According to all _modern_ mathematical formulations of quantum mechanics, quantum systems are modeled with [state vectors.](State%20vector.md) (For a discussion of "pre-modern" formulations see [old quantum theory.](Quantum%20Mechanics%20(index).md#Old%20Quantum%20Theory)) ^587867 There are several standard [non-relativistic formulations](Mathematical%20Formulations%20of%20Quantum%20Mechanics.md#Non-relativistic%20formulations%20of%20quantum%20mechanics) of quantum mechanics that have been shown to be mathematically equivalent and thus each allow us to extract the same information associated with a state vector. All mathematical formulations rely on the same [postulates](Postulates%20of%20Quantum%20Mechanics.md). In order to fully describe quantum systems, these formulations are extended in [quantum field theory.](Mathematical%20Formulations%20of%20Quantum%20Mechanics.md#Quantum%20Field%20Theory) # Non-relativistic formulations of quantum mechanics The [modern](Mathematical%20formulations%20of%20Quantum%20Mechanics#^587867) non-relativistic formulations of quantum mechanics are as follows and are given below roughly in chronological order of their development. The first couple are also typically where we start in our understanding of modern quantum theory when teaching it: * the initial formulation by Schrödinger where we model [quantum systems](Quantum%20systems.md) as [states](State%20vector) and [wavefunction](Wavefunction.md)s which are solutions to [Schrödinger's equation](Quantum%20Mechanics/Quantum%20Dynamics/Schrödinger%20Equation.md). This also gives rise to the [Schrödinger picture](Schrödinger%20picture) in [quantum dynamics.](Quantum%20Dynamics%20(index).md) * the formulation by Heisenberg where we describe the [quantum measurement](Quantum%20measurement%20(index).md) in terms of [hermitian operators](Observable) being applied to [state vectors](State%20vector.md) and [time evolution](time%20evolution%20operators.md) as [unitary operators](time%20evolution%20operators.md#Derivation) being applied to state vectors. Dirac extended this formulation with the introduction of a dynamical equation known as the [Heisenberg equation of motion.](Heisenberg%20Equation%20of%20Motion.md) This formulation gives rise to the [Heisenberg picture](Heisenberg%20picture) in quantum dynamics. The following formulations still give equivalent results to the ones above, but may be more useful for specific problems with complicated conditions rather than as a way to start using quantum mechanics: * the [path integral formulation](Path%20integral%20formulation) by Feynman, which gives the [probability amplitude](probability%20amplitude) in terms of an weighted sum of trajectories. This formulation becomes foundational when extended to [quantum field theory.](Mathematical%20Formulations%20of%20Quantum%20Mechanics.md#Quantum%20Field%20Theory) * the _phase-space formulation,_ developed independently by Moyal^[[Moyal J. E., _Quantum mechanics as a statistical theory_, Mathematical Proceedings of the Cambridge Philosophical Society, 1949.](Moyal%20J.%20E.,%20Quantum%20mechanics%20as%20a%20statistical%20theory,%20Mathematical%20Proceedings%20of%20the%20Cambridge%20Philosophical%20Society,%201949..md)] and Groenwold^[[Groenwold, H.J., _On the principles of elementary quantum mechanics_, Physica, vol. 12., 1946.](Groenwold,%20H.J.,%20On%20the%20principles%20of%20elementary%20quantum%20mechanics,%20Physica,%20vol.%2012.,%201946..md)], which models [quantum systems](Quantum%20systems.md) as [quasiprobability distributions](Phase%20space%20distributions%20(quantum%20mechanics).md#Quasiprobability%20distributions) in [phase space.](Phase%20space%20distributions%20(quantum%20mechanics).md) This formalation used to model certain systems in [quantum optics](Quantum%20Optics%20(index).md) and [many body quantum physics.](Multi-particle%20quantum%20systems%20(index).md) %%since there are names of people all of these must have footnote citations.%% # Quantum Field Theory The [above](Mathematical%20Formulations%20of%20Quantum%20Mechanics.md#Non-relativistic%20formulations%20of%20quantum%20mechanics) formulations alone are limited in two important ways: * They cannot account for [special relativity](Special%20Relativity%20(Index).md) * Modeling interactions between two or more particles becomes a major computational challenge. %%It would be interesting to show how much of a challenge it becomes. There was a UMass problem set that aimed to do that.%% Therefore we need to add something to these [non-relativistic formulations](Mathematical%20Formulations%20of%20Quantum%20Mechanics.md). That is, we need to treat quantum particles as [quantized fields](Field%20Quantization.md) while expanding the above formulations to account for that. This is the purpose of [quantum field theory.](Quantum%20Field%20Theory%20(Index)) #QuantumMechanics/MathematicalFoundations