Editing Wave function (section)

== Definitions (other cases) ==
Following are the general forms of the wave function for systems in higher dimensions and more particles, as well as including other degrees of freedom than position coordinates or momentum components.

=== Finite dimensional Hilbert space ===
While [[Hilbert space]]s originally refer to infinite dimensional [[Complete metric space|complete]] [[inner product space]]s they, by definition, include finite dimensional [[Complete metric space|complete]] [[inner product space]]s as well.{{sfn | Treves | 2006 | p=112-125}}
In physics, they are often referred to as ''finite dimensional Hilbert spaces''.<ref name=":0">{{Cite web |last=B. Griffiths |first=Robert |author-link=Robert B. Griffiths |title=Hilbert Space Quantum Mechanics |url=https://quantum.phys.cmu.edu/QCQI/qitd114.pdf |page=1}}</ref> For every finite dimensional Hilbert space there exist [[orthonormal basis]] kets that [[Span (mathematics)|span]] the entire Hilbert space.

If the {{math|''N''}}-dimensional set <math display="inline">\{ |\phi_i\rangle \}</math> is orthonormal, then the projection operator for the space spanned by these states is given by:

<math display="block">P = \sum_i |\phi_i\rangle\langle \phi_i | = I </math>where the projection is equivalent to identity operator since <math display="inline">\{ |\phi_i\rangle \}</math> spans the entire Hilbert space, thus leaving any vector from Hilbert space unchanged. This is also known as completeness relation of finite dimensional Hilbert space.

The wavefunction is instead given by:

<math display="block">|\psi\rangle = I|\psi\rangle = \sum_i |\phi_i\rangle\langle \phi_i |\psi\rangle  </math>where <math display="inline">\{ \langle \phi_i |\psi\rangle \} </math>, is a set of complex numbers which can be used to construct a wavefunction using the above formula.

==== Probability interpretation of inner product ====
If the set <math display="inline">\{ |\phi_i\rangle \}</math> are eigenkets of a non-[[Degenerate energy levels|degenerate]] [[observable]] with eigenvalues <math display="inline">\lambda_i</math>, by the [[postulates of quantum mechanics]], the probability of measuring the observable to be <math display="inline">\lambda_i</math> is given according to [[Born rule]] as:{{sfn | Landsman | 2009}}

<math display="block">P_\psi(\lambda_i) = |\langle \phi_i|\psi \rangle|^2    </math>

For non-degenerate <math display="inline">\{ |\phi_i\rangle \}</math> of some observable, if eigenvalues <math display="inline">\lambda</math> have subset of eigenvectors labelled as <math display="inline">\{ |\lambda^{(j)}\rangle \}</math>, by the [[Mathematical formulation of quantum mechanics|postulates of quantum mechanics]], the probability of measuring the observable to be <math display="inline">\lambda</math> is given by:

<math display="block">P_\psi(\lambda) =\sum_j |\langle \lambda^{(j)}|\psi \rangle|^2 = |\widehat P_\lambda |\psi \rangle |^2          </math>where <math display="inline">\widehat P_\lambda =\sum_j|\lambda^{(j)}\rangle\langle\lambda^{(j)}|         </math> is a projection operator of states to subspace spanned by <math display="inline">\{ |\lambda^{(j)}\rangle \}</math>. The equality follows due to orthogonal nature of <math display="inline">\{ |\phi_i\rangle \}</math>.

Hence, <math display="inline">\{ \langle \phi_i |\psi\rangle \} </math> which specify state of the quantum mechanical system, have magnitudes whose square gives the probability of measuring the respective <math display="inline">|\phi_i\rangle   </math> state.

==== Physical significance of relative phase ====
While the relative phase has observable effects in experiments, the global phase of the system is experimentally indistinguishable. For example in a particle in superposition of two states, the global phase of the particle cannot be distinguished by finding expectation value of observable or probabilities of observing different states but relative phases can affect the expectation values of observables.

While the overall phase of the system is considered to be arbitrary, the relative phase for each state <math display="inline">|\phi_i\rangle   </math> of a prepared state in superposition can be determined based on physical meaning of the prepared state and its symmetry. For example, the construction of spin states along x direction as a superposition of spin states along z direction, can done by applying appropriate rotation transformation on the spin along z states which provides appropriate phase of the states relative to each other.

==== Application to include spin ====
An example of finite dimensional Hilbert space can be constructed using spin eigenkets of <math display="inline">s</math>-spin particles which forms a <math display="inline">2s+1</math> dimensional [[Hilbert space]]. However, the general wavefunction of a particle that fully describes its state, is always from an infinite dimensional [[Hilbert space]] since it involves a tensor product with [[Hilbert space]] relating to the position or momentum of the particle. Nonetheless, the techniques developed for finite dimensional Hilbert space are useful since they can either be treated independently or treated in consideration of linearity of tensor product.

Since the [[spin operator]] for a given <math display="inline">s</math>-spin particles can be represented as a finite <math display="inline">(2s+1)^2    </math> [[Matrix (mathematics)|matrix]] which acts on <math display="inline">2s+1</math> independent spin vector components, it is usually preferable to denote spin components using matrix/column/row notation as applicable.

For example, each {{math|{{ket|''s<sub>z</sub>''}}}} is usually identified as a column vector:<math display="block">|s\rangle \leftrightarrow \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} \,, \quad |s-1\rangle \leftrightarrow \begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} \,, \ldots \,, \quad |-(s-1)\rangle \leftrightarrow \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \\ 0 \\ \end{bmatrix} \,,\quad |-s\rangle \leftrightarrow  \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \\ \end{bmatrix}</math>

but it is a common abuse of notation, because the kets {{math|{{ket|''s<sub>z</sub>''}}}} are not synonymous or equal to the column vectors. Column vectors simply provide a convenient way to express the spin components.

Corresponding to the notation, the z-component spin operator can be written as:<math display="block">\frac{1}{\hbar}\hat{S}_z = \begin{bmatrix} s & 0 & \cdots & 0 & 0 \\ 0 & s-1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & -(s-1) & 0 \\ 0 & 0 & \cdots & 0 & -s \end{bmatrix} </math>

since the [[eigenvector]]s of z-component spin operator are the above column vectors, with eigenvalues being the corresponding spin quantum numbers.

Corresponding to the notation, a vector from such a finite dimensional Hilbert space is hence represented as:

<math display="block">|\phi\rangle  =  \begin{bmatrix} \langle s| \phi\rangle \\ \langle s-1| \phi\rangle \\ \vdots \\ \langle -(s-1)| \phi\rangle \\ \langle -s| \phi\rangle \\ \end{bmatrix} =\begin{bmatrix} \varepsilon_s \\  \varepsilon_{s-1}\\ \vdots \\ \varepsilon_{-s+1} \\ \varepsilon_{-s} \\ \end{bmatrix} </math>where <math display="inline"> \{ \varepsilon_i \}       </math> are corresponding complex numbers.

In the following discussion involving spin, the complete wavefunction is considered as tensor product of spin states from finite dimensional Hilbert spaces and the wavefunction which was previously developed. The basis for this Hilbert space are hence considered: <math>  |\mathbf{r}, s_z\rangle   = |\mathbf{r}\rangle |s_z\rangle   </math>.

=== 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.

===Many-particle states in 3d position space===
[[File:Two particle wavefunction.svg|upright=1.4|thumb|Traveling waves of two free particles, with two of three dimensions suppressed. Top is position-space wave function, bottom is momentum-space wave function, with corresponding probability densities.]]

If there are many particles, in general there is only one wave function, not a separate wave function for each particle. The fact that ''one'' wave function describes ''many'' particles is what makes [[quantum entanglement]] and the [[EPR paradox]] possible. The position-space wave function for {{math|''N''}} particles is written:{{sfn|Atkins|1974}}
<math display="block">\Psi(\mathbf{r}_1,\mathbf{r}_2 \cdots \mathbf{r}_N,t)</math>
where {{math|'''r'''<sub>''i''</sub>}} is the position of the {{mvar|i}}-th particle in three-dimensional space, and {{mvar|t}} is time. Altogether, this is a complex-valued function of {{math|3''N'' + 1}} real variables.

In quantum mechanics there is a fundamental distinction between ''[[identical particles]]'' and ''distinguishable'' particles. For example, any two electrons are identical and fundamentally indistinguishable from each other; the laws of physics make it impossible to "stamp an identification number" on a certain electron to keep track of it.{{sfn|Griffiths|2004}} This translates to a requirement on the wave function for a system of identical particles:
<math display="block">\Psi \left ( \ldots \mathbf{r}_a, \ldots , \mathbf{r}_b, \ldots \right ) = \pm \Psi \left ( \ldots \mathbf{r}_b, \ldots , \mathbf{r}_a, \ldots \right )</math>
where the {{math|+}} sign occurs if the particles are ''all bosons'' and {{math|−}} sign if they are ''all fermions''. In other words, the wave function is either totally symmetric in the positions of bosons, or totally antisymmetric in the positions of fermions.{{sfn|Zettili|2009|p=463}} The physical interchange of particles corresponds to mathematically switching arguments in the wave function. The antisymmetry feature of fermionic wave functions leads to the [[Pauli exclusion principle|Pauli principle]]. Generally, bosonic and fermionic symmetry requirements are the manifestation of [[particle statistics]] and are present in other quantum state formalisms.

For {{math|''N''}} ''distinguishable'' particles (no two being [[identical particles|identical]], i.e. no two having the same set of quantum numbers), there is no requirement for the wave function to be either symmetric or antisymmetric.

For a collection of particles, some identical with coordinates {{math|'''r'''<sub>1</sub>, '''r'''<sub>2</sub>, ...}} and others distinguishable {{math|'''x'''<sub>1</sub>, '''x'''<sub>2</sub>, ...}} (not identical with each other, and not identical to the aforementioned identical particles), the wave function is symmetric or antisymmetric in the identical particle coordinates {{math|'''r'''<sub>''i''</sub>}} only:
<math display="block">\Psi \left ( \ldots \mathbf{r}_a, \ldots , \mathbf{r}_b, \ldots , \mathbf{x}_1, \mathbf{x}_2, \ldots \right ) = \pm \Psi \left ( \ldots \mathbf{r}_b, \ldots , \mathbf{r}_a, \ldots , \mathbf{x}_1, \mathbf{x}_2, \ldots \right )</math>

Again, there is no symmetry requirement for the distinguishable particle coordinates {{math|'''x'''<sub>''i''</sub>}}.

The wave function for ''N'' particles each with spin is the complex-valued function
<math display="block">\Psi(\mathbf{r}_1, \mathbf{r}_2 \cdots \mathbf{r}_N, s_{z\,1}, s_{z\,2} \cdots s_{z\,N}, t)</math>

Accumulating all these components into a single vector,

<math display="block">| \Psi \rangle = \overbrace{\sum_{s_{z\,1},\ldots,s_{z\,N}}}^{\text{discrete labels}} \overbrace{\int_{R_N} d^3\mathbf{r}_N \cdots \int_{R_1} d^3\mathbf{r}_1}^{\text{continuous labels}} \; \underbrace{{\Psi}( \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} )}_{\begin{array}{c}\text{wave function (component of } \\ \text{ state vector along basis state)}\end{array}} \; \underbrace{| \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} \rangle }_{\text{basis state (basis ket)}}\,.</math>

For identical particles, symmetry requirements apply to both position and spin arguments of the wave function so it has the overall correct symmetry.

The formulae for the inner products are integrals over all coordinates or momenta and sums over all spin quantum numbers. For the general case of {{math|''N''}} particles with spin in 3-d,
<math display="block"> ( \Psi_1 , \Psi_2 ) = \sum_{s_{z\,N}} \cdots \sum_{s_{z\,2}} \sum_{s_{z\,1}} \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_1 \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_2\cdots \int\limits_{\mathrm{ all \, space}} d ^3 \mathbf{r}_N \Psi^{*}_1 \left(\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right )\Psi_2 \left(\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) </math>
this is altogether {{mvar|N}} three-dimensional [[volume integral]]s and {{mvar|N}} sums over the spins. The differential volume elements {{math|''d''<sup>3</sup>'''r'''<sub>''i''</sub>}} are also written "{{math|''dV''<sub>''i''</sub>}}" or "{{math|''dx<sub>i</sub> dy<sub>i</sub> dz<sub>i</sub>''}}".

The multidimensional Fourier transforms of the position or position–spin space wave functions yields momentum or momentum–spin space wave functions.

====Probability interpretation====
For the general case of {{mvar|N}} particles with spin in 3d, if {{math|Ψ}} is interpreted as a probability amplitude, the probability density is
<math display="block">\rho\left(\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) =  \left | \Psi\left (\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) \right |^2</math>

and the probability that particle 1 is in region {{math|''R''<sub>1</sub>}} with spin {{math|1=''s''<sub>''z''1</sub> = ''m''<sub>1</sub>}} ''and'' particle 2 is in region {{math|''R''<sub>2</sub>}} with spin {{math|1=''s''<sub>''z''2</sub> = ''m''<sub>2</sub>}} etc. at time {{math|''t''}} is the integral of the probability density over these regions and evaluated at these spin numbers:

:<math>P_{\mathbf{r}_1\in R_1,s_{z\,1} = m_1, \ldots, \mathbf{r}_N\in R_N,s_{z\,N} = m_N} (t) = \int_{R_1} d ^3\mathbf{r}_1 \int_{R_2} d ^3\mathbf{r}_2\cdots \int_{R_N} d ^3\mathbf{r}_N \left | \Psi\left (\mathbf{r}_1 \cdots \mathbf{r}_N,m_1\cdots m_N,t \right ) \right |^2</math>

==== Physical significance of phase ====
In non-relativistic quantum mechanics, it can be shown using Schrodinger's time dependent wave equation that the equation:

<math display="block">\frac{\partial \rho}{\partial t} + \nabla\cdot\mathbf J = 0  </math>is satisfied, where <math display="inline">\rho(\mathbf x,t) = | \psi(\mathbf x,t)|^2   </math> is the probability density and <math display="inline">\mathbf J(\mathbf x,t) = \frac{\hbar}{2im}(\psi^* \nabla\psi-\psi\nabla\psi^*) = \frac{\hbar}{m} \text{Im}(\psi^* \nabla\psi)  </math>, is known as the [[Probability current|probability flux]] in accordance with the continuity equation form of the above equation.

Using the following expression for wavefunction:<math display="block">\psi(\mathbf x,t)= \sqrt{\rho(\mathbf x,t)}\exp{\frac{iS(\mathbf x,t )}{\hbar}} </math>where <math display="inline">\rho(\mathbf x,t) = | \psi(\mathbf x,t)|^2   </math> is the probability density and <math display="inline">S(\mathbf x,t) </math> is the phase of the wavefunction, it can be shown that:

<math display="block">\mathbf J(\mathbf x,t) = \frac{\rho \nabla S}{m} </math>

Hence the spacial variation of phase characterizes the [[Probability current|probability flux]].

In classical analogy, for <math display="inline">\mathbf J = \rho \mathbf v </math>, the quantity <math display="inline"> \frac{\nabla S}{m} </math> is analogous with velocity. Note that this does not imply a literal interpretation of <math display="inline"> \frac{\nabla S}{m} </math> as velocity since velocity and position cannot be simultaneously determined as per the [[uncertainty principle]]. Substituting the form of wavefunction in Schrodinger's time dependent wave equation, and taking the classical limit, <math display="inline"> \hbar |\nabla^2 S| \ll |\nabla S|^2 </math>:

<math display="block">\frac{1}{2m} |\nabla S(\mathbf x, t)|^2 + V(\mathbf x) + \frac{\partial S}{\partial t} = 0 </math>

Which is analogous to [[Hamilton–Jacobi equation|Hamilton-Jacobi equation]] from classical mechanics. This interpretation fits with [[Hamilton–Jacobi theory]], in which <math display="inline"> \mathbf{P}_{\text{class.}} = \nabla S </math>, where ''{{mvar|S}}'' is [[Hamilton's principal function]].<ref>{{Cite book |last1=Sakurai |first1=Jun John |title=Modern quantum mechanics |last2=Napolitano |first2=Jim |date=2021 |publisher=Cambridge University Press |isbn=978-1-108-47322-4 |edition=3rd |location=Cambridge |pages=94–97}}</ref>