Here we describe postulates that are axiomatic principles with which quantum mechanics is understood. They came about mainly as a result of experimentation and conclusions made by those who analyzed experimental data. Therefore, they cannot be proven mathematically, though they fit within mathematically rigorous models of real world systems. The following three postulates apply to any _[closed quantum system](Closed%20quantum%20systems.md)_ That is, they apply to any hypothetical [quantum system](Quantum%20systems.md) that is completely isolated from any other other quantum system. 1) A [closed quantum system](Closed%20quantum%20systems.md) is completely described by its [state vector](State%20vector.md). ^b25268 2) There exist [observables](Observable.md), which are operators that each relate a [state vector](State%20vector.md) with a measurable quantity. ^fbbd01 3) The evolution of [closed quantum systems](Closed%20quantum%20systems.md) is modeled deterministically by [equations of motion](Equations%20of%20motion%20for%20closed%20quantum%20systems.md) that are equivalent to [unitary transformations](Unitary%20transformations%20in%20quantum%20mechanics.md) of [state vectors](State%20vector.md) by [time evolution operators](time%20evolution%20operators.md). ^5b8fd6 Any interaction between two systems in quantum mechanics results in a _[quantum measurement](Quantum%20measurement%20(index).md)_, which leads to the emergence [open quantum systems](Open%20quantum%20systems.md). The following two postulates describe this observation. 4) [quantum measurement](Quantum%20measurement%20(index).md)s are probabilistic, following the [[Born rule]]. ^d45426 5) The resulting [state vector](State%20vector.md) following a measurement is defined in terms of a set of _[[measurement operator]]s_. ^5d87a8 These same postulates may be combined are broken down further, however, regardless of how they are enumerated, they are built around three core concepts: that of [state vectors](State%20vector.md), the [evolution of state vectors in time](Quantum%20Dynamics%20(index).md), and [quantum measurement.](Quantum%20measurement%20(index).md) %%There seem to be two roughly equivalent ways of defining the 3rd postulate - either in terms of time evolution operators and equations of motions, or more generally through the notion of defining unitary operators in quantum mechanics in the broadest sense. It is best to define this in terms of time evolution operators since that reflects the physical reality and also any physical realization of a quantum gate, which are unitary operators that are a bit more general than time evolution operators.%% # Possible derivations of the measurement postulates We may in fact argue in favor of [the 4th and 5th axiom](Postulates%20of%20Quantum%20Mechanics.md) based on [the first 3](Postulates%20of%20Quantum%20Mechanics.md), while the first 3 can be considered absolutely fundamental. However, the exact way in which these measurement postulates relate to the first three postulates of quantum mechanics is an open question subject to the [[measurement problem]]. --- # Recommended reading For a concise explanation of these [5 postulates](Postulates%20of%20Quantum%20Mechanics.md) in [quantum mechanics](Quantum%20Mechanics%20(index).md) see: * [McGreevy, John. A., Physics 212A Lecture Notes, Fall 2015.](McGreevy,%20John.%20A.,%20Physics%20212A%20Lecture%20Notes,%20Fall%202015..md), pgs. 19-23. The way we present these postulates here follows most closely from these notes. Additional discussions of the axiomatic principles in quantum mechanics are found below. * [Commins, E.D. _Quantum Mechanics: And Experimentalist Approach_](Quantum%20Mechanics,%20An%20Experimentalist%20Approach%20(E.%20D.%20Commins).pdf), pgs. 28-34. Here the author chooses to enumerate the foundational axioms in quantum mechanics as 7 "rules." * [Nielson, M. A., Chuang, I. L. _Quantum Computation and Quantum Information_, Cambridge University Press, 2010](Nielsen,%20M.%20A.,%20Chuang,%20I.%20L.%20Quantum%20Computation%20and%20Quantum%20Information,%20Cambridge%20University%20Press,%202010.md) pgs. 80-97. Here the authors choose to enumerate these axioms as 2 postulates pertaining to isolated systems and a 3rd postulate pertaining to the idea of [quantum measurement,](Quantum%20measurement%20(index).md) which is is an important focus for Nielson and Chuang given its role in quantum computation. * [Shankar, R., _Principles of Quantum Mechanics_, Plenum Press, 2nd edition, 1994.](Shankar,%20R.,%20Principles%20of%20Quantum%20Mechanics,%20Plenum%20Press,%202nd%20edition,%201994..md), pgs. 115-122. Here four postulates are presented - three pertaining to state vectors, observables, and time evolution, and one pertaining to measurement. These postulates are compared side by side with corresponding postulates from classical physics. #QuantumMechanics/FoundationsOfQuantumMechanics #QuantumMechanics/MathematicalFoundations