# Planck's assumption on blackbody radiation

I have just started basic quantum mechanics and have come across this expression for energy density of blackbody radiations; Planck's formulation. $$u_{\nu} \ d\nu = g(\nu) \ \langle E \rangle \ d\nu$$

The textbook says that:

To model the absorption and emission of electromagnetic radiation by these oscillators, he argued that the spectral energy density of the black body radiation over a range of frequency dv is nothing but the product of the number of oscillators having a frequency between v and v + dv , g(v) and the average energy ⟨E⟩, of an oscillator having a frequency v :

What I don't understand is: if u(v) is the energy density (per unit v), isn't $$u(\nu)\ d\nu$$ the total energy instead of energy density? And one more question; if $$g(\nu)$$ is the number of oscillators having frequency in the range $$d\nu$$, and $$⟨E⟩$$ is the average energy; then isn't their product the total energy instead of being the energy density, and if that is supposed to be so then why do we have a $$d\nu$$ part on the right side of equation also?

## 1 Answer

$$g(\nu)$$ would be the number density of oscillators per unit frequency at a frequency $$\nu$$.

$$u(\nu)\ d\nu$$ would be the energy density in a frequency band $$d\nu$$ at a frequency $$\nu$$.

The $$d\nu$$ is written on both sides of the equation since otherwise you are saying $$0 = 0$$, because no oscillators have a frequency of exactly $$\nu$$.