Some books like Griffith's begin quantum mechanics with the Schrödinger equation as a postulate while some other text books derive it and state $[x,p]=i \hbar$ as an axiom. I'm not sure which one came first as the operator $\hat{p}$ can be derived from both. Also I'd like to know how did the commutation relation came about because the textbooks I have don't mention about it.
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2$\begingroup$ Basically, Matrix mechanics formulation of quantum mechanics, due to Werner Heisenberg, Max Born, and Pascual Jordan in 1925, and wave mechanics formulated in late 1925, and published in 1926, by Erwin Schrödinger, were independent mathematical formulations of q.m. 1/2 $\endgroup$– Mauro ALLEGRANZADec 7, 2015 at 18:45
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$\begingroup$ ... "Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to Paul Dirac, who wrote a lucid account in his 1930 classic The Principles of Quantum Mechanics. " 2/2 $\endgroup$– Mauro ALLEGRANZADec 7, 2015 at 18:45
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If you are asking "which came first" historically, the answer is clear:
Commutation relation was written for the first time in the paper of Born and Jordan, On Quantum Mechanics, submitted on September 27, 1925. It is formula (38) in this paper. English translation is available in Van der Waerden's book, Sources on quantum mechanics.
Schrodinger equation was published in Annalen der Physik by Scrodinger in his paper Quantisierung als Eigenwertproblem, submitted in January 1926. (I don't know whether there is an English translation).
If you are asking "what comes first" logically, it depends on the system of exposition. In every theory we are free what to take as axioms and how to develop it from the axioms. In most modern expositions, Schrodinger equation is taken as an axiom. The system where the commutation relation "comes first" is called "matrix" mechanics, and you can read about it in Dirac'a book.
answered Dec 7, 2015 at 22:35
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$\begingroup$ Is there a derivation for the commutation relation in his book? $\endgroup$– WeezyDec 8, 2015 at 8:29
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2$\begingroup$ Axioms are not derived, by definition. There are physical reasons behind the "axioms" in physical theories. $\endgroup$ Dec 8, 2015 at 13:49
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$\begingroup$ I believe that the non trivial commutation relation between x and p appeared first in the seminal Heisenberg's work which is previous to Born-Jordan's one. Perhaps in a disguised form and certainly with unclear interpretation. $\endgroup$ Nov 19, 2017 at 1:21