# Wave function

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Some trajectories of a harmonic oscillator (a ball attached to a spring) in classical mechanics (A–B) and quantum mechanics (C–H). In quantum mechanics (C–H), the ball has a wave function, which is shown with real part in blue and imaginary part in red. The trajectories C, D, E, F, (but not G or H) are examples of standing waves, (or "stationary states"). Each standing wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have any energy.

A wave function (or wavefunction) in quantum mechanics describes the quantum state of a system of one or more particles, and contains all the information about the system considered in isolation. Quantities associated with measurements, such as the average momentum of a particle, are derived from the wave function by mathematical operations that describe its interaction with observational devices. Thus it is a central entity in quantum mechanics. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi). The Schrödinger equation determines how the wave function evolves over time, that is, the wave function is the solution of the Schrödinger equation. The wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality. The wave of the wave function, however, is not a wave in physical space; it is a wave in an abstract mathematical "space", and in this respect it differs fundamentally from water waves or waves on a string.[1][2][3][4][5][6][7]

The wave function may be represented as a complex vector-valued function over its domain, but for a given system such a representation is not unique. It may be taken to be a function of time and all position coordinates of the particles, that is, the wave function is in position space. A wave function may similarly be considered over momentum space; a function instead of time and the momenta of all the particles. In general, the wave function of a system is a function of continuous and discrete variables characterizing the system's degrees of freedom, and there is one wave function for the entire system, not a separate wave function for each particle in the system. Some particles, like electrons and photons, have nonzero spin, and the wave function must include this fundamental property as an intrinsic degree of freedom. In general, for a particle with half-integer spin the wave function is a spinor, for a particle with integer spin the wave function is a tensor. Particles with spin zero are called scalar particles, those with spin 1 vector particles, and more generally for higher integer spin, tensor particles. No elementary particle with spin Template:Frac or higher is known, except for the hypothesized spin 2 graviton.

In the Copenhagen interpretation, an interpretation of quantum mechanics, the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle as being at a given place at a given time. Since the wave function is complex valued, only its relative phase and relative magnitude can be measured. Its value does not in isolation tell anything about the magnitudes or directions of measurable observables; one has to apply quantum operators to the wave function ψ and find the eigenvalues which correspond to sets of possible results of measurement.

The unit of measurement for ψ depends on the system. For one particle in three dimensions, its units are [length]−3/2. These units are required so that an integral of |ψ|2 over a region of three-dimensional space is a dimensionless probability (the probability that the particle is measured as being in that region). For different numbers of particles and/or dimensions, the units vary and can be found by dimensional analysis.[8]

## Historical background

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In the 1920s and 1930s, quantum mechanics was developed using calculus and linear algebra. Those who used the techniques of calculus included Louis de Broglie, Erwin Schrödinger, and others, developing "wave mechanics". Those who applied the methods of linear algebra included Werner Heisenberg, Max Born, and others, developing "matrix mechanics". Schrödinger subsequently showed that the two approaches were equivalent.[9]

In 1905 Planck postulated the proportionality between the frequency of a photon and its energy, in the Planck-Einstein relation, E = hf. In 1925, De Broglie published the corresponding relation between momentum and wavelength, λ = h/p, now called the De Broglie relation. These equations represent wave–particle duality. In 1926, Schrödinger published the famous wave equation now named after him, indeed the Schrödinger equation, based on classical energy conservation using quantum operators and the de Broglie relations such that the solutions of the equation are the wave functions for the quantum system. Later Pauli invented the Pauli equation that adds a description of electron's spin and magnetic dipole. However, no one, even Schrödinger or De Broglie, was clear on how to interpret it.[10] Around 1924–27, Max Born, Heisenberg, Bohr and others provided the perspective of probability amplitude.[11] This is the Copenhagen interpretation of quantum mechanics. There are many other interpretations of quantum mechanics, but this relates calculations of quantum mechanics directly to probabilistic experimental observations.

In 1927, Hartree and Fock made the first step in an attempt to solve the N-body wave function, and developed the self-consistency cycle: an iterative algorithm to approximate the solution. Now it is also known as the Hartree–Fock method.[12] The Slater determinant and permanent (of a matrix) was part of the method, provided by John C. Slater.

Schrödinger did encounter an equation for the wave function that satisfied relativistic energy conservation before he published the non-relativistic one, but discarded it as it predicted negative probabilities and negative energies. In 1927, Klein, Gordon and Fock also found it, but incorporated the electromagnetic interaction and proved that it was Lorentz invariant. De Broglie also arrived at the same equation in 1928. This relativistic wave equation is now most commonly known as the Klein–Gordon equation.[13]

In 1927, Pauli phenomenologically found a non-relativistic equation to describe spin-1/2 particles in electromagnetic fields, now called the Pauli equation. Pauli found the wave function was not described by a single complex function of space and time, but needed two complex numbers, which respectively correspond to the spin +1/2 and −1/2 states of the fermion. Soon after in 1928, Dirac found an equation from the first successful unification of special relativity and quantum mechanics applied to the electron, now called the Dirac equation. In this, the wave function is a spinor represented by four complex-valued components:[12] two for the electron and two for the electron's antiparticle, the positron. In the non-relativistic limit, the Dirac wave function resembles the Pauli wavefunction for the electron. Later, other relativistic wave equations were found.

## Wave functions and function spaces

Functional analysis is commonly used to formulate the wave function with a necessary mathematical precision; usually they are quadratically integrable functions (at least locally) because it is compatible with the Hilbert space formalism mentioned below. The set on which their function space is defined is the configuration space of the system. In many situations it is a Euclidean space, that implies that wave functions are functions of several real variables. Superficially, this formalism is simple to understand for the following reasons.

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi }$
• Functions can easily describe wave-like motion, using periodic functions, and Fourier analysis can be readily done.
• Functions are easy to produce, visualize, and interpret, because of the pictorial nature of the graph of a function. One can plot curves, surfaces, contour lines, more generally any level sets. If the situation is in a high number of dimensions – one can analyze the function in a lower-dimensional slice to see the behavior of the function within that confined region.

For concreteness and simplicity, in this article, when coordinates are needed we use Cartesian coordinates so that r is short for (x, y, z), although spherical polar coordinates and other orthogonal coordinates are often useful to solve the Schrödinger equation for potentials with certain geometric symmetries, in which case the position and wave function is expressed in these coordinates.

One does not have to define wave functions necessarily on real spaces: appropriate function spaces can be defined wherever a measure can provide integration. Operator theory and linear algebra, as shown below, can deal with situations where the real analysis is not applicable.

### Requirements

Continuity of the wave function and its first spatial derivative (in the x direction, y and z coordinates not shown), at some time t.

The following constraints on the wave function are formulated for the calculations and physical interpretation to make sense:[14][15]

A requirement less restrictive is that the wave function must belong to the Sobolev space W1,2. It means that it is differentiable in the sense of distributions, and its gradient is square-integrable. This relaxation is necessary for potentials that are not functions but are distributions, such as the Dirac delta function.

If these requirements are not met, it is not possible to interpret the wave function as a probability amplitude.[16]

## Definition (one spinless particle in 1d)

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For now, consider the simple case of a single particle, without spin, in one spatial dimension. (More general cases are discussed below).

### Position-space wave function

The state of such a particle is completely described by its wave function:

${\displaystyle \Psi (x,t)\,,}$

where x is position and t is time. This is a complex-valued function of two real variables x and t.

If interpreted as a probability amplitude, the square modulus of the wave function is the positive real number

${\displaystyle \left|\Psi (x,t)\right|^{2}={\Psi (x,t)}^{*}\Psi (x,t)=\rho (x,t)}$

interpreted as the probability density that the particle is at x, rather than some other location. The star * indicates complex conjugate. If the particle's position is measured, its location is not deterministic, but is described by a probability distribution. The probability that its position x will be in the interval axb is the integral of the density over this interval:

${\displaystyle P_{a\leq x\leq b}(t)=\int \limits _{a}^{b}dx\,|\Psi (x,t)|^{2}}$

where t is the time at which the particle was measured. This leads to the normalization condition:

${\displaystyle \int \limits _{-\infty }^{\infty }dx\,|\Psi (x,t)|^{2}=1\,,}$

because if the particle is measured, there is 100% probability that it will be somewhere.

Since the Schrödinger equation is linear, if any number of wave functions Ψn for n = 1, 2, ... are solutions of the equation, then so is their sum, and their scalar multiples by complex numbers an. Taking scalar multiplication and addition together is known as a linear combination:

${\displaystyle \sum _{n}a_{n}\Psi _{n}(x,t)=a_{1}\Psi _{1}(x,t)+a_{2}\Psi _{2}(x,t)+\cdots }$

This is the superposition principle. Multiplying a wave function Ψ by any nonzero constant complex number c to obtain cΨ does not change any information about the quantum system, because c cancels in the Schrödinger equation for cΨ. All that happens is that any normalization constants will be rescaled.

Since linear combinations of wave functions obtain more wave functions, the set of all wave functions W = {Ψ(x, t)} is an infinite dimensional vector space over the field of complex numbers. To form a vector space basis B, we need a maximal set of wave functions ψ1, ψ2, ... in W which are linearly independent: each one of them is not a linear combination of the others, for example ψ1z2ψ2 + z3ψ3 + ... and ψ2z1ψ1 + z3ψ3 + ..., etc., for any complex numbers zn and every function in W is a linear combination of functions in B. This linear independence allows a linear combination of ψ1, ψ2, ... to uniquely construct an arbitrary wave function in W:

${\displaystyle \Psi (x,t)=\sum _{n}a_{n}\psi _{n}(x,t)}$

In this way, Ψ(x, t) can be viewed as an infinite dimensional vector, where the complex-valued coefficients an are the components of the vector. The choice of which wave functions to use as a basis is not unique, but if a change of basis is made, the components an need to change to compensate.

### Momentum-space wave function

The particle also has a wave function in momentum space:

${\displaystyle \Phi (p,t)}$

where p is the momentum in one dimension, which can be any value from −∞ to +∞, and t is time.

All the previous remarks on superposition, normalization, etc. apply similarly. In particular, if the particle's momentum is measured, the result is not deterministic, but is described by a probability distribution:

${\displaystyle P_{a\leq p\leq b}(t)=\int \limits _{a}^{b}dp\,|\Phi (p,t)|^{2}\,,}$

and the normalization condition is:

${\displaystyle \int \limits _{-\infty }^{\infty }dp\,\left|\Phi \left(p,t\right)\right|^{2}=1\,.}$

### Relation between wave functions

The position-space and momentum-space wave functions are Fourier transforms of each other, therefore both contain the same information, and either one alone is sufficient to calculate any property of the particle. As elements of abstract physical Hilbert space, whose elements are the possible states of the system under consideration, they represent the same object, but they are not equal when viewed as square-integrable functions. (A function and its Fourier transform are not equal.) For one dimension:[17]

${\displaystyle \Phi (p,t)={\frac {1}{\sqrt {2\pi \hbar }}}\int \limits _{-\infty }^{\infty }dx\,e^{-ipx/\hbar }\Psi (x,t)\quad \rightleftharpoons \quad \Psi (x,t)={\frac {1}{\sqrt {2\pi \hbar }}}\int \limits _{-\infty }^{\infty }dp\,e^{ipx/\hbar }\Phi (p,t)}$

Sometimes the wave-vector k is used in place of momentum p, since they are related by the de Broglie relation

${\displaystyle p=\hbar k}$

and the equivalent space is referred to as k-space. Again it makes no difference which is used since p and k are equivalent – up to a constant. In practice, the position-space wave function is used much more often than the momentum-space wave function. The potential entering the Schrödinger equation determines in which basis the description is easiest. For the harmonic oscillator, x and p enter symmetrically, so there it doesn't matter which description one uses.

### Phase space

In the common formulations of quantum mechanics, the wave function is never a function of both the position and momentum of a particle at any instant, because of the Heisenberg uncertainty principle; if the position of the particle is known exactly, the momentum is not known at all, and vice versa. For a particle in 1d we can never write a wave function as Ψ(x, p, t). Taken together, x and p are called phase space variables. However, it is possible to construct a phase space formulation of quantum mechanics, using different mathematics and physical interpretations, in a way that does not violate the uncertainty principle.

## Definitions (other cases)

The electron probability density for the first few hydrogen atom electron orbitals shown as cross-sections. These orbitals form an orthonormal basis for the wave function of the electron. Different orbitals are depicted with different scale.
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.

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.

The position-space wave function of a single particle in three spatial dimensions is similar to the case of one spatial dimension above:

${\displaystyle \Psi (\mathbf {r} ,t)}$

where r is the position vector in three-dimensional space, and t is time. As always Ψ(r, t) is a complex number, for this case a complex-valued function of four real variables.

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 N particles is written:[12]

${\displaystyle \Psi (\mathbf {r} _{1},\mathbf {r} _{2}\cdots \mathbf {r} _{N},t)}$

where ri is the position of the ith particle in three-dimensional space, and t is time. Altogether, this is a complex-valued function of 3N + 1 real variables.

For a particle with spin, the wave function can be written in "position–spin space" as:

${\displaystyle \Psi (\mathbf {r} ,t,s_{z})}$

which is a complex-valued function of position r in three-dimensional space, time t, and sz, the spin projection quantum number along the z axis. (The z axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) The sz parameter, unlike r and t, is a discrete variable. For example, for a spin-1/2 particle, sz can only be +1/2 or −1/2, and not any other value. (In general, for spin s, sz can be s, s − 1, ... , −s + 1, −s.) The particle is generally a superposition of all the possible spin states

{\displaystyle {\begin{aligned}\Psi (\mathbf {r} ,t,s_{z})&=\psi _{-s}(\mathbf {r} ,t)\xi _{-s}(s_{z})+\psi _{-s+1}(\mathbf {r} ,t)\xi _{-s+1}(s_{z})+\cdots \\&+\psi _{s-1}(\mathbf {r} ,t)\xi _{s-1}(s_{z})+\psi _{s}(\mathbf {r} ,t)\xi _{s}(s_{z})\,,\end{aligned}}}

where ψs, ψs + 1, ..., ψs − 1, ψs are space functions and ξs, ξs + 1, ..., ξs − 1, ξs are spin functions, each of them are complex-valued. The space functions take in space and time, the spin functions take in the spin quantum numbers, and the subscripts on the functions label the spin states of the particle.

Often, the complex values of the wave function for all the spin numbers are arranged into a column vector, in which there are as many entries in the column vector as there are allowed values of sz. In this case, the spin dependence is placed in indexing the entries and the wave function is a complex vector-valued function of space and time only:

${\displaystyle \Psi (\mathbf {r} ,t)={\begin{bmatrix}\Psi (\mathbf {r} ,t,s)\\\Psi (\mathbf {r} ,t,s-1)\\\vdots \\\Psi (\mathbf {r} ,t,-(s-1))\\\Psi (\mathbf {r} ,t,-s)\\\end{bmatrix}}}$

The wave function for N particles each with spin is:

${\displaystyle \Psi (\mathbf {r} _{1},\mathbf {r} _{2}\cdots \mathbf {r} _{N},s_{z\,1},s_{z\,2}\cdots s_{z\,N},t)}$

a complex-valued function. By analogy to the case of one particle with spin, this is also a sum over all the spin states of the system, and the complex values of the many particle case could be arranged into a multidimensional array. (However, for identical particles, symmetry requirements are needed, see also below).

Concerning the general case of N particles with spin in 3d, if Ψ is interpreted as a probability amplitude, the probability density is:

${\displaystyle \rho \left(\mathbf {r} _{1}\cdots \mathbf {r} _{N},s_{z\,1}\cdots s_{z\,N},t\right)=\left|\Psi \left(\mathbf {r} _{1}\cdots \mathbf {r} _{N},s_{z\,1}\cdots s_{z\,N},t\right)\right|^{2}}$

and the probability that particle 1 is in region R1 with spin sz1 = m1 and particle 2 is in region R2 with spin sz2 = m2 etc. at time t is the integral of the probability density over these regions and spins:

${\displaystyle P_{\mathbf {r} _{1}\in R_{1},s_{z\,1}=m_{1},\ldots ,\mathbf {r} _{N}\in R_{N},s_{z\,N}=m_{N}}(t)=\int \limits _{R_{1}}d^{3}\mathbf {r} _{1}\int \limits _{R_{2}}d^{3}\mathbf {r} _{2}\cdots \int \limits _{R_{N}}d^{3}\mathbf {r} _{N}\left|\Psi \left(\mathbf {r} _{1}\cdots \mathbf {r} _{N},m_{1}\cdots m_{N},t\right)\right|^{2}}$

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

## Inner product

### Position-space inner products

The inner product of two wave functions Ψ1 and Ψ2 is useful and important for a number of reasons given below. For the case of one spinless particle in 1d, it can be defined as the complex number (at time t)[nb 1]

${\displaystyle \left\langle \Psi _{1},\Psi _{2}\right\rangle =\int \limits _{-\infty }^{\infty }dx\,\Psi _{1}^{*}(x,t)\Psi _{2}(x,t).}$

More generally, the formulae for the inner products are integrals over all coordinates or momenta and sums over all spin quantum numbers. That is, for one spinless particle in 3d the inner product of two wave functions can be defined as the complex number:

${\displaystyle \langle \Psi _{1},\Psi _{2}\rangle =\int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} \,\Psi _{1}^{*}(\mathbf {r} ,t)\Psi _{2}(\mathbf {r} ,t)\,,}$

while for many spinless particles in 3d:

${\displaystyle \langle \Psi _{1},\Psi _{2}\rangle =\int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{1}\int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{2}\cdots \int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{N}\,\Psi _{1}^{*}(\mathbf {r} _{1}\cdots \mathbf {r} _{N},t)\Psi _{2}(\mathbf {r} _{1}\cdots \mathbf {r} _{N},t)}$

(altogether, this is N three-dimensional volume integrals with differential volume elements d3ri, also written "dVi" or "dxi dyi dzi"). For one particle with spin in 3d:

${\displaystyle \langle \Psi _{1},\Psi _{2}\rangle =\sum _{\mathrm {all\,} s_{z}}\int \limits _{\mathrm {all\,space} }\,d^{3}\mathbf {r} \Psi _{1}^{*}(\mathbf {r} ,t,s_{z})\Psi _{2}(\mathbf {r} ,t,s_{z})\,,}$

and for the general case of N particles with spin in 3d:

${\displaystyle \langle \Psi _{1},\Psi _{2}\rangle =\sum _{s_{z\,N}}\cdots \sum _{s_{z\,2}}\sum _{s_{z\,1}}\int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{1}\int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{2}\cdots \int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{N}\Psi _{1}^{*}\left(\mathbf {r} _{1}\cdots \mathbf {r} _{N},s_{z\,1}\cdots s_{z\,N},t\right)\Psi _{2}\left(\mathbf {r} _{1}\cdots \mathbf {r} _{N},s_{z\,1}\cdots s_{z\,N},t\right)}$

(altogether, N three-dimensional volume integrals followed by N sums over the spins).

In the Copenhagen interpretation, the modulus squared of the inner product (a complex number) gives a real number

${\displaystyle \left|\left\langle \Psi _{1},\Psi _{2}\right\rangle \right|^{2}=P\left(\Psi _{2}\rightarrow \Psi _{1}\right)\,,}$

which is interpreted as the probability of the wave function Ψ2 "collapsing" to the new wave function Ψ1 upon measurement of an observable, whose eigenvalues are the possible results of the measurement, with Ψ1 being an eigenvector of the resulting eigenvalue.

Although the inner product of two wave functions is a complex number, the inner product of a wave function Ψ with itself,

${\displaystyle \langle \Psi ,\Psi \rangle =\|\Psi \|^{2}\,,}$

is always a positive real number. The number ||Ψ|| (not ||Ψ||2) is called the norm of the wave function Ψ, and is not the same as the modulus |Ψ|.

A wave function is normalized if:

${\displaystyle \left\langle \Psi ,\Psi \right\rangle =1\,.}$

If Ψ is not normalized, then dividing by its norm gives the normalized function Ψ/||Ψ||.

Two wave functions Ψ1 and Ψ2 are orthogonal if their inner product is zero:

${\displaystyle \left\langle \Psi _{1},\Psi _{2}\right\rangle =0\,.}$

A set of wave functions Ψ1, Ψ2, ... are orthonormal if they are each normalized and are all orthogonal to each other:

${\displaystyle \langle \Psi _{m},\Psi _{n}\rangle =\delta _{mn}\,,}$

where m and n each take values 1, 2, ..., and δmn is the Kronecker delta (+1 for m = n and 0 for mn). Orthonormality of wave functions is instructive to consider since this guarantees linear independence of the functions. (However, the wave functions do not have to be orthonormal and can still be linearly independent, but the inner product of Ψm and Ψn is more complicated than the mere δmn).

Returning to the superposition above:

${\displaystyle \Psi =\sum _{n}a_{n}\psi _{n}}$

if the basis wave functions ψn are orthonormal, then the coefficients have a particularly simple form:

${\displaystyle a_{n}=\langle \psi _{n},\Psi \rangle }$

If the basis wave functions were not orthonormal, then the coefficients would be different.

### Momentum-space inner products

Analogous to the position case, the inner product of two wave functions Φ1(p, t) and Φ2(p, t) can be defined as:

${\displaystyle \langle \Phi _{1},\Phi _{2}\rangle =\int \limits _{-\infty }^{\infty }dp\,\Phi _{1}^{*}(p,t)\Phi _{2}(p,t)\,,}$

and similarly for more particles in higher dimensions.

One particular solution to the time-independent Schrödinger equation is

${\displaystyle \Psi _{p}(x)=e^{ipx/\hbar },}$

a plane wave, which can be used in the description of a particle with momentum exactly p, since it is an eigenfunction of the momentum operator. These functions are not normalizable to unity (they aren't square-integrable), so they are not really elements of physical Hilbert space. The set

${\displaystyle \{\Psi _{p}(x,t),-\infty \leq p\leq \infty \}}$

forms what is called the momentum basis. This "basis" is not a basis in the usual mathematical sense. For one thing, since the functions aren't normalizable, they are instead normalized to a delta function,

${\displaystyle \langle \Psi _{p},\Psi _{p'}\rangle =\delta (p-p').}$

For another thing, though they are linearly independent, there are too many of them (they form an uncountable set) for a basis for physical Hilbert space. They can still be used to express all functions in it using Fourier transforms as described above.

## Units of the wave function

Although wave functions are complex numbers, both the real and imaginary parts each have the same units (the imaginary unit i is a pure number without physical units). The units of ψ depend on the number of particles N the wave function describes, and the number of spatial or momentum dimensions n of the system.

When integrating |ψ|2 over all the coordinates, the volume element dnr1dnr2...dnrN has units of [length]Nn. Since the normalization conditions require the integral to be the unitless number 1, |ψ|2 must have units of [length]Nn, thus the units of |ψ| and hence ψ are [length]Nn/2. Likewise, in momentum space, length is replaced by momentum, and the units are [momentum]Nn/2. These results are true for particles of any spin, since for particles with spin, the summations are over dimensionless spin quantum numbers.

## Distinguishable and identical particles

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.[18] This translates to a requirement on the wave function for a system of N identical particles all of common spin s; the wave function of the system is either totally symmetric or totally antisymmetric in all the positions of the particles:[19]

${\displaystyle \Psi \left(\ldots \mathbf {r} _{a},\ldots ,\mathbf {r} _{b},\ldots \right)=\left(-1\right)^{2s}\Psi \left(\ldots \mathbf {r} _{b},\ldots ,\mathbf {r} _{a},\ldots \right)}$

where the + sign occurs if the particles are all bosons (s = 0, 1, 2,...), and sign if they are all fermions (s = 1/2, 3/2,...). For N identical particles there is no such thing as "mixed symmetry": the wave function cannot be symmetric for some of the particles and antisymmetric for others. Notice 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 principle. Generally, bosonic and fermionic symmetry requirements are the manifestation of particle statistics and are present in other quantum state formalisms.

For N distinguishable particles (no two being identical), there is no requirement for the wave function to be either symmetric or antisymmetric.

For a collection of particles, some identical all of spin s with coordinates r1, r2, ... and others distinguishable x1, x2, ... (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 ri only:

${\displaystyle \Psi \left(\ldots \mathbf {r} _{a},\ldots ,\mathbf {r} _{b},\ldots ,\mathbf {x} _{1},\mathbf {x} _{2},\ldots \right)=\left(-1\right)^{2s}\Psi \left(\ldots \mathbf {r} _{b},\ldots ,\mathbf {r} _{a},\ldots ,\mathbf {x} _{1},\mathbf {x} _{2},\ldots \right)}$

Again, there is no symmetry requirement for the distinguishable particle coordinates xi.

## Wave functions as elements of an abstract vector space

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The set of all possible wave functions (at any given time) forms an abstract mathematical vector space. This vector space is infinite-dimensional, because there is no finite set of functions which can be added together in various combinations to create every possible function. Specifically, the entire wave function is treated as a single abstract vector:

${\displaystyle \Psi (\mathbf {r} )\leftrightarrow |\Psi \rangle }$

where Template:Ket is a "ket" (a vector) written in bra–ket notation. As always, the state vector for the system is solved from the Schrödinger equation (or other dynamical pictures of quantum mechanics):

${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$

The statement that "wave functions form an abstract vector space" means that it is possible to multiply wave functions by complex numbers and add together different wave functions in a coherent superposition. If Template:Ket and Template:Ket are two states in the vector space, and a and b are two complex numbers, then the linear combination

${\displaystyle |\Psi \rangle =a|\psi \rangle +b|\phi \rangle }$

(subject to normalization) is also in the same vector space. The dual vectors are denoted as "bras", Template:Bra, which do not live in the same space as Template:Ket, but instead the dual space:

${\displaystyle \langle \Psi |=a^{*}\langle \psi |+b^{*}\langle \phi |}$

where * denotes complex conjugate.

The inner product of two wave functions Template:Ket and Template:Ket can be defined by

${\displaystyle \langle \Psi _{1},\Psi _{2}\rangle =\langle \Psi _{1}|\,|\Psi _{2}\rangle \equiv \langle \Psi _{1}|\Psi _{2}\rangle \,.}$

For these reasons, wave functions are elements of a Hilbert space. See the quantum state article for more explanation of the Hilbert space formalism and its consequences to quantum physics.

There are several advantages to understanding wave functions as elements of an abstract vector space:

• All the powerful tools of linear algebra can be used to manipulate and understand wave functions. For example:
• Linear algebra explains how a vector space can be given a basis, and then any vector in the vector space can be expressed in this basis. This explains the relationship between a wave function in position space and a wave function in momentum space, and suggests that there are other possibilities too.
• Bra–ket notation can be used to manipulate wave functions.
• The idea that quantum states are vectors in an abstract vector space (technically, a complex projective Hilbert space) is completely general in all aspects of quantum mechanics and quantum field theory, whereas the idea that quantum states are complex-valued "wave" functions of space is only true in certain situations.

Following is a summary of the bra–ket formalism applied to wave functions, with general discrete or continuous bases.

### Discrete and continuous bases

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A Hilbert space with a discrete basis Template:Ket for i = 1, 2...n is orthonormal if the inner product of all pairs of basis kets are given by the Kronecker delta:

${\displaystyle \langle \varepsilon _{i}|\varepsilon _{j}\rangle =\delta _{ij}\,.}$

Orthonormal bases are convenient to work with because the inner product of two vectors have simple expressions. A wave function Template:Ket expressed in this discrete basis of the Hilbert space, and the corresponding bra in the dual space, are respectively given by:

${\displaystyle |\Psi \rangle =\sum _{i=1}^{n}c_{i}|\varepsilon _{i}\rangle ={\begin{bmatrix}c_{1}\\\vdots \\c_{n}\end{bmatrix}}\,,\quad \langle \Psi |=|\Psi \rangle ^{\dagger }=\sum _{i=1}^{n}c_{i}^{*}\langle \varepsilon _{i}|={\begin{bmatrix}c_{1}^{*}&\cdots &c_{n}^{*}\end{bmatrix}}\,,}$

where the complex numbers

${\displaystyle c_{i}=\langle \varepsilon _{i}|\Psi \rangle }$

are the components of the vector. The column vector is a useful way to list the numbers, and operations on the entire vector can be done according to matrix addition and multiplication. The entire vector Template:Ket is independent of the basis, but the components depend on the basis. If a change of basis is made, the components of the vector must also change to compensate.

A Hilbert space with a continuous basis } is orthonormal if the inner product of all pairs of basis kets are given by the Dirac delta function:

${\displaystyle \langle \varepsilon |\varepsilon '\rangle =\delta (\varepsilon -\varepsilon ')\,.}$

As with the discrete bases, a symbol ε is used in the basis states, two common notations are Template:Ket and sometimes Template:Ket. A particular basis ket may be subscripted or primed , or simply given another symbol in place of ε.

While discrete basis vectors are summed over a discrete index, continuous basis vectors are integrated over a continuous index (a variable of a function). In what follows, all integrals are with respect to the real-valued basis variable ε (not complex-valued), over the required range. Usually this is just the real line or subsets of it. The state Template:Ket in the continuous basis of the Hilbert space, with the corresponding bra in the dual space, are respectively given by:[20]

${\displaystyle |\Psi \rangle =\int d\varepsilon |\varepsilon \rangle \Psi (\varepsilon )\,,\quad \langle \Psi |=\int d\varepsilon \langle \varepsilon |{\Psi (\varepsilon )}^{*}\,,}$

where the components are the complex-valued functions

${\displaystyle \Psi (\varepsilon )=\langle \varepsilon |\Psi \rangle }$

of a real variable ε.

### Completeness conditions

The completeness conditions (also called closure relations) are

${\displaystyle \sum _{i=1}^{n}|\varepsilon _{i}\rangle \langle \varepsilon _{i}|=1\,,\quad \int d\varepsilon \,|\varepsilon \rangle \langle \varepsilon |=1\,}$

for discrete and continuous orthonormal bases, respectively. An orthonormal set of kets form bases if and only if they satisfy these relations.[20] In each case, the equality to unity means this is an identity operator; its action on any state leaves it unchanged. Multiplying any state on the right of these gives the representation of the state Template:Ket in the basis. The inner product of a first state Template:Ket with a second Template:Ket can also be obtained by multiplying Template:Ket on the left and Template:Ket on the right of the relevant completeness condition.

### Inner product

Physically, the nature of the inner product is dependent on the basis in use, because the basis is chosen to reflect the quantum state of the system.

If Template:Ket is a state in the above basis with components c1, c2, ..., cn and Template:Ket is another state in the same basis with components z1, z2, ..., zn, the inner product is the complex number:

${\displaystyle \langle \Psi _{1}|\Psi _{2}\rangle =\left(\sum _{i}z_{i}^{*}\langle \varepsilon _{i}|\right)\left(\sum _{j}c_{j}|\varepsilon _{j}\rangle \right)=\sum _{ij}z_{i}^{*}c_{j}\langle \varepsilon _{i}|\varepsilon _{j}\rangle =\sum _{i}z_{i}^{*}c_{i}\,.}$

If Template:Ket is a state in the above continuous basis with components Ψ1(ε′), and Template:Ket is another state in the same basis with components Ψ2(ε), the inner product is the complex number:

${\displaystyle \langle \Psi _{1}|\Psi _{2}\rangle =\left(\int d\varepsilon '{\Psi _{1}(\varepsilon ')}^{*}\langle \varepsilon '|\right)\left(\int d\varepsilon \Psi _{2}(\varepsilon )|\varepsilon \rangle \right)=\int d\varepsilon '\int d\varepsilon {\Psi _{1}(\varepsilon ')}^{*}\Psi _{2}(\varepsilon )\langle \varepsilon '|\varepsilon \rangle =\int d\varepsilon {\Psi _{1}(\varepsilon )}^{*}\Psi _{2}(\varepsilon )\,.}$

where the integrals are taken over all ε and ε.

The square of the norm (magnitude) of the state vector Template:Ket is given by the inner product of Template:Ket with itself, a real number:

${\displaystyle \|\Psi \|^{2}=\langle \Psi |\Psi \rangle =\sum _{j=1}^{n}|c_{j}|^{2}\,,\quad \|\Psi \|^{2}=\langle \Psi |\Psi \rangle =\int d\varepsilon \,|\Psi (\varepsilon )|^{2}}$

for the discrete and continuous bases, respectively. Each say the projection of a complex probability amplitude onto itself is real. If Template:Ket is normalized, these expressions would be each separately equal to 1. If the state is not normalized, then dividing by its magnitude normalizes the state:

${\displaystyle |\Psi _{N}\rangle ={\frac {1}{\|\Psi \|}}|\Psi \rangle }$

### Normalized components and probabilities

In the literature, the following results are often presented with normalized wavefunctions. Here, we keep the normalization factors to show where they appear if the wavefunction is not already normalized.

For the discrete basis, projecting the normalized state Template:Ket onto a particular state the system may collapse to, Template:Ket, gives the complex number;

${\displaystyle \langle \varepsilon _{q}|\Psi _{N}\rangle =\langle \varepsilon _{q}|{\frac {1}{\|\Psi \|}}\left(\sum _{i=1}^{n}c_{i}|\varepsilon _{i}\rangle \right)={\frac {c_{q}}{\|\Psi \|}}\,,}$

so the modulus squared of this gives a real number;

${\displaystyle P(\varepsilon _{q})=\left|\langle \varepsilon _{q}|\Psi _{N}\rangle \right|^{2}={\frac {\left|c_{q}\right|^{2}}{\|\Psi \|^{2}}}\,,}$

In the Copenhagen interpretation, this is the probability of state Template:Ket occurring.

In the continuous basis, the projection of the normalized state onto some particular basis Template:Ket is a complex-valued function;

${\displaystyle \langle \varepsilon '|\Psi _{N}\rangle =\langle \varepsilon '|\left({\frac {1}{\|\Psi \|}}\int d\varepsilon |\varepsilon \rangle \Psi (\varepsilon )\right)={\frac {1}{\|\Psi \|}}\int d\varepsilon \langle \varepsilon '|\varepsilon \rangle \Psi (\varepsilon )={\frac {1}{\|\Psi \|}}\int d\varepsilon \delta (\varepsilon '-\varepsilon )\Psi (\varepsilon )={\frac {\Psi (\varepsilon ')}{\|\Psi \|}}\,,}$

so the squared modulus is a real-valued function

${\displaystyle \rho (\varepsilon ')=\left|\langle \varepsilon '|\Psi _{N}\rangle \right|^{2}={\frac {\left|\Psi (\varepsilon ')\right|^{2}}{\|\Psi \|^{2}}}}$

In the Copenhagen interpretation, this function is the probability density function of measuring the observable ε, so integrating this with respect to ε between aε′ ≤ b gives:

${\displaystyle P_{a\leq \varepsilon \leq b}={\frac {1}{\|\Psi \|^{2}}}\int _{a}^{b}d\varepsilon '|\Psi (\varepsilon ')|^{2}={\frac {1}{\|\Psi \|^{2}}}\int _{a}^{b}d\varepsilon '|\langle \varepsilon '|\Psi \rangle |^{2}\,,}$

the probability of finding the system with ε between ε′ = a and ε′ = b.

### Wave function collapse

The physical meaning of the components of Template:Ket is given by the wave function collapse postulate, also known as wave function collapse. If the observable(s) ε (momentum and/or spin, position and/or spin, etc.) corresponding to states Template:Ket has distinct and definite values, λi, and a measurement of that variable is performed on a system in the state Template:Ket then the probability of measuring λi is . If the measurement yields λi, the system "collapses" to the state Template:Ket irreversibly and instantaneously.

### Time dependence

In the Schrödinger picture, the states evolve in time, so the time dependence is placed in Template:Ket according to:[21]

${\displaystyle |\Psi (t)\rangle =\sum _{i}\,|\varepsilon _{i}\rangle \langle \varepsilon _{i}|\Psi (t)\rangle =\sum _{i}c_{i}(t)|\varepsilon _{i}\rangle }$

for discrete bases, or

${\displaystyle |\Psi (t)\rangle =\int d\varepsilon \,|\varepsilon \rangle \langle \varepsilon |\Psi (t)\rangle =\int d\varepsilon \,\Psi (\varepsilon ,t)|\varepsilon \rangle }$

for continuous bases. However, in the Heisenberg picture the states Template:Ket are constant in time and time dependence is placed in the Heisenberg operators, so Template:Ket is not written as Template:Ket.

### Tensor product

It is useful to introduce another operation with the physical interpretation of forming composite states from a collection of other states. This is the tensor product. Given two systems described by states Template:Ket and Template:Ket, the tensor product of the states forms the composite state denoted by or simply without any operation symbol , and the new system includes both of the original systems together. The tensor product state lives in a new space; the tensor product of the original Hilbert spaces. The bases spanning this space are the tensor products of the original bases. The product is not commutative in general, so . If Template:Ket has components ci and Template:Ket has components zj, each in a discrete orthonormal basis Template:Ket, then:

${\displaystyle |\Psi \rangle |\Phi \rangle =\left(\sum _{i}c_{i}|\varepsilon _{i}\rangle \right)\left(\sum _{j}z_{j}|\varepsilon _{j}\rangle \right)=\sum _{i,j}c_{i}z_{j}|\varepsilon _{i}\rangle |\varepsilon _{j}\rangle }$

and the notation can be simplified by abbreviating , Aij = cizj, and , so that

${\displaystyle |A\rangle =\sum _{i,j}A_{ij}|E_{ij}\rangle }$

The same procedure follows for continuous bases using integration. This can also be extended to any number of states, however taking tensor products for fermions and bosons is complicated by the symmetry requirements, see identical particles for general results.

## Position representations

### State space for one spin-0 particle in 1d

For a spinless particle in one spatial dimension (the x-axis or real line), the state Template:Ket can be expanded in terms of a continuum of basis states; Template:Ket, also written Template:Ket, corresponding to the set of all position coordinates x. The completeness condition for this basis is

${\displaystyle 1=\int \limits _{-\infty }^{\infty }dx\,|x\rangle \langle x|}$

and the orthogonality relation is

${\displaystyle \langle x'|x\rangle =\delta (x'-x)}$

The state Template:Ket is expressed by:

${\displaystyle |\Psi \rangle =\left(\int \limits _{-\infty }^{\infty }dx\,|x\rangle \langle x|\right)|\Psi \rangle =\int \limits _{-\infty }^{\infty }dx\,|x\rangle \langle x|\Psi \rangle =\int \limits _{-\infty }^{\infty }dx\,\Psi (x)|x\rangle }$

in which the "wave function" described as a function is a component of the complex state vector.

${\displaystyle \Psi (x)=\langle x|\Psi \rangle }$

The inner product as stated at the beginning of this article is:

${\displaystyle \langle \Psi _{1}|\Psi _{2}\rangle =\langle \Psi _{1}|\left(\int \limits _{-\infty }^{\infty }dx\,|x\rangle \langle x|\right)|\Psi _{2}\rangle =\int \limits _{-\infty }^{\infty }dx\,\langle \Psi _{1}|x\rangle \langle x|\Psi _{2}\rangle =\int \limits _{-\infty }^{\infty }dx\,\Psi _{1}(x)^{*}\Psi _{2}(x)\,.}$

If the particle is confined to a region R (a subset of the x-axis), the integrals in the inner product and completeness condition would be integrals over R.

### State space (other cases)

The previous example can be extended to more particles in higher dimensions, and include spin.

For one spinless particle in 3d, the basis states are Template:Ket and any state vector Template:Ket in this space is expressed in terms of the basis vectors as Template:Ket:

${\displaystyle |\Psi \rangle =\int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} |\mathbf {r} \rangle \langle \mathbf {r} |\Psi \rangle }$

with components:

${\displaystyle \langle \mathbf {r} |\Psi \rangle =\Psi (\mathbf {r} )}$

For N spinless particles in 3d, the basis states are Template:Ket. This is the tensor product of the one-particle position bases Template:Ket, Template:Ket, ..., Template:Ket, each of which spans the separate one-particle Hilbert spaces, so Template:Ket are the basis states for the tensor product of the one-particle Hilbert spaces (the Hilbert space for the composite many particle system). Any state vector Template:Ket in this space is

${\displaystyle |\Psi \rangle =\int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{N}\cdots \int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{2}\int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{1}|\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N}\rangle \langle \mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N}|\Psi \rangle }$

with components:

${\displaystyle \langle \mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N}|\Psi \rangle =\Psi (\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N})}$

For one particle with spin in 3d, the basis states are Template:Ket, the tensor product of the position basis Template:Ket and spin basis Template:Ket, which exists in a new space from the spin space and position space alone. Any state Template:Ket in this space is:

${\displaystyle |\Psi \rangle =\sum _{s_{z}}\int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} |\mathbf {r} ,s_{z}\rangle \langle \mathbf {r} ,s_{z}|\Psi \rangle }$

with components:

${\displaystyle \langle \mathbf {r} ,s_{z}|\Psi \rangle =\Psi (\mathbf {r} ,s_{z})}$

For N particles with spin in 3d, the basis states are Template:Ket, the tensor product of the position basis Template:Ket and spin basis Template:Ket, which exists in a new space from the spin space and position space alone. Any state in this space is:

${\displaystyle |\Psi \rangle =\sum _{s_{z\,1},\ldots ,s_{z\,N}}\int \limits \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{1}\,|\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}\rangle \langle \mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}|\Psi \rangle }$

with components:

${\displaystyle \langle \mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}|\Psi \rangle =\Psi (\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N})}$

If the particles are restricted to regions of position space, then the integrals in the completeness relations are taken over those regions, rather than the entire coordinate space. For the general case of many particles with spin in 3d, if particle 1 is in region R1, particle 2 is in region R2, and so on, the state in this position–spin representation is:

${\displaystyle |\Psi \rangle =\sum _{s_{z\,1},\ldots ,s_{z\,N}}\int \limits \limits _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{R_{1}}d^{3}\mathbf {r} _{1}\,\Psi (\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N})|\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}\rangle }$

The orthogonality relation for this basis is: