  The space- and temperature-dependent electron distribution $n(\mathbf r,T)$
is essential for the theoretical description of the opto-electronic properties
of disordered semiconductors. We present two powerful techniques to access
$n(\mathbf r,T)$ without solving the Schr\"odinger equation. First, we derive
the density for non-degenerate electrons by applying the Hamiltonian
recursively to random wave functions (RWF). Second, we obtain a
temperature-dependent effective potential from the application of a universal
low-pass filter (ULF) to the random potential acting on the charge carriers in
disordered media. Thereby, the full quantum-mechanical problem is reduced to
the quasi-classical description of $n(\mathbf r,T)$ in an effective potential.
We numerically verify both approaches by comparison with the exact
quantum-mechanical solution. Both approaches prove superior to the widely used
localization landscape theory (LLT) when we compare our approximate results for
the charge carrier density and mobility at elevated temperatures obtained by
RWF, ULF, and LLT with those from the exact solution of the Schr\"odinger
equation.
