Is the Divisor (2π)^3 Necessary in Photon Gas Equations?

[Sky Darmos]

Hello everybody,

In this Wikipedia article we find an equation for a photon gas which contradicts an equation given by Stefan Weinberg in his book "The first three minutes":

https://en.wikipedia.org/wiki/Photon_gas
The equation given here has 16 π k^3 ζ(3) T^3 in the numerator and c^3 h^3 (2 π)^3 in the denominator.
In Weinberg's version of this "(2 π)^3" is missing. Obviously "h^3 (2 π)^3" is "h_quer".
The difference between the two equations is significant.
There is a discussion going on on this Wikipedia page about if this divisor "(2 π)^3" is necessary or not.
I don't understand how this has been noticed since 2011, but nobody made the afford to correct or clarify.

I hope you can help me to end this confusion.

Thanks in advance.

 

[Orodruin]

Please don’t write parts of an equation separately and ask people to put the pieces together. Write out the actual equations (both of them!)

 

[Sky Darmos]

Here are the equations as picture files:

 

[Ibix]

You can use LaTeX in this forum - see the link below the reply box. Your formula is $$\frac{16 \pi k^3 \zeta(3) T^3}{ c^3 h^3 (2 \pi)^3}$$and you say the $(2\pi)^3$ is not present in Weinberg's version.

 

[kimbyd]

A simple way to approach the issue is to recognize that this problem is likely all about confusion of the use of $\hbar$ vs. $h$. If you replace $h$ with $\hbar$ in one of the two equations, they will agree with one another. Only one choice will work correctly.

Which means it looks like Weinberg's version is probably the right one. You'd have to check the derivation to be sure, however.

 

[Orodruin]

I think it is pretty clear that someone just forgot to change the $h$ to an $\hbar$ in Wikipedia. All other equations in that table are using $\hbar$ and not $h$.

Kolb-Turner, which I trust, has
$$
n = \frac{\zeta(3) gT^3}{\pi^2}
$$
in natural units ($k = c = \hbar = 1$), which agrees with the Wikipedia entry if you take $h \to \hbar$ there.

See kids, this is why you should not take Wikipedia on face value.

 

[Sky Darmos]

Oh, I see, they tried to use ℏ everywhere, that is why they had to add the (2π)^3 here.
Makes sense, thank you Orodruin.