Black bodies, energy per unit volume

Question

While reading about black bodies, I found that the average energy per mode per unit volume is $KT$ classically, but I don't understand why this is per unit volume. The linear oscillators are considering to be in the walls of a cavity modeling a blackbody, so shouldn't it be modes per area?

Answer 1

The harmonic (not linear) oscillators are living in the volume: they are Fourier modes of the electromagnetic fields like $$\int dx\,dy\,dz\,\exp(-i\vec k\cdot \vec x) \vec E(x,y,z) $$

For a periodic cubic box of volume $L_xL_yL_z$, the wave vector $\vec k =(k_x,k_y,k_z)$ belongs to a lattice that has spacings $2\pi/L_x$ etc. in the three directions, respectively, for the complex exponential to be single-valued. For a box with walls, $\exp(ik_x x)$ is simply replaced by $\sin(kx)$ etc. but the counting doesn't change much (except for factors of two and the equivalence between $k$ and $-k$).

You see that in the limit of a large box, the lattice in the $\vec k$-space becomes infinitely dense and summing over allowed values of $\vec k$ in the lattice becomes the same thing as integrating over $d^3 k$ but divided by the volume of the "cell" in the $k$-space which is $(2\pi)^3/(L_xL_yL_z)$. Dividing by that is the same thing as multiplying the integral by $$\frac{L_x L_y L_z}{(2\pi)^3}$$ So the energy stored in the harmonic oscillators etc. is obtained by integrating over the $\vec k$-space with a simple multiplicative factor of the volume in the $x$-space.

Walls aren't supposed to carry energy energy in the black body. Well, they can if it is a thick wall in thermal equilibrium with the radiation but the black-body radiation is computed from the thermal state of the electromagnetic field itself, not from the walls.

Answer 2

There's another twist on Lubos's answer (since his amazing answers are likely to be upvoted more than mine, you'll likely need to look above rather than below for it :) ) that I find extremely helpful and it certainly shows that energy per unit volume rather than area is the quantity that makes sense.

In second quantised field theories, there are two "divergences to infinity" that one might find offputting (at least I did at first) to do with the quantum ground state. One thinks of the universe as a "box" and then the (nonrelativistically) second quantised field is essentially a collection of nonrelativistic quantum harmonic oscillators, one for each of the box's modes with wavevector $(k_x, k_y, k_z)$ as in Lubos's answer. As you think of the box getting bigger and bigger, the packing density in momentum ($\mathbf{k}$)-space gets denser and denser.

Now the quantum ground state of each oscillator of frequency $\omega$ has energy $\hbar \omega / 2$ so that it fulfills the Heisenberg uncertainty principle. Therefore, the total ground state energy for all the modes with frequency less than some "cutoff" frequency $\omega_0$ in a box of sidelengths $(L_x, L_y, L_z)$ is:

$$\sum\limits_{4\pi^2\,c^2\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\right) <\omega_0^2} \hbar \;c\;\sqrt{\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}}$$

Now this of course diverges as the mode spacing in momentum space approaches nought and we replace the ground state energy sum with an integral, even if we limit the frequencies to ones below a cutoff $\omega_0$! Oh dear: Planck makes his postulate to clear up an ultraviolet catastrophe in Rayleigh Jeans theory only to lead ultimately to another divergence.

But this particular divergence IS altogether physically reasonable, because the divergent energy is in proportion to the box volume. The energy "cost" per unit volume to set up a quantum electromagnetic field is a perfectly well defined, finite number. Imagine being some god who wants to create a universe: so you log into, say Odin's account, at "myuniverse.com" and it will tell you you need to pay an amount of energy to establish a quantum electromagnetic field in proportion to the volume of the universe you want to build! Less flippantly, modern physics doesn't believe empty space is empty: space IS the quantum fields that fill it. If the universe is big, then the packing density of its momentum space is high and therefore so is the energy needed to establish it.

Notice how you would not have this reasonable physical interpretation for the divergence if the blackbody radiators had energy per unit surface area of the box.

The other divergence one comes across is the ultraviolet divergence one lets $\omega_0\to\infty$. This is handled by "renormalisation" procedures, which are a way of asserting that we dont really know what happens at very short length scales, so we set up our calculations so that other known physical data can implicitly set the integral value as $\omega_0\to\infty$. It is an assertion of our ignorance so far, and means that other physical data (such as electron self energies) must be put into the theory by hand rather than naturally being derived from it.