Editing Wave function (section)

=== One-particle states in 3d position space ===

The position-space wave function of a single particle without spin in three spatial dimensions is similar to the case of one spatial dimension above: <math display="block">\Psi(\mathbf{r},t)</math> where {{math|'''r'''}} is the [[position vector]] in three-dimensional space, and {{math|''t''}} is time. As always {{math|Ψ('''r''', ''t'')}} is a complex-valued function of real variables. As a single vector in [[Dirac notation]]
<math display="block">|\Psi(t)\rangle = \int d^3\! \mathbf{r}\, \Psi(\mathbf{r},t) \,|\mathbf{r}\rangle </math>

All the previous remarks on inner products, momentum space wave functions, Fourier transforms, and so on extend to higher dimensions.

For a particle with [[Spin (physics)|spin]], ignoring the position degrees of freedom, the wave function is a function of spin only (time is a parameter);
<math display="block">\xi(s_z,t)</math>
where {{math|''s''<sub>z</sub>}} is the [[Spin (physics)|spin projection quantum number]] along the {{mvar|z}} axis. (The {{mvar|z}} axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) The {{math|''s<sub>z</sub>''}} parameter, unlike {{math|'''r'''}} and {{mvar|t}}, is a [[Continuous or discrete variable#Discrete variable|discrete variable]]. For example, for a [[spin-1/2]] particle, {{math|''s''<sub>z</sub>}} can only be {{math|+1/2}} or {{math|−1/2}}, and not any other value. (In general, for spin {{mvar|s}}, {{math|''s<sub>z</sub>''}} can be {{math|''s'', ''s'' − 1, ..., −''s'' + 1, −''s''}}). Inserting each quantum number gives a complex valued function of space and time, there are {{math|2''s'' + 1}} of them. These can be arranged into a [[column vector]]

<math display="block">\xi = \begin{bmatrix} \xi(s,t) \\ \xi(s-1,t) \\ \vdots \\ \xi(-(s-1),t) \\ \xi(-s,t) \\ \end{bmatrix} = \xi(s,t) \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} + \xi(s-1,t)\begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} + \cdots + \xi(-(s-1),t) \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \\ 0 \\ \end{bmatrix} + \xi(-s,t) \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \\ \end{bmatrix} </math>

In [[bra–ket notation]], these easily arrange into the components of a vector:
<math display="block">|\xi (t)\rangle = \sum_{s_z=-s}^s \xi(s_z,t) \,| s_z \rangle </math>

The entire vector {{math|''ξ''}} is a solution of the Schrödinger equation (with a suitable Hamiltonian), which unfolds to a coupled system of {{math|2''s'' + 1}} ordinary differential equations with solutions {{math|''ξ''(''s'', ''t''), ''ξ''(''s'' − 1, ''t''), ..., ''ξ''(−''s'', ''t'')}}. The term "spin function" instead of "wave function" is used by some authors. This contrasts the solutions to position space wave functions, the position coordinates being continuous degrees of freedom, because then the Schrödinger equation does take the form of a wave equation.

More generally, for a particle in 3d with any spin, the wave function can be written in "position–spin space" as:
<math display="block">\Psi(\mathbf{r},s_z,t)</math>
and these can also be arranged into a column vector
<math display="block">\Psi(\mathbf{r},t) = \begin{bmatrix} \Psi(\mathbf{r},s,t) \\ \Psi(\mathbf{r},s-1,t) \\ \vdots \\ \Psi(\mathbf{r},-(s-1),t) \\ \Psi(\mathbf{r},-s,t) \\ \end{bmatrix}</math>
in which the spin dependence is placed in indexing the entries, and the wave function is a complex [[vector-valued function]] of space and time only.

All values of the wave function, not only for discrete but [[Continuous or discrete variable#Continuous variable|continuous variables]] also, collect into a single vector
<math display="block">|\Psi(t)\rangle =
\sum_{s_z}\int d^3\!\mathbf{r} \,\Psi(\mathbf{r},s_z,t)\, |\mathbf{r}, s_z\rangle </math>

For a single particle, the [[Bra–ket notation#Composite bras and kets|tensor product]] {{math|&otimes;}} of its position state vector {{math|{{ket|''ψ''}}}} and spin state vector {{math|{{ket|''ξ''}}}} gives the composite position-spin state vector
<math display="block">|\psi(t)\rangle\! \otimes\! |\xi(t)\rangle =
\sum_{s_z}\int d^3\! \mathbf{r}\, \psi(\mathbf{r},t)\,\xi(s_z,t) \,|\mathbf{r}\rangle \!\otimes\! |s_z\rangle </math>
with the identifications
<math display="block">|\Psi (t)\rangle = |\psi(t)\rangle
\!\otimes\!
 |\xi(t)\rangle </math>
<math display="block">\Psi(\mathbf{r},s_z,t) = \psi(\mathbf{r},t)\,\xi(s_z,t) </math>
<math display="block">|\mathbf{r},s_z \rangle= |\mathbf{r}\rangle \!\otimes\! |s_z\rangle </math>

The tensor product factorization of energy eigenstates is always possible if the orbital and spin angular momenta of the particle are separable in the [[Hamiltonian operator]] underlying the system's dynamics (in other words, the Hamiltonian can be split into the sum of orbital and spin terms{{sfn|Shankar|1994|pp=378–379}}). The time dependence can be placed in either factor, and time evolution of each can be studied separately. Under such Hamiltonians, any tensor product state evolves into another tensor product state, which essentially means any unentangled state remains unentangled under time evolution. This is said to happen when there is no physical interaction between the states of the tensor products. In the case of non separable Hamiltonians, energy eigenstates are said to be some linear combination of such states, which need not be factorizable; examples include a particle in a [[magnetic field]], and [[spin–orbit coupling]].

The preceding discussion is not limited to spin as a discrete variable, the total [[angular momentum operator|angular momentum]] ''J'' may also be used.{{sfn|Landau|Lifshitz|1977}} Other discrete degrees of freedom, like [[isospin]], can expressed similarly to the case of spin above.