A complete description of a [quantum system](Quantum%20systems.md) may be given either as a single [vector](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vectors) or multiple vectors in a [Hilbert Space.](Hilbert%20Spaces%20in%20Quantum%20Mechanics.md) These vectors are called _state vectors._ State vectors are written in [bra-ket notation](Quantum%20Mechanics%20(index).md#Bra-ket%20notation) where by default we use the convention of defining them in terms of _kets_.
State vectors are in an [equivalence class](Equivalence%20class.md), $|\psi \rangle \simeq c|\psi \rangle$ where $c\in \mathbb{C}$ is a [global phase factor](State%20vector.md#Phases%20of%20state%20vectors). ^065a4b
Otherwise:
# Phases of state vectors
For a perfectly isolated system, one may freely include a _global phase factor_,
$c=|c|e^{i\phi}, \;\;\mathrm{where}\;\; c\in\mathbb{C}.$
A global phase factor has no physical significance on its own. However, if the state vector comes into contact with a larger quantum system or exists in a [Quantum superposition](Quantum%20superposition.md) with another state, the phase factor would be local relative to the new state vector describing the broader system. _local phase factors_ (or _relative_ phase factors) do have physical significance since they exist alongside other local phases, and thus need to be accounted for.
%%Give an example here of why that's the case.%%
# Eigenstate decomposition of state vectors
State vectors almost always can be decomposed into eigenstates that form an [orthonormal basis](Orthonormal%20bases.md) and such eigenstates each correspond to an [observable](Observable.md) quantity. Such states are elaborated on [here.](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates) There are a few exceptions, which are discussed [here.](State%20vector.md#non-orthonormal%20states)
These [eigenstate decompositions](State%20vector.md#Eigenstate%20decomposition%20of%20state%20vectors) may be either [discrete](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates) or [continuous,](State%20vector.md#State%20vectors%20with%20continuous%20eigenstates) corresponding to eigenstates that belong in [finite](Hilbert%20space%20dimension%20in%20quantum%20mechanics.md#Finite%20dimensional%20Hilbert%20spaces%20in%20quantum%20mechanics) and [infinite dimensional](Hilbert%20space%20dimension%20in%20quantum%20mechanics.md#Infinite%20dimensional%20Hilbert%20spaces%20in%20quantum%20mechanics) [Hilbert spaces](Hilbert%20Spaces%20in%20Quantum%20Mechanics.md) respectively. [Below](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates) we first introduce this notion first for [discrete](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates) decompositions and then we extend this notion for [continuous sets.](State%20vector.md#State%20vectors%20with%20continuous%20eigenstates)
## Decomposing state vectors into sets of orthonormal eigenstates
A state vector, $|\psi\rangle,$ that is an [eigenvector](Eigenvalues%20and%20eigenvectors.md) of a particular [observable](Observable.md) and can be written in terms of an [orthonormal basis](Orthonormal%20bases.md), $\{|a_i\rangle\}$, consisting of _normalized_ eigenvectors that are themselves also state vectors called _eigenstates_. This means that for any pair, $|a_i\rangle, |a_j\rangle$ in that set, $\langle a_i|a_j\rangle =\delta_{a_i,a_j}$ where $\delta_{a_i,a_j}$ is a [Kronecker delta.](Kronecker%20delta.md) ^ba778b
Here we start by considering situations where we have discrete decompositions rather than decompositions where [the eigenstates exist on a continuum.](State%20vector.md#State%20vectors%20with%20continuous%20eigenstates) The decomposition of $|\psi\rangle$ takes he form of the following sum: $|\psi\rangle=\sum_i |a_i\rangle\langle a_i|\psi\rangle = \sum_i c_i|a_i\rangle.$Here $|a\rangle$ are state vectors representing possible states the physical system can be in following a [measurement](Wave%20function%20collapse). In the above sum that gives $|\psi\rangle$ they form a [[Quantum superposition]]. The [inner product](Inner%20products.md), $c_i=\langle a_i|\psi\rangle$ is a _complex [[probability amplitude]]_ associated with a particular quantum state. Notice the [completeness relation](State%20vector.md#State%20vector%20completeness%20relation) relation appears in the intermediary expression. ^248983
### State vector completeness relation
Here $\sum_i |a_i\rangle\langle a_i|=\mathbb{1}$ is the [completeness relation](Orthonormal%20bases.md#Completeness%20relation) for an orthonormal basis. The completeness relation consists of the sum of all [Projection operators in quantum mechanics](Projection%20operators%20in%20quantum%20mechanics.md) that can be composed from the orthonormal basis. ^18c186
The completeness relation also satisfies the requirement that the [Hilbert Space](Hilbert%20Spaces%20in%20Quantum%20Mechanics.md) is [complete.](Hilbert%20Spaces%20in%20Quantum%20Mechanics.md#^bb8d84)
### Vector representations of state vectors
For a finite $n$ dimensional state we can infer the following vector representation $|\psi\rangle = \sum_j^n |a_j\rangle\langle a_j|\psi\rangle = \begin{pmatrix} \langle a_1|\psi\rangle\\ \vdots \\ \langle a_n|\psi\rangle \end{pmatrix}.$ Thus a ket state is a column vector of [probability amplitude](probability%20amplitude)s.
### State vectors with continuous eigenstates
The [eigenstate decomposition](State%20vector.md#Eigenstate%20decomposition%20of%20state%20vectors) may also be continuous, meaning that the [orthonormal basis contains an infinite number of elements on a continuous rather than discrete spectrum](Orthonormal%20bases.md#Orthonormal%20functions) Rather than considering pairs of eigenstates, $|a_i\rangle, |a_j\rangle$ where the subscripts $i$ and $j$ are used to label discrete quantities, let's consider the case where instead we have eigenstates corresponding to continuous variables $a''$ and $a'.$ The [Kronecker delta](Kronecker%20delta.md) in [$\langle a_i|a_j\rangle =\delta_{a_i,a_j}$](state%20vector#^ba778b) is [switched](Kronecker%20delta.md#Relationship%20with%20the%20Dirac%20delta%20function) with a [Dirac delta function](Dirac%20delta%20function.md) such that $\langle a''|a'\rangle =\delta(a'-a'').$
Our sums [are switched](Riemann%20integral.md) to integrals such that the decomposition, [$|\psi\rangle=\sum_i |a_i\rangle\langle a_i|\psi\rangle = \sum_i c_i|a_i\rangle.$ ](state%20vector#^248983) becomes $|\psi\rangle = \int da'\, |a'\rangle\langle a'|\psi\rangle.$
The [state vector completeness relation,](State%20vector.md#State%20vector%20completeness%20relation) [$\sum_i |a_i\rangle\langle a_i|=\mathbb{1}$](state%20vector#^18c186) becomes $\int da\,|a\rangle\langle a|=1.$
%%There is a distinction between a basis being continuous and having an infinite number of vectors that span it. this needs to be accounted for.%%
## State vector normalization
[(... see more)](state%20vector%20normalization.md)
### non-orthonormal states
[coherent state](coherent%20state.md)
# Measurements on state vectors
The probability of measuring a particular result from a [[state vector]] is governed by the [[Born rule]].
The process of measurement maps the state vector to a particular subspace such that the state vector after a measurement is determined by the [[measurement operator]]. Unless otherwise stated, this is always going to be a [projection operator](Projection%20operators%20in%20quantum%20mechanics.md) following from the [[von Neumann postulate]]. However, where we may also model the measurement process with more detail using a more detailed appraoch referred to as [_generalized measurement._](POVM.md)
# Unitary transformations on state vectors
## Dynamics with state vectors
The dynamics of a [closed quantum system](Closed%20quantum%20systems.md) described with a [state vector](State%20vector.md) is modeled by the [Schrödinger Equation,](Schrödinger%20Equation.md) [$i\hbar\frac{\partial}{\partial t}|\psi(\mathbf{r},t)\rangle = \hat{H}|\psi(\mathbf{r},t)\rangle$](Schrödinger%20Equation#^8ec45d)
%%this is not complete explain more and link to the article on equations of motion in quantum mechanics%%
# Ensembles and state vectors
If we consider only a single quantum particle its quantum state is considered a [Pure state](Pure%20state.md) and may be represented with a single state vector. However statistical [ensembles](Ensembles%20of%20quantum%20systems.md) involve sets of multiple state vectors that can only be fully described using a [[density matrix]]. Such ensemble systems are called [[mixed state]]s.
[(... see more)](Ensembles%20of%20quantum%20systems.md)
## State vectors of composite systems
([... see more](Composite%20quantum%20systems.md))
#QuantumMechanics/QuantumStateRepresentations/StateVectors