3D Cavity Standing Waves & Calculated Degrees of Freedom

Exact Degree of Freedom Mathematical Formulas for 3D Cavities

• 1. Spatial Standing Waves in 3D: Inside a cubic cavity $[L \times L \times L]$, boundary conditions require $k_x = \frac{n_x\pi}{L}$, $k_y = \frac{n_y\pi}{L}$, $k_z = \frac{n_z\pi}{L}$. Unlike 1D points or 2D lines, nodes in 3D are entire 2D planes slicing through the cavity where $E = 0$ constantly. The volume partitions into $n_x \times n_y \times n_z$ oscillating cells.
• 2. Polarization Multiplicity ($g_{\text{pol}} = 2$): Electromagnetic radiation is transverse ($\nabla \cdot \vec{E} = 0$). For every wavevector $\vec{k}$, there are two independent orthogonal transverse polarization states ($\vec{\varepsilon}_1 \perp \vec{\varepsilon}_2 \perp \vec{k}$).
• 3. Total Degrees of Freedom per Mode ($f = 4$): Each polarization acts as a harmonic oscillator with 2 quadratic energy terms (electric energy $\propto \iiint \varepsilon_0 E^2 dV$ and magnetic energy $\propto \iiint \mu_0^{-1} B^2 dV$). Therefore, each spatial triple $(n_x, n_y, n_z)$ accounts for exactly $2 \times 2 = 4$ degrees of freedom, carrying an average thermal energy of:
  \langle E_{(n_x, n_y, n_z)} \rangle = 4 \times \left(\frac{1}{2} k_B T\right) = 2 k_B T
• 4. 3D Mode Density Derivation: Valid modes occupy the positive octant ($1/8$th) of a sphere of radius $k$ in 3D $k$-space:
  N(k) = 2 \times \frac{1}{8} \times \frac{\frac{4}{3}\pi k^3}{(\pi / L)^3} = \frac{V k^3}{3\pi^2} \implies g(\nu)d\nu = \frac{8\pi \nu^2}{c^3} V d\nu This quadratic growth $g(\nu) \propto \nu^2$ is the root of the ultraviolet catastrophe when classical equipartition assigns constant energy per degree of freedom across all frequencies.