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Kelly and Leff demonstrated and discussed formal and conceptual similaritiesbetween basic thermodynamic formulas for the classical ideal gas and black bodyphoton gas. Leff pointed out that thermodynamic formulas for the photon gascannot be deduced completely by thermodynamic methods since these formulas holdtwo characteristic parameters, {\it r} and {\it b}, whose accurate values canbe obtained exclusively by accurate methods of the quantum statistics (byexplicit use of the Planck's or Bose-Einstein distribution). In this work weprove that the complete quantum thermodynamics of the black body photon gas canbe done by simple, thermodynamic (non-statistical) methods. We prove that bothmentioned parameters and corresponding variables (photons number and pressure)can be obtained very simply and practically exactly (with relative error aboutfew percent), by non-statistical (without any use of the Planck's orBose-Einstein distribution), quantum thermodynamic methods. Corner-stone ofthese methods represents a quantum thermodynamic stability condition that is,in some degree, very similar to quantum stability condition in the Bohr quantumatomic theory (de Broglie's interpretation of the Bohr quantization postulate).Finally, we discuss conceptual similarities between black body photon gasentropy and Bekenstein-Hawking black hole entropy.

Complete Quantum Thermodynamics of the Black Body Photon Gas