Generalized Einstein relations between absorption and emission spectra at thermodynamic equilibrium

Significance Einstein’s relationships between absorption and emission conflict with the time-energy uncertainty principle by ascribing a finite radiative lifetime to the upper state of an infinitely narrow spectroscopic line. We derive generalized Einstein relations for transitions between broadened levels that are compatible with the uncertainty principle, detailed balance, and causality. For transitions between two bands, four Einstein coefficient spectra replace the three Einstein coefficients. A single underlying lineshape generates the four different spectra. The change in thermodynamic standard free energy between two bands can be determined from absorption and emission spectra, enabling measurement of excited state thermodynamic properties with spectroscopic accuracy.