kT-1dϵ

We can write the equation as:

localid="1647752992962">U=∫0∞ϵ8πVϵ2(hc)31eϵ/kT-1dϵ(1)

Distribution function for Planck's constant is given as:

n¯Pl=1eϵ/kT-1

Substituting this into (1)

U=∫0∞ϵ8πVϵ2(hc)3n¯Pldϵ

Hence the energy density for Planck's constant is

g(ϵ)=8πVϵ2(hc)3

Using Python to solve this function, the code is:

The graph is:

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Most popular questions from this chapter

In analogy with the previous problem, consider a system of identical spin‐0bosonstrapped in a region where the energy levels are evenly spaced. Assume that Nis a large number, and again let qbe the number of energy units.

(a) Draw diagrams representing all allowed system states from q=0up to q=6.Instead of using dots as in the previous problem, use numbers to indicate the number of bosons occupying each level.

(b) Compute the occupancy of each energy level, for q=6. Draw a graph of the occupancy as a function of the energy at each level.

(c) Estimate values of μand Tthat you would have to plug into the Bose-Einstein distribution to best fit the graph of part(b).

(d) As in part (d) of the previous problem, draw a graph of entropy vs energy and estimate the temperature at q=6from this graph.

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(b) Calculate the condensation temperature, and compare kTctoε0.

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For a system obeying Boltzmann statistics, we know what μis from Chapter 6. Suppose, though, that you knew the distribution function (equation 7.31) but didn't know μ. You could still determine μ by requiring that the total number of particles, summed over all single-particle states, equal N. Carry out this calculation, to rederive the formula μ=-kTlnZ1/N. (This is normally how μ is determined in quantum statistics, although the math is usually more difficult.)

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(c) Sketch (or use a computer to plot) the spectrum of light given off by the filament. Indicate the region on the graph that corresponds to visible wavelengths, between400and700nm.

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( e) To increase the efficiency of an incandescent bulb, would you want to raise or lower the temperature? (Some incandescent bulbs do attain slightly higher efficiency by using a different temperature.)

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