The notion _quantum superposition_ follows directly from the [superposition principle](Superposition%20principle.md) in classical mechanics, which itself follows from [the superposition for vectors in linear algebra,](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Superposition%20principle) since quantum [states](State%20vector.md) are also vectors. Thus if [state vectors](State%20vector.md) $|\psi_1\rangle$ and $|\psi_2\rangle$ are possible quantum states that describe a particular [[Pure state]] then $a|\psi_1\rangle+b|\psi_2\rangle$ is also a possible quantum state where $a=|a|e^{i\phi_1}\;\; \mathrm{and}\;\; b=|b|e^{i\phi_2}.$ ^664d95 Here $a$ and $b$ are [local phase factors.](State%20vector.md#Phases%20of%20state%20vectors) The existence of such factors follows from the fact that for a any single state vector we may write [$|\psi \rangle \simeq c|\psi \rangle$](State%20vector.md#^065a4b) where $c\in \mathbb{C}$ is a global phase factor. Thus we may define, for example, a [two-state system](Two-Level%20Systems.md)$|\psi\rangle = c_1|\psi_1\rangle+c_2|\psi_2\rangle$ where $c_1, c_2 \in \mathbb{C}$ and $|c_1|^2+|c_2|^2=1.$^687da0 or indeed any higher order superposition $|\psi\rangle = c_1|\psi_1\rangle+c_2|\psi_2\rangle+c_3|\psi_2\rangle+...,$ which may describe an [n-level system](n-level%20system) where likewise $c_1, c_2, ... c_n \in \mathbb{C}.$ and $\sum_{c=1}^{c=n} |c_i|=1.$ Here we've yet to say anything about the [physical interpretation](Quantum%20superposition.md#Physical%20interpretation) of a quantum superposition, which differs significantly from superposition in classical physics in that it is derived from the observation of [randomness in quantum measurements.](Quantum%20measurement%20(index).md#Randomness%20in%20quantum%20measurements) However, as with a classical superposition, we are modeling our system as sum of vectors. # Physical interpretation ## Measurement [measurement on a quantum superposition](measurement%20on%20a%20quantum%20superposition.md) %%You should state the relationship between those local phase factors and measurement probabilities probably.%% # Infinite order superpositions [coherent state](coherent%20state.md) %%This may be misleading, momentum and position states are also continuous, meaning they have infinite terms. this has to be explained along with coherent states.%% # Ensembles of quantum superpositions The [[density matrix]] of a quantum superposition is that of a [pure state.](Pure%20state.md) --- # Recommended reading For a concise mathematical summary of the [superposition principle in quantum mechanics](Quantum%20superposition.md) see: * [McGreevy, John. A., Physics 212A Lecture Notes, Fall 2015.](McGreevy,%20John.%20A.,%20Physics%20212A%20Lecture%20Notes,%20Fall%202015..md) For an more in-depth discussion that delves into the physical meaning of [Quantum superposition](Quantum%20superposition.md) see: * [Dirac, P. A. M., _The Principles of Quantum Mechanics_,]([International%20series%20of%20monographs%20on%20physics]%20P.%20A.%20M.%20Dirac%20-%20Principles%20of%20Quantum%20Mechanics,%20The%20(1978,%20Oxford%20University%20Press)%20-%20libgen.lc.pdf), Oxford Univ Press, Revised 4th edition (1978), pgs. 4-18. Here Paul Dirac discusses the physical meaning of superposition grounded in an experimental example followed by the mathematical formulation. #QuantumMechanics/StationaryStateQuantumSystems #QuantumMechanics/QuantumStateRepresentations/StateVectors