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Very early and important scientific equations in the hand of Albert Einstein related to his 1925 paper, "Quantum Theory of the Ideal Gas", a foundational publication in the history of quantum mechanics which not only established the field of quantum statistics (called Bose-Einstein Statistics), but also predicted Bose-Einstein Condensation and laid the groundwork for Erwin Schrodinger's quantum wave theory a year later in 1926.
In "Quantum Theory of the Ideal Gas", Einstein applies to atoms Louis de Broglie's then radical hypothesis of the nature of photons: that they exhibit both static and wave-like properties. This would lead Erwin Schrodinger to formulate the Schrodinger Equation in 1926, considered one of the most important tenets of quantum mechanics. Einstein also theoretically predicts the quantum state of atoms at near zero temperatures in this paper, a hypothesis which wouldn't be proven in a laboratory setting for 70 years, earning the three physicists who confirmed Einstein's hypothesis a Nobel Prize in 2001. This discovery has led to modern technologies like atomic clocks, quantum sensors, and laser cooling technologies.
In addition to the equations, Einstein handwrites the following on this page, "Taking into account the particle's equations of motion [equations] / From this follows, taking into account the particle's equations of motion [equations] / in the known way [equation] / p is therefore constant along the path. In addition, because p can only contain the p, in the combination L, due to the condition that the distribution in the thermodynamic equilibrium should be an isotope, it must be possible to express p in the form [equation] / Because equilibrium distributions prevail at our gas's various locations, corresponding to the various values of the volume V at the same temperature, a relationship will thus exist between II and V when the temperature is held fixed."
On the other side of the same sheet, Einstein writes and then signs a draft letter to the Reichsdruckerei, the German state printing office who published many of Einstein's papers. From Berlin on 20 February 1925, Einstein writes in German, translated, "Please find enclosed what you wished for. I had overlooked the proofs because they arrived during my absence. There is one more paper of mine in press, for which I ask to get the proofs quickly, as I will have to be away for three months from the beginning of March. In this connection, I would like to note that the slowness with which the Academy's proceedings are currently being printed can scarcely be endured any longer. I am seriously considering complaining to the Academy if there is no improvement. / Sincerely, / A. Einstein". Einstein then crosses out the missive, perhaps changing his mind about sending it.
Single sheet measures 8.5'' x 11''. Folds, partially affecting Einstein's signature, toning and scattered foxing with small holes. Very good condition.