Editing Bra–ket notation (section)

===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]].