# Wave function

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 (also named a state function) in quantum mechanics describes the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of space and time. The Schrödinger equation describes how the wave function evolves over time. The wave function behaves qualitatively like other waves, like 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 most common symbols for a wave function are ψ or Ψ (lower-case and capital psi).

Although values of ψ are complex numbers, |ψ|2 is real corresponding by Max Born's proposal to the probability density of finding a particle in a given place at a given time, if the particle's position is to be measured. Louis de Broglie in his later years proposed a real-valued wave function connected to the complex wave function by a proportionality constant and developed the de Broglie–Bohm theory.

The unit of measurement for ψ depends on the system. For one particle in three dimensions, its units are [length]−3/2. These unusual units are required so that an integral of |ψ|2 over a region of three-dimensional space is a unitless probability (i.e., the probability that the particle is in that region). For different numbers of particles and/or dimensions, the units may be different and can be found by dimensional analysis.[1]

The wave function is central to quantum mechanics as the most direct way to describe the motion of a particle.

Although the wavefunction contains information, it is a complex-valued quantity; only its relative phase and relative magnitude can be measured. It does not directly tell anything about the magnitudes or directions of measurable observables. An operator extracts this information by acting on the wavefunction ψ. For details and examples on how quantum mechanical operators act on the wave function, commutation of operators, and expectation values of operators; see Operator (physics).

## Historical background

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.[2] In each case, the wavefunction was at the centre of attention in two forms, giving quantum mechanics its unity.

In 1905 Planck postulated the proportionality between the frequency of a photon and its energy, in the Planck–Einstein equation, E = hf. In 1925, De Broglie published the symmetric relation between momentum and wavelength, p = h/λ, 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 wavefunctions 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 and De Broglie, were clear on how to interpret it.[3] Around 1924–27, Max Born, Heisenberg, Bohr and others provided the perspective of probability amplitude.[4] This is the Copenhagen interpretation of quantum mechanics. There are many other interpretations of quantum mechanics, but this is considered the most important – since quantum calculations can be understood.

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.[5] The Slater determinant and permanent (of a matrix) was part of the method, provided by John C. Slater.

Interestingly, Schrödinger did encounter an equation for which the wave function satisfied relativistic energy conservation before he published the non-relativistic one, but it led to unacceptable consequences; negative probabilities and negative energies, so he discarded it.[6]:3 In 1927, Klein, Gordon and Fock also found it, but taking a step further: incorporated the electromagnetic interaction into it and proved it was Lorentz-invariant. De Broglie also arrived at exactly the same equation in 1928. This relativistic wave equation is now known most commonly as the Klein–Gordon equation.[7]

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 wavefunction was not a single complex number, but two complex numbers, which 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. He found the wavefunction for this equation could not be a single complex number, but a four-component spinor.[5] Spin automatically entered into the properties of the wavefunction. Later other wave equations were developed: see relativistic wave equations for further information.

## Wave functions and function spaces

Functional analysis is commonly used to formulate the wavefunction 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 an Euclidean space, that implies that wavefunctions are functions of several real variables. Superficially, this formalism is simple to understand for the following reasons.

$i\hbar\frac{\partial }{\partial t}\Psi = \hat{H} \Psi$
• 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 wavefunction is expressed in these coordinates.

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

### Requirements

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

The following constraints on the wavefunction are formulated for the calculations and physical interpretation to make sense:[8][9]

A requirement less restrictive is that the wavefunction 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 wavefunction as a probability amplitude.[10]

## Definitions (spin-0 particles)

### One spin-0 particle in one spatial dimension

Travelling waves of a free particle.
The real parts of position and momentum wave functions Ψ(x) and Φ(p), and corresponding probability densities |Ψ(x)|2 and |Φ(p)|2, for one spin-0 particle in one x or p dimension. The wavefunctions shown are continuous, finite, single-valued and normalized. The colour opacity (%) of the particles corresponds to the probability density (not the wavefunction) of finding the particle at position x or momentum p.

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

#### Position-space wavefunction

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

$\Psi(x,t)$,

where x is position and t is time. This function is complex-valued, meaning that Ψ(x, t) is a complex number.

Interpreted as a probability amplitude, 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 [a, b] (meaning axb) is:

$P_{a\le x\le b} (t) = \int\limits_a^b d x\,|\Psi(x,t)|^2$

where t is the time at which the particle was measured. In other words, |Ψ(x, t)|2 is the probability density that the particle is at x, rather than some other location; see probability amplitude for details. This leads to the normalization condition:

$\int\limits_{-\infty}^\infty d x \, |\Psi(x,t)|^2 = 1\,,$

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

The inner product of two wave functions Ψ1(x, t) and Ψ2(x, t) is useful and important for a number of reasons, and can be defined as the complex number (at time t):

$\langle \Psi_1 , \Psi_2 \rangle = \int\limits_{-\infty}^\infty d x \, \Psi_1^*(x, t)\Psi_2(x, t) \,,$

In the Copenhagen interpretation, the modulus squared of this complex number gives a real number, |Ψ1, Ψ2|2, which is interpreted as the probability of the wavefunction Ψ2 "collapsing" to the new wavefunction Ψ1.

Although the inner product of two wavefunctions is a complex number, the inner product of a wavefunction Ψ with itself, Ψ, Ψ, is always a positive real number. A wavefunction is normalized if that real number is 1: Ψ, Ψ = 1.

The number $\| \Psi \| = \sqrt{\langle \Psi, \Psi \rangle}$ is called the norm of the wavefunction. If the wavefunction Ψ is not normalized, then dividing by its norm will normalize it. Note that the complex modulus $| \Psi | = \sqrt{ \Psi^{*} \Psi }$ and the norm $\| \Psi \| = \sqrt{\langle \Psi, \Psi \rangle}$ are not the same.

A set of wavefunctions Ψ1, Ψ2, ... are orthonormal if they are each normalized (inner product each wavefunction with itself is 1) and are all orthogonal to each other (inner product of any two different wavefunctions is zero):

$\langle \Psi_m , \Psi_n \rangle = \int\limits_{-\infty}^\infty d x \, \Psi_m^*(x, t)\Psi_n(x, t) = \delta_{mn} \,,$

where m and n each take values 1, 2, ..., and δmn is the Kronecker delta.

Since the Schrödinger equation is linear, if any number of wavefunctons 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):

$\Psi(x,t) = \sum_n a_n \Psi_n(x,t) = a_1 \Psi_1(x,t) + a_2 \Psi_2(x,t) + \cdots$

This is the superposition principle. Note that multiplying a wavefunction by any nonzero but constant complex number c (also called a phase factor in this context) does not change any information about the quantum system, because cΨ satisfies exactly the same Schrödinger equation Ψ (the constant c cancels).

Since linear combinations of wavefunctions obtain more wavefunctions, the set of all wavefunctions forms a complex vector space over the field of complex numbers (more details are given later).

#### Momentum-space wavefunction

The particle also has a wave function in momentum space:

$\Phi(p,t)$

where p is the momentum in one dimension, which can be any value from −∞ to +∞, and t is time. Analogous to the position case, the inner product of two wave functions Φ1(p, t) and Φ2(p, t) can be defined as:

$\langle \Phi_1 , \Phi_2 \rangle = \int\limits_{-\infty}^\infty d p \, \Phi_1^*(p, t)\Phi_2(p, t) \,,$

Interpreted as a probability amplitude, if the particle's momentum is measured, the result is not deterministic, but is described by a probability distribution:

$P_{a\le p\le b} (t) = \int\limits_a^b d p \, |\Phi(p,t)|^2$,

and the normalization condition is similar:

$\langle \Phi , \Phi \rangle = \int\limits_{-\infty}^{\infty} d p \, \left | \Phi \left ( p, t \right ) \right |^2 = 1.$

#### Relation between wavefunctions

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. For one dimension:[11]

$\Phi(p,t) = \frac{1}{\sqrt{2\pi\hbar}}\int\limits_{-\infty}^\infty d x \, e^{-ipx/\hbar} \Psi(x,t) \upharpoonleft \downharpoonright \Psi(x,t) = \frac{1}{\sqrt{2\pi\hbar}}\int\limits_{-\infty}^\infty d p \, e^{ipx/\hbar} \Phi(p,t)$

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

$p = \hbar k,$

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 wavefunction is used much more often than the momentum-space wavefunction.

#### Example of normalization

A particle is restricted to a 1D region between x = 0 and x = L; its wave function is:

\begin{align} \Psi (x,t) & = Ae^{i(kx-\omega t)}, & 0 \leq x \leq L \\ \Psi (x,t) & = 0, & x < 0, x > L \\ \end{align}.

To normalize the wave function we need to find the value of the arbitrary constant A; solved from

$\int\limits_{-\infty}^{\infty} dx \, |\Psi|^2 = 1 .$

From Ψ, we have |Ψ|2 = A2, so the integral becomes;

$\int\limits_{-\infty}^0 dx \cdot 0 + \int\limits_0^L dx \, A^2 + \int\limits_L^\infty dx \cdot 0 = 1 ,$

Solving this equation gives A = 1/L, so the normalized wave function in the box is;

$\Psi (x,t) = \frac{1}{\sqrt{L}} e^{i(kx-\omega t)}, \quad 0 \leq x \leq L.$

### One spin-0 particle in three spatial dimensions

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.

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

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

where r is the position vector in three-dimensional space, and t is time. As always Ψ(r, t) is a complex number.

The inner product of two wave functions Ψ1(r, t) and Ψ2(r, t) can be defined as the complex number:

$\langle \Psi_1 , \Psi_2 \rangle = \int\limits_{\mathrm{ all \, space}} d^3\mathbf{r} \, \Psi_1^*(\mathbf{r},t)\Psi_2(\mathbf{r},t) \,,$

Interpreted as a probability amplitude, if the particle's position is measured at time t, the probability that it is in a region R is:

$P_{\mathbf{r}\in R} (t) = \int\limits_R d^3\mathbf{r} \, \left |\Psi(\mathbf{r},t) \right |^2$

(a three-dimensional integral over the region R, with differential volume element d3r, also written "dV" or "dx dy dz"). The normalization condition is:

$\langle \Psi , \Psi \rangle = \int\limits_{\mathrm{ all \, space}} d^3\mathbf{r} \, \left | \Psi(\mathbf{r},t)\right |^2 = 1,$

where the integrals are taken over all of three-dimensional space.

The 3d Fourier transform of the position space wavefunction gives the corresponding momentum space wavefunction[12]

$\Phi(\mathbf{p},t) = \frac{1}{\sqrt{\left(2\pi\hbar\right)^3}}\int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r} \, e^{-i \mathbf{r}\cdot \mathbf{p} /\hbar} \Psi(\mathbf{r},t)$

where p is the momentum in 3-dimensional space, and t is time.

The inner product of two wave functions Φ1(p,t) and Φ2(p, t) can be defined as the complex number:

$\langle \Phi_1 , \Phi_2 \rangle = \int\limits_{\mathrm{ all \, space}} d^3\mathbf{p} \, \Phi_1^*(\mathbf{p},t)\Phi_2(\mathbf{p},t) \,,$

Interpreted as a probability amplitude, the probability of measuring the momentum vector in a region of momentum space M is given by:

$P_{\mathbf{p} \in M} (t) = \int_M d^3 \mathbf{p} \left | \Phi \left ( \mathbf{p}, t \right ) \right |^2 ,$

where, analogous to position space, d3p = dpx dpy dpz is a differential 3-momentum volume element in momentum space. The normalization condition is:

$\langle \Phi , \Phi \rangle = \int\limits_{\mathrm{ all \, space}} d^3\mathbf{p} \, \left | \Phi \left ( \mathbf{p}, t \right ) \right |^2 = 1.$

To get back to the position space wavefunction, we apply the inverse Fourier transform on the momentum space wavefunction:

$\Psi(\mathbf{r},t) = \frac{1}{\sqrt{\left(2\pi\hbar\right)^3}}\int\limits_{\mathrm{ all \, space}} d^3\mathbf{p} \, e^{i \mathbf{r}\cdot \mathbf{p} /\hbar} \Phi(\mathbf{p},t).$

### Many spin-0 particles in three spatial dimensions

Travelling waves of two free particles, with two of three dimensions suppressed. Top is position space wavefunction, bottom is momentum space wavefunction, 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 N particles is written:[5]

$\Psi(\mathbf{r}_1,\mathbf{r}_2 \cdots \mathbf{r}_N,t)$

where ri is the position of the ith particle in three-dimensional space, and t is time.

The inner product of two wave functions Ψ1(r1, r2, ..., rN, t) and Ψ2(r1, r2, ..., rN, t) can be defined as the complex number:

$\langle \Psi_1 , \Psi_2 \rangle = \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_1 \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_2\cdots \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_N \, \Psi_1^*(\mathbf{r}_1 \cdots \mathbf{r}_N,t)\Psi_2(\mathbf{r}_1 \cdots \mathbf{r}_N,t) \,,$

(altogether, this is 3N one-dimensional integrals).

Interpreted as a probability amplitude, if the particles' positions are all measured simultaneously at time t, the probability that particle 1 is in region R1 and particle 2 is in region R2 and so on is:

$P_{\mathbf{r}_1\in R_1,\mathbf{r}_2\in R_2 \cdots \mathbf{r}_N\in R_N}(t) = \int\limits_{R_1} d ^3\mathbf{r}_1 \int\limits_{R_2} d ^3\mathbf{r}_2\cdots \int\limits_{R_N} d ^3\mathbf{r}_N |\Psi(\mathbf{r}_1 \cdots \mathbf{r}_N,t)|^2$

The normalization condition is:

$\langle \Psi , \Psi \rangle = \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_1 \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_2\cdots \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_N |\Psi(\mathbf{r}_1 \cdots \mathbf{r}_N,t)|^2 = 1$

## Definitions (particles with spin)

### One particle with spin in three dimensions

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

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

where r is a position in three-dimensional space, t is time, and sz is 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.)

The inner product of two wave functions Ψ1(r1, sz, t) and Ψ2(r1, sz, t) can be defined as the complex number:

$\langle \Psi_1 , \Psi_2 \rangle = \sum_{\mathrm{all\, }s_z} \int\limits_{\mathrm{ all \, space}} \, d^3\mathbf{r} \Psi^{*}_1(\mathbf{r},t,s_z)\Psi_2(\mathbf{r},t,s_z)$

Interpreted as a probability amplitude, if the particle's position and spin is measured simultaneously at time t, the probability that its position is in R1 and its spin projection quantum number is a certain value sz = m is:

$P_{\mathbf{r}\in R,s_z=m} (t) = \int\limits_{R} \, d ^3\mathbf{r} |\Psi(\mathbf{r},t,m)|^2$

The normalization condition is:

$\langle \Psi , \Psi \rangle = \sum_{\mathrm{all\, }s_z} \int\limits_{\mathrm{ all \, space}} \, d^3\mathbf{r} |\Psi(\mathbf{r},t,s_z)|^2 = 1$.

Since the spin quantum number has discrete values, it must be written as a sum rather than an integral, taken over all possible values.

It is convenient to write the wavefunction as a column vector, in which there are as many entries in the column vector as there are allowed values of sz, and the entries are indexed by the spin quantum number:[13]

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

and the normalization condition is equivalent to:

$\int\limits_{\mathrm{ all \, space}} \, d ^3\mathbf{r} \Psi^\dagger (\mathbf{r},t)\Psi(\mathbf{r},t) = 1$.

where the dagger denotes the Hermitian conjugate (complex conjugate transpose of the column vector into a row vector).

### Many particles with spin in three dimensions

Likewise, the wavefunction for N particles each with spin is:

$\Psi(\mathbf{r}_1, \mathbf{r}_2 \cdots \mathbf{r}_N, s_{z\,1}, s_{z\,2} \cdots s_{z\,N}, t)$

The inner product of two wave functions Ψ1(r1, r2, ..., rN, sz 1, sz 2, ..., sz N, t) and Ψ2(r1, r2, ..., rN, sz 1, sz 2, ..., sz N, t) can be defined as the complex number:

$\langle \Psi_1 , \Psi_2 \rangle = \sum_{s_{z\,N}} \cdots \sum_{s_{z\,2}} \sum_{s_{z\,1}} \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_1 \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_2\cdots \int\limits_{\mathrm{ all \, space}} d ^3 \mathbf{r}_N \Psi^{*}_1 \left(\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right )\Psi_2 \left(\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right )$

Now there are 3N one-dimensional integrals followed by N sums.

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. reads:

$P_{\mathbf{r}_1\in R_1,s_{z\,1} = m_1, \ldots, \mathbf{r}_N\in R_N,s_{z\,N} = m_N} (t) = \int\limits_{R_1} d ^3\mathbf{r}_1 \int\limits_{R_2} d ^3\mathbf{r}_2\cdots \int\limits_{R_N} d ^3\mathbf{r}_N \left | \Psi\left (\mathbf{r}_1 \cdots \mathbf{r}_N,m_1\cdots m_N,t \right ) \right |^2$

The normalization condition is:

$\langle \Psi , \Psi \rangle = \sum_{s_{z\,N}} \cdots \sum_{s_{z\,2}} \sum_{s_{z\,1}} \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_1 \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_2\cdots \int\limits_{\mathrm{ all \, space}} d ^3 \mathbf{r}_N \left | \Psi \left (\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) \right |^2 = 1$

## Distinguishable and identical particles

In quantum mechanics there is a fundamental distinction between identical particles and distinguishable particles. For example, any two electrons are 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.[14] This translates to a requirement on the wavefunction: For example, if particles 1 and 2 are indistinguishable, then:

$\Psi \left ( \mathbf{r},\mathbf{r'},\mathbf{r}_3,\mathbf{r}_4,\cdots \right ) = \left ( -1 \right )^{2s} \Psi \left ( \mathbf{r'},\mathbf{r},\mathbf{r}_3,\mathbf{r}_4,\cdots \right )$

where s is the spin quantum number of the particle: integer for bosons (s = 1, 2, 3, ...) and half-integer for fermions (s = 1/2, 3/2, ...).

The wavefunction is said to be symmetric (no sign change) under boson interchange and antisymmetric (sign changes) under fermion interchange. The latter feature of the wavefunction 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 interacting distinguishable particles (all the particles are different, no two are identical), which do not interact mutually and move independently in a time-independent potential, the (spatial part of the) wavefunction can be separated into a product of separate wavefunctions for each particle:[15]

$\Psi (\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N) = \prod_{i=1}^N\psi(\mathbf{r}_i) = \psi(\mathbf{r}_1)\psi(\mathbf{r}_2)\cdots\psi(\mathbf{r}_N).$

This separation of variables is a simple method for solving partial differential equations like the Schrödinger equation. If the potential is time-dependent, then the wavefunction cannot be separated into the separate wavefunctions of the particles.

For non-interacting distinguishable particles in a time-independent potential, the spatial part of the wavefunction is the product of separate wavefunctions for each particle:

$\Psi (\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}, s_{z\,2} \cdots s_{z\,N} ) = \prod_{i=1}^N\psi(\mathbf{r}_i,s_{z\,i}) = \psi(\mathbf{r}_1,s_{z\,1})\psi(\mathbf{r}_2,s_{z\,2})\cdots\psi(\mathbf{r}_N,s_{z\,N}).$

## Units of the wavefunction

Even though wavefunctions are complex numbers, both the real and imaginary parts each have the same units (the imaginary unit i is a number without unit). The units of ψ depend on the number of particles the wavefunction describes, and the number of spatial or momentum dimensions of the system. In general, for N particles with positions r1, r2, ..., rN in n spatial dimensions, the normalization conditions require ψ to have units of [length]Nn/2. The square root of the length unit is removed when one finds |ψ|2, which has units of [length]Nn.

In momentum space, length is replaced by momentum, and the units are [momentum]Nn/2.

These results are true for particles with or without spin, since for particles with spin, the summations are over dimensionless spin quantum numbers.

## Wave functions as elements of an abstract vector space

The set of all possible wave functions (at any given time) forms an abstract mathematical vector space. Specifically, the entire wave function is treated as a single abstract vector:

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

where 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):

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

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. It is a Hilbert space, for the following reasons.

• The statement that "wave functions form an abstract vector space" means that it is possible multiply wave functions by complex numbers and add together different wave functions in a coherent superposition. If |ψ and |ϕ are two states in the Hilbert space, and a and b are two complex numbers, then the linear combination
$|\Psi \rangle = a|\psi\rangle + b|\phi\rangle$
(subject to normalization, see below) is also in the Hilbert space. The dual vectors are denoted as "bras", Ψ|, which do not live in the same space as , but instead the dual space:
$\langle \Psi | = a^{*} \langle \psi | + b^{*} \langle \phi |$
where * denotes complex conjugate.
• The inner product of wave functions can be defined.

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 breakdown of the bra–ket formalism. Kets are analogous to the more elementary Euclidean vectors, although the components are complex-valued.

### Discrete and continuous bases

Discrete components Ak of a complex vector |A = ∑k Ak|ek, which belongs to a countably infinite-dimensional Hilbert space; there are countably infinitely many k values and basis vectors |ek.
Continuous components ψ(x) of a complex vector |ψ = ∫dx ψ(x)|ψ, which belongs to an uncountably infinite-dimensional Hilbert space; there are uncountably infinitely many x values and basis vectors |x.
Components of complex vectors plotted against index number; discrete k and continuous x. Two probability amplitudes out of infinitely many are highlighted.

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

$\langle \varepsilon_i | \varepsilon_j \rangle = \delta_{ij}\,.$

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

$| \psi \rangle = \sum_{i = 1}^n c_i | \varepsilon_i \rangle = \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix} \,,\quad \langle \psi | = | \psi \rangle^{*} = \sum_{i = 1}^n c^{*}_i \langle \varepsilon_i | = \begin{bmatrix} c_1^{*} & \cdots & c_n^{*} \end{bmatrix} \,,$

where the complex numbers ci = εi|ψ are the components of the vector. The column vector is a useful representation in terms of matrices. The entire vector |ψ 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 { |ε } is orthonormal if the inner product of all pairs of basis kets are given by the Dirac delta function:

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

As with the discrete bases, a symbol ε is used in the basis states, two common notations are |ε and sometimes |ψε. A particular basis ket may be subscripted |ε0|ψε0 or primed |ε|ψε.

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 |ψ in the continuous basis of the Hilbert space, with the corresponding bra in the dual space, are respectively given by:[16][17]

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

where the components are the complex-valued functions ψ(ε) = ε|ψ of a real variable ε.

### Completeness conditions

The completeness conditions (also called closure relations) are

$\sum_{i=1}^n | \varepsilon_i \rangle \langle \varepsilon_i | = 1 \,,\quad \int d\varepsilon \, | \varepsilon \rangle \langle \varepsilon | = 1 \,.$

for the discrete and continuous orthonormal bases, respectively. An orthonormal set of kets form bases if and only if they satisfy these relations.[17] 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 |ψ in the basis. The inner product of a first state |χ with a second |ψ can also be obtained by multiplying χ| on the left and |ψ 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 |ψ is a state in the above basis with components c1, c2, ..., cn and |χ is another state in the same basis with components z1, z2, ..., zn, the inner product is the complex number:

$\langle \chi | \psi \rangle = \left(\sum_i z^*_i \langle\varepsilon_i | \right)\left(\sum_j c_j |\varepsilon_j\rangle \right) =\sum_{ij} z^*_i c_j \delta_{ij} =\sum_i z^*_i c_i \,.$

If |ψ is a state in the above continuous basis with components ψ(ε), and |χ is another state in the same basis with components χ(ε′), the inner product is the complex number:

$\langle \chi | \psi \rangle = \left( \int d\varepsilon' {\chi(\varepsilon')}^{*} \langle \varepsilon' | \right)\left(\int d\varepsilon \psi(\varepsilon) |\varepsilon \rangle \right) = \int d\varepsilon' \int d\varepsilon {\chi(\varepsilon')}^{*} \psi(\varepsilon) \delta (\varepsilon' - \varepsilon ) = \int d\varepsilon {\chi(\varepsilon)}^{*} \psi(\varepsilon) \,.$

where the integrals are taken over all ε and ε.

### Normalization

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

$\|\psi\|^2 = \langle \psi | \psi \rangle = \sum_{j=1}^n | c_j |^2 \,,\quad \|\psi\|^2 = \langle \psi | \psi \rangle = \int d \varepsilon \, | \psi(\varepsilon) |^2$

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

$| \psi_N \rangle = \frac{1}{\| \psi \|} | \psi \rangle$

### Normalized components and probabilities

For the discrete basis, projecting the normalized state |ψN onto a particular state the system may collapse to, |εq, gives the complex number;

$\langle \varepsilon_q | \psi_N \rangle = \langle \varepsilon_q | \frac{1}{\|\psi\|} \left ( \sum_{i = 1}^n c_i | \varepsilon_i \rangle \right ) = \frac{c_q}{\|\psi\|} \,,$

so the modulus squared of this gives a real number;

$P(\varepsilon_q) = \left| \langle \varepsilon_q | \psi_N \rangle \right|^2 = \frac{\left|c_q\right|^2}{\|\psi\|^2} \, ,$

In the Copenhagen interpretation, this is the probability of state |εq occurring.

In the continuous basis, the projection of the normalized state onto some particular basis |ε is a complex-valued function;

$\langle \varepsilon' | \psi_N \rangle = \langle \varepsilon' | \left( \frac{1}{\|\psi\|}\int d \varepsilon | \varepsilon \rangle \psi(\varepsilon) \right) = \frac{1}{\|\psi\|}\int d \varepsilon \langle \varepsilon' | \varepsilon \rangle \psi(\varepsilon) = \frac{1}{\|\psi\|}\int d \varepsilon \delta( \varepsilon' - \varepsilon ) \psi(\varepsilon) = \frac{\psi(\varepsilon')}{\|\psi\|} \,,$

so the squared modulus is a real-valued function

$\rho(\varepsilon') = \left| \langle \varepsilon' | \psi_N \rangle \right|^2 = \frac{\left|\psi(\varepsilon')\right|^2}{\|\psi\|^2}$

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:

$P_{a \leq \varepsilon \leq b} = \frac{1}{\|\psi\|^2}\int_a^b d\varepsilon' | \psi(\varepsilon') |^2 = \frac{1}{\|\psi\|^2}\int_a^b d\varepsilon' | \langle \varepsilon' | \psi \rangle |^2 \,,$

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

### Wave function collapse

The physical meaning of the components of |ψ 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 |εi has distinct and definite values, λi, and a measurement of that variable is performed on a system in the state |ψ then the probability of measuring λi is |εi|ψ|2. If the measurement yields λi, the system "collapses" to the state |εi, irreversibly and instantaneously.

### Time dependence

In the Schrödinger picture, the states evolve in time, so the time dependence is placed in |ψ according to:[18]

$|\psi(t)\rangle = \sum_i \, | \varepsilon_i \rangle \langle \varepsilon_i | \psi(t)\rangle = \sum_i c_i(t) | \varepsilon \rangle$

for discrete bases, or

$|\psi(t)\rangle = \int d\varepsilon \, | \varepsilon \rangle \langle \varepsilon | \psi(t)\rangle = \int d\varepsilon \, \psi(\varepsilon,t) | \varepsilon \rangle$

for continuous bases. However, in the Heisenberg picture the states |ψ are constant in time and time dependence is placed in the Heisenberg operators, so |ψ is not written as |ψ(t).

## Position representations (spinless particles)

### 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 |ψ can be expanded in terms of a continuum of states; |x, also written |ψx, corresponding to the set of all position coordinates x.

If the particle is confined to a region R (a subset of the x-axis), the state is:

$| \psi \rangle = \int\limits_R d x \, | x \rangle \langle x | \psi \rangle = \int\limits_R d x \, \psi(x) | x \rangle$

$1 = \int\limits_R d x \, | x \rangle \langle x |$

and the inner product as stated at the beginning of this article (in that case R = (−∞, ∞)):

$\langle \chi | \psi \rangle = \int\limits_R d x \, \langle \chi | x \rangle \langle x | \psi \rangle = \int\limits_R d x \, \chi(x)^{*} \psi(x) \,.$.

The "wavefunction" described previously is simply a component of the complex state vector. Projecting |ψ onto a particular position state |x0, where x0 is in R:

$\langle x_0 | \psi \rangle = \int\limits_R d x \, \langle x_0 | x \rangle \psi(x) = \int\limits_R d x \, \delta( x_0 - x ) \psi(x) = \psi(x_0) \,.$

### State space for one spin-0 particle in 3d

In three dimensions, |ψ can be expanded in terms of a continuum of states with definite position, |r, also written |x, y, z or |ψr, corresponding to each r = (x, y, z).

If the particle is confined to a region R (a subset of 3d space), the state is;

$| \psi \rangle = \int\limits_R d^3\mathbf{r} \, | \mathbf{r} \rangle \langle \mathbf{r} | \psi\rangle = \int\limits_R d^3\mathbf{r} \, \psi(\mathbf{r}) | \mathbf{r} \rangle$

The closure relation is

$1 = \int\limits_R d^3\mathbf{r} \, | \mathbf{r} \rangle \langle \mathbf{r} |$

leading to the inner product of |ψ with itself leads to the normalization conditions in the three-dimensional definitions above:

$\langle \chi | \psi \rangle = \int\limits_R d^3\mathbf{r} \, \langle \chi | \mathbf{r} \rangle \langle \mathbf{r} | \psi \rangle = \int\limits_R d^3\mathbf{r} \, \chi(\mathbf{r})^{*} \psi(\mathbf{r})$.

Projecting |ψ onto a particular position state |r0, where r0 is in R:

$\langle \mathbf{r}_0 | \psi \rangle = \int\limits_R d^3 \mathbf{r} \, \langle \mathbf{r}_0 | \mathbf{r} \rangle \psi(\mathbf{r}) = \int\limits_R d^3 \mathbf{r} \, \delta( \mathbf{r}_0 - \mathbf{r} ) \psi(\mathbf{r}) = \psi(\mathbf{r}_0)$

### State space for many spin-0 particles in 3d

In three dimensions, |ψ can be expanded in terms of a continuum of states for all the particles each with definite position, |r1, r2, ..., rN, corresponding to each rj = (xj, yj, zj).

If particle 1 is in region R1, particle 2 is in region R2, and so on, the state in this position representation is:

\begin{align} | \psi \rangle & = \int\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits_{R_2} d^3\mathbf{r}_2 \int\limits_{R_1} 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 \\ & = \int\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits_{R_2} d^3\mathbf{r}_2 \int\limits_{R_1} d^3\mathbf{r}_1 \, \psi ( \mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N ) | \mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N \rangle \end{align}

The closure relation is

$1 = \int\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits_{R_2} d^3\mathbf{r}_2 \int\limits_{R_1} d^3\mathbf{r}_1 \, | \mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N \rangle \langle \mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N |$

leading to the inner product of |ψ with itself leads to the normalization conditions in the three-dimensional definitions above:

$\langle \chi | \psi \rangle = \int\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits_{R_2} d^3\mathbf{r}_2 \int\limits_{R_1} d^3\mathbf{r}_1 \chi(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N)^{*} \psi(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N)$.

Projecting |ψ onto a particular position state |r1 0, r2 0, ..., rN 0, where r1 0 is in R1, r2 0 is in R2, etc., gives ψ(r1 0, r2 0, ..., rN 0).

## Position and spin representations

### State space for one particle with spin in 3d

In Dirac notation, for a particle with spin s, in all three spatial dimensions, the basis states |r, sz are a combination of the discrete variable sz and the continuous variable r,[13] more specifically a tensor product of the spin basis |sz and position basis |r, which exists in a new space from the spin space and position space alone.[19] Applying the above formalism, the state can be written:

$| \Psi \rangle = \sum_{s_z} \int\limits_R d^3 \, \mathbf{r} \Psi(\mathbf{r},s_z) | \mathbf{r}, s_z \rangle$

and therefore the closure relation is:

$1 = \sum_{s_z} \int\limits_R d^3 \, \mathbf{r} | \mathbf{r},s_z\rangle \langle \mathbf{r} , s_z |$

Projecting Ψ onto a particular position-spin state |r0, m, where r0 is in R:

$\langle \mathbf{r}_0, m | \Psi \rangle = \sum_{s_z}\int\limits_R d^3 \mathbf{r} \, \langle \mathbf{r}_0, m | \mathbf{r}, s_z \rangle \Psi(\mathbf{r}, s_z) = \sum_{s_z}\int\limits_R d^3 \mathbf{r} \, \delta_{m \, s_z}\delta( \mathbf{r}_0 - \mathbf{r} ) \Psi(\mathbf{r}, s_z) = \Psi(\mathbf{r}_0, m) \,.$

where the joint orthogonality relation

$\langle \mathbf{r}_0, m | \mathbf{r}, s_z \rangle = \delta_{m\,s_z}\delta( \mathbf{r}_0 - \mathbf{r} )$

has been employed. For more particles, there are sums over the allowed spins for each particle, and integrals over the position vector for each particle.

## Ontology

Whether the wave function really exists, and what it represents, are major questions in the interpretation of quantum mechanics. Many famous physicists of a previous generation puzzled over this problem, such as Schrödinger, Einstein and Bohr. Some advocate formulations or variants of the Copenhagen interpretation (e.g. Bohr, Wigner and von Neumann) while others, such as Wheeler or Jaynes, take the more classical approach[20] and regard the wave function as representing information in the mind of the observer, i.e. a measure of our knowledge of reality. Some, including Schrödinger, Einstein, Bohm and Everett and others, argued that the wave function must have an objective, physical existence. The latter argument is consistent with the fact that whenever two observers both think that a system is in a pure quantum state, they will always agree on exactly what state it is in (but this may not be true if one or both of them thinks the system is in a mixed state).[21] For more on this topic, see Interpretations of quantum mechanics.

## Examples

### One-dimensional quantum tunnelling

Scattering at a finite potential barrier of height $V_0$. The amplitudes and direction of left and right moving waves are indicated. In red, those waves used for the derivation of the reflection and transmission amplitude. $E > V_0$ for this illustration.

One of most prominent features of the wave mechanics is a possibility for a particle to reach a location with a prohibitive (in classical mechanics) force potential. In the one-dimensional case of particles with energy less than $V_0$ in the square potential

$V(x)=\begin{cases}V_0 & |x|

the steady-state solutions to the wave equation have the form (for some constants $k, \kappa$)

$\psi (x) = \begin{cases} A_{\mathrm{r}}\exp(ikx)+A_{\mathrm{l}}\exp(-ikx) & x<-a, \\ B_{\mathrm{r}}\exp(\kappa x)+B_{\mathrm{l}}\exp(-\kappa x) & |x|\le a, \\ C_{\mathrm{r}}\exp(ikx)+C_{\mathrm{l}}\exp(-ikx) & x>a. \end{cases}$

Note that these wave functions are not normalized; see scattering theory for discussion.

The standard interpretation of this is as a stream of particles being fired at the step from the left (the direction of negative x): setting Ar = 1 corresponds to firing particles singly; the terms containing Ar and Cr signify motion to the right, while Al and Cl – to the left. Under this beam interpretation, put Cl = 0 since no particles are coming from the right. By applying the continuity of wave functions and their derivatives at the boundaries, it is hence possible to determine the constants above.

### Other

Here are examples of wavefunctions for specific applications:

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