Editing Bra–ket notation (section)

==Usage in quantum mechanics==

The mathematical structure of quantum mechanics is based in large part on [[linear algebra]]:
*[[Wave function]]s and other quantum states can be represented as vectors in a [[Separable space|separable]] [[complex number|complex]] [[Hilbert space]]. (The exact structure of this Hilbert space depends on the situation.) In bra–ket notation, for example, an electron might be in the "state" {{math|{{ket|''ψ''}}}}. (Technically, the quantum states are ''[[Ray (quantum theory)|ray]]s'' of vectors in the Hilbert space, as {{math|''c''{{ket|''ψ''}}}} corresponds to the same state for any nonzero complex number {{math|''c''}}.)
*[[Quantum superposition]]s can be described as vector sums of the constituent states. For example, an electron in the state {{math|{{sfrac|1|√2}}{{ket|1}} + {{sfrac|''i''|√2}}{{ket|2}}}} is in a quantum superposition of the states {{math|{{ket|1}}}} and {{math|{{ket|2}}}}.
*[[Measurement in quantum mechanics|Measurements]] are associated with linear operators (called [[observable]]s) on the Hilbert space of quantum states.
*Dynamics are also described by linear operators on the Hilbert space. For example, in the [[Schrödinger picture]], there is a linear [[time evolution]] operator {{math|''U''}} with the property that if an electron is in state {{math|{{ket|''ψ''}}}} right now, at a later time it will be in the state {{math|''U''{{ket|''ψ''}}}}, the same {{math|''U''}} for every possible {{math|{{ket|''ψ''}}}}.
*[[Normalizable wave function|Wave function normalization]] is scaling a wave function so that its [[norm (mathematics)|norm]] is 1.

Since virtually every calculation in quantum mechanics involves vectors and linear operators, it can involve, and often does involve, bra–ket notation. A few examples follow:

===Spinless position–space wave function===

<div class="skin-invert-image">
{{multiple image
 | left
 | footer    = Components of complex vectors plotted against index number; discrete {{math|''k''}} and continuous {{math|''x''}}. Two particular components out of infinitely many are highlighted.
 | width1    = 225
 | image1    = Discrete complex vector components.svg 
 | caption1  = Discrete components {{math|''A''<sub>''k''</sub>}} of a complex vector {{math|1={{ket|''A''}} = Σ<sub>''k''</sub> ''A''<sub>''k''</sub> {{ket|''e<sub>k</sub>''}}}}.
 | width2    = 230
 | image2    = Continuous complex vector components.svg
 | caption2  = Continuous components {{math|''ψ''(''x'')}} of a complex vector {{math|1={{ket|''ψ''}} = ∫ d''x'' ''ψ''(''x''){{ket|''x''}}}}.
}}
</div>

The Hilbert space of a [[Spin (physics)|spin]]-0 point particle  can be represented in terms of a "position [[basis (linear algebra)|basis]]" {{math|{ {{ket|'''r'''}} }<nowiki/>}}, where the label {{math|'''r'''}} extends over the set of all points in [[position space]]. These states satisfy the eigenvalue equation for the [[position operator]]:
<math display="block"> \hat{\mathbf{r}}|\mathbf{r}\rangle = \mathbf{r}|\mathbf{r}\rangle.</math>
The position states are "[[Dirac_delta_function#Quantum_mechanics|generalized eigenvectors]]", not elements of the Hilbert space itself, and do not form a countable orthonormal basis. However, as the Hilbert space is separable, it does admit a countable dense subset within the [[domain of definition]] of its wavefunctions. That is, starting from any ket {{math|{{ket|Ψ}}}} in this Hilbert space, one may ''define'' a complex scalar function of {{math|'''r'''}}, known as a [[wavefunction]],
<math display="block">\Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r}|\Psi\rang \,.</math>

On the left-hand side, {{math|Ψ('''r''')}} is a function mapping any point in space to a complex number; on the right-hand side,
<math display="block">\left|\Psi\right\rangle = \int d^3\mathbf{r} \, \Psi(\mathbf{r}) \left|\mathbf{r}\right\rangle</math> 
is a ket consisting of a superposition of kets with relative coefficients specified by that function.

It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by
<math display="block">\hat A(\mathbf{r}) ~ \Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r}|\hat A|\Psi\rang \,.</math>

For instance, the [[momentum]] operator <math>\hat \mathbf {p}</math> has the following coordinate representation,
<math display="block">\hat{\mathbf{p} } (\mathbf{r}) ~ \Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r} |\hat \mathbf{p}|\Psi\rang = - i \hbar \nabla \Psi(\mathbf{r}) \,.</math>

One occasionally even encounters an expression such as <math >\nabla |\Psi\rang </math>, though this is something of an [[abuse of notation]]. The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is projected onto the position basis, <math>\nabla \lang\mathbf{r}|\Psi\rang \,,</math>
even though, in the momentum basis, this operator amounts to a mere multiplication operator (by {{math|''iħ'''''p'''}}). That is, to say,
<math display="block"> \langle \mathbf{r} |\hat \mathbf{p} = - i \hbar \nabla \langle \mathbf{r}| ~,</math>
or
<math display="block"> \hat \mathbf{p} = \int d^3 \mathbf{r} ~| \mathbf{r}\rangle ( - i \hbar \nabla) \langle \mathbf{r}| ~.</math>

===Overlap of states===
In quantum mechanics the expression {{math|{{bra-ket|''φ''|''ψ''}}}} is typically interpreted as the [[probability amplitude]] for the state {{math|''ψ''}} to [[wavefunction collapse|collapse]] into the state {{math|''φ''}}. Mathematically, this means the coefficient for the projection of {{math|''ψ''}} onto {{math|''φ''}}. It is also described as the projection of state {{math|''ψ''}} onto state {{math|''φ''}}.

===Changing basis for a spin-1/2 particle===
A stationary [[spin-1/2|spin-{{1/2}}]] particle has a two-dimensional Hilbert space. One [[orthonormal basis]] is:
<math display="block">|{\uparrow}_z \rangle \,, \; |{\downarrow}_z \rangle</math>
where {{math|{{ket|↑<sub>''z''</sub>}}}} is the state with a definite value of the [[angular momentum operator|spin operator {{math|''S<sub>z</sub>''}}]] equal to +{{1/2}} and {{math|{{ket|↓<sub>''z''</sub>}}}} is the state with a definite value of the [[angular momentum operator|spin operator {{math|''S<sub>z</sub>''}}]] equal to −{{1/2}}.

Since these are a basis, ''any'' quantum state of the particle can be expressed as a [[linear combination]] (i.e., [[quantum superposition]]) of these two states:
<math display="block">|\psi \rangle = a_{\psi} |{\uparrow}_z \rangle + b_{\psi} |{\downarrow}_z \rangle</math>
where {{math|''a<sub>ψ</sub>''}} and {{math|''b<sub>ψ</sub>''}} are complex numbers.

A ''different'' basis for the same Hilbert space is:
<math display="block">|{\uparrow}_x \rangle \,, \; |{\downarrow}_x \rangle</math>
defined in terms of {{math|''S<sub>x</sub>''}} rather than {{math|''S<sub>z</sub>''}}.

Again, ''any'' state of the particle can be expressed as a linear combination of these two:
<math display="block">|\psi \rangle = c_{\psi} |{\uparrow}_x \rangle + d_{\psi} |{\downarrow}_x \rangle</math>

In vector form, you might write
<math display="block">|\psi\rangle \doteq \begin{pmatrix} a_\psi \\ b_\psi \end{pmatrix} \quad \text{or} \quad |\psi\rangle \doteq \begin{pmatrix} c_\psi \\ d_\psi \end{pmatrix} </math>
depending on which basis you are using. In other words, the "coordinates" of a vector depend on the basis used.

There is a mathematical relationship between <math>a_\psi</math>, <math>b_\psi</math>, <math>c_\psi</math> and <math>d_\psi</math>; see [[change of basis]].