Editing Wave function (section)

===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>