Wave function

Not to be confused with Wave equation.

Comparison of classical and quantum harmonic oscillator conceptions for a single spinless particle. The two processes differ greatly. The classical process (A–B) is represented as the motion of a particle along a trajectory. The quantum process (C–H) has no such trajectory. Rather, it is represented as a wave. Panels (C–F) show four different standing wave solutions of the Schrödinger equation. Panels (G–H) further show two different wave functions that are solutions of the Schrödinger equation but not standing waves.

A wave function in quantum mechanics describes the quantum state of an isolated system of one or more particles. There is one wave function containing all the information about the entire system, not a separate wave function for each particle in the system. Its interpretation is that of a probability amplitude. Quantities associated with measurements, such as the average momentum of a particle, can be derived from the wave function. It is a central entity in quantum mechanics and is important in all modern theories, like quantum field theory incorporating quantum mechanics, while its interpretation may differ. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi).

For a given system, once a representation corresponding to a maximal set of commuting observables and a suitable coordinate system is chosen, the wave function is a complex-valued function of the system's degrees of freedom corresponding to the chosen representation and coordinate system, continuous as well as discrete. Such a set of observables, by a postulate of quantum mechanics, are Hermitian linear operators on the space of states representing a set of physical observables, like position, momentum and spin that can, in principle, be simultaneously measured with arbitrary precision. Wave functions can be added together and multiplied by complex numbers to form new wave functions, and hence are elements of a vector space. This is the superposition principle of quantum mechanics. This vector space is endowed with an inner product such that it is a complete metric topological space with respect to the metric induced by the inner product. In this way the set of wave functions for a system form a function space that is a Hilbert space. The inner product is a measure of the overlap between physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The actual space depends on the system's degrees of freedom (hence on the chosen representation and coordinate system) and the exact form of the Hamiltonian entering the equation governing the dynamical behavior. In the non-relativistic case, disregarding spin, this is the Schrödinger equation.

The Schrödinger equation determines the allowed wave functions for the system and how they evolve over time. A 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.

For a given system, the choice of which relevant degrees of freedom to use are not unique, and correspondingly the domain of the wave function is not unique. It may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space, the two are related by a Fourier transform. These descriptions are the most important, but they are not the only possibilities. Just like in classical mechanics, canonical transformations may be used in the description of a quantum system. Some particles, like electrons and photons, have nonzero spin, and the wave function must include this fundamental property as an intrinsic discrete 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. The terminology derives from how the wave functions transform under a rotation of the coordinate system. No elementary particle with spin ​³⁄₂ or higher is known, except for the hypothesized spin 2 graviton. Other discrete variables can be included, such as isospin. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g. a point in space) assigns a complex number for each possible value of the discrete degrees of freedom (e.g. z-component of spin). These values are often displayed in a column matrix (e.g. a 2 × 1 column vector for a non-relativistic electron with spin ​¹⁄₂).

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 or having a definite momentum, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation, this general requirement a wave function must satisfy is called the normalization condition. 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, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.

The unit of measurement for ψ depends on the system, and can be found by dimensional analysis of the normalization condition for the system. For one particle in three dimensions, its units are [length]^−3/2, because an integral of | ψ |² over a region of three-dimensional space is a dimensionless probability.

Historical background

In 1905 Einstein postulated the proportionality between the frequency of a photon and its energy, E = hf, and in 1916 the corresponding relation between photon momentum and wavelength, λ = h/p. In 1923, De Broglie was the first to suggest that the relation λ = h/p, now called the De Broglie relation, holds for massive particles, the chief clue being Lorentz invariance, and this can be viewed as the starting point for the modern development of quantum mechanics. The equations represent wave–particle duality for both massless and massive particles.

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.

In 1926, Schrödinger published the famous wave equation now named after him, indeed the Schrödinger equation, based on classical Conservation of energy using quantum operators and the de Broglie relations such that the solutions of the equation are the wave functions for the quantum system. However, no one was clear on how to interpret it. At first, Schrödinger and others thought that wave functions represent particles that are spread out with most of the particle being where the wave function is large. This was shown to be incompatible with how elastic scattering of a wave packet representing a particle off a target appears; it spreads out in all directions. While a scattered particle may scatter in any direction, it does not break up and take off in all directions. In 1926, Born provided the perspective of probability amplitude. This relates calculations of quantum mechanics directly to probabilistic experimental observations. It is accepted as part of the Copenhagen interpretation of quantum mechanics. There are many other interpretations of quantum mechanics. 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. 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.

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: 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 wave function for the electron. Later, other relativistic wave equations were found.

Wave functions and wave equations in modern theories

All these wave equations are of enduring importance. The Schrödinger equation and the Pauli equation are under many circumstances excellent approximations of the relativistic variants. They are considerably easier to solve in practical problems than the relativistic equations. The Klein-Gordon equation and the Dirac equation, while being relativistic, do not represent full reconciliation of quantum mechanics and special relativity. The branch of quantum mechanics where these equations are studied the same way as the Schrödinger equation, often called relativistic quantum mechanics, while very successful, has its limitations (see e.g. Lamb shift) and conceptual problems (see e.g. Dirac sea).

Relativity makes it inevitable that the number of particles in a system is not constant. For full reconciliation, quantum field theory is needed. In this theory, the wave equations and the wave functions have their place, but in a somewhat different guise. The main objects of interest are not the wave functions, but rather operators, so called field operators (or just fields where "operator" is understood) on the Hilbert space of states (to be described next section). It turns out that the original relativistic wave equations and their solutions are still needed to build the Hilbert space. Moreover, the free fields operators, i.e. when interactions are assumed not to exist, turn out to (formally) satisfy the same equation as do the fields (wave functions) in many cases.

Thus the Klein-Gordon equation (spin 0) and the Dirac equation (spin ​¹⁄₂) in this guise remain in the theory. Higher spin analogues include the Proca equation (spin 1), Rarita–Schwinger equation (spin ​³⁄₂), and, more generally, the Bargmann–Wigner equations. For massless free fields two examples are the free field Maxwell equation (spin 1) and the free field Einstein equation (spin 2) for the field operators. All of them are essentially a direct consequence of the requirement of Lorentz invariance. Their solutions must transform under Lorentz transformation in a prescribed way, i.e. under a particular representation of the Lorentz group and that together with few other reasonable demands, e.g. the cluster decomposition principle, with implications for causality is enough to fix the equations.

It should be emphasized that this applies to free field equations; interactions are not included. It should also be noted that the equations and their solutions, though needed for the theories, are not the central objects of study.

Wave functions and function spaces

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 concept of Function spaces enters naturally in the discussion about wave functions. A function space is a set of functions, usually with some defining requirements on the functions, together with a topology on that set. The latter will sparsely be used here, it is only needed to obtain a precise definition of what it means for a subset of a function space to be closed. A wave function is an element of a function space partly characterized by the following concrete and abstract descriptions.

• The Schrödinger equation is linear. This means that the solutions to it, wave functions, can be added and multiplied by scalars to form a new solution.
• The superposition principle of quantum mechanics. If Ψ and Φ are two states in the abstract space of states of a quantum mechanical system, then aΨ + bΦ is a valid state as well.

The first item says that the set of solutions to the Schrödinger equation is a vector space. The second item says that the set of allowable states is a vector space. This similarity is of course not accidental. Not all properties of the respective spaces have been given so far. There are also a distinctions between the spaces to keep in mind.

• Basic states are characterized by a set of quantum numbers. This is a set of eigenvalues of a maximal set of commuting observables. A choice of such a set may be called a choice of representation. It is a postulate of quantum mechanics that a physically observable quantity of a system, like position, momentum and spin, is represented by a linear Hermitian operator on the state space. The possible outcomes of measurement of the quantity are the eigenvalues of the operator. Maximality refers to that no more algebraically independent linear Hermitian operator can be added to the set that commutes with the ones already present. The physical interpretation is that such a set represents what can – in theory – be simultaneously be measured with arbitrary position at a given time. The set is non-unique. It may for a one-particle system, for example, be position and spin z-projection, (x, S_z), or it may be momentum and spin y-projection, (p, S_y). At a deeper level, most observables, perhaps all, arise as generators of symmetries.
• Once a representation is chosen, there is still arbitrariness. It remains to choose a coordinate system. This may, for example, correspond to a choice of x, y- and z-axis, or a choice of curvlinear coordinates as exemplified by the spherical coordinates used for the atomic wave functions illustrated below. This final choice also fixes a basis in abstract Hilbert space. The basic states are labeled by the quantum numbers corresponding to the maximal set of commuting observables and an appropriate coordinate system.
• Wave functions corresponding to a state are accordingly not unique. This has been exemplified already with momentum and position space wave functions describing the same abstract state. This non-uniqueness reflects the non-uniqueness in the choice of a maximal set of commuting observables.
• The abstract states are "abstract" only in that an arbitrary choice necessary for a particular explicit description of it is not given. This is the same as saying that no choice of maximal set of commuting observables has been given. This is analogous to a vector space without a specified basis.
• The wave functions of position and momenta, respectively, can be seen as a choice of representation yielding two different, but entirely equivalent, explicit descriptions of the same state for a system with no discrete degrees of freedom.
• Corresponding to the two examples in the first item, to a particular state there corresponds two wave functions, Ψ(x, S_z) and Ψ(p, S_y), both describing the same state. For each choice of maximal commuting sets of observables for the abstract state space, there is a corresponding representation that is associated to a function space of wave functions.
• Each choice of representation should be thought of as specifying a unique function space in which wave functions corresponding to that choice of representation lives. This distinction is best kept, even if one could argue that two such function spaces are mathematically equal, e.g. being the set of square integrable functions. One can then think of the function spaces as two distinct copies of that set.
• Between all these different function spaces and the abstract state space, there are one-to-one correspondences (here disregarding normalization and unobservable phase factors), the common denominator here being a particular abstract state. The relationship between the momentum and position space wave functions, for instance, describing the same state is the Fourier transform.

To make this concrete, in the figure to the right, the 19 sub-images are images of wave functions in position space (their norm squared). The wave functions each represent the abstract state characterized by the triple of quantum numbers (n, l, m), in the lower right of each image. These are the principal quantum number, the orbital angular momentum quantum number and the magnetic quantum number. Together with one spin-projection quantum number of the electron, this is a complete set of observables.

The figure can serve to illustrate some further properties of the function spaces of wave functions.

• In this case, the wave functions are square integrable. One can initially take the function space as the space of square integrable functions, usually denoted L².
• The displayed functions are solutions to the Schrödinger equation. Obviously, not every function in L² satisfies the Schrödinger equation for the hydrogen atom. The function space is thus a subspace of L².
• The displayed functions form part of a basis for the function space. To each triple (n, l, m), there corresponds a basis wave function. If spin is taken into account, there are two basis functions for each triple. The function space thus has a countable basis.
• The basis functions are mutually orthonormal. For this concept to have a meaning, there must exist an inner product. The function space is thus an inner product space. The inner product between two states intuitively measures the "overlap" between the states. The physical interpretation is that the norm squared is proportional to the transition probability between the states. That is.

$P(\Psi \rightarrow \Phi_i) = |(\Psi, \Phi_i)|^2$,

where the i is an index composed of quantum numbers corresponding to a representation and the probabilities are the probabilities of finding the state Ψ in the definite state represented by Φ_i upon measurement of the physical observables corresponding to the representation, for instance, i could be the quadruple (n, l, m, S_z). This is the Born rule, and is one of the fundamental postulates of quantum mechanics.

These observations encapsulate the essence of the function spaces of which wave functions are elements. Mathematically, this is expressed (in one spatial dimension, disregarding here unimportant issues of normalization) for a particle with no internal degrees of freedom as

$\Psi = I\Psi = \int \Phi_x (\Phi_x, \Psi)dx = \int \Psi(x) \Phi_x dx = \int \Phi_p (\Phi_p, \Psi)dp = \int \Psi(p) \Phi_p dp,$

where Ψ is any "abstract" state, Φ_x is an eigenfunction of the position operator representing a particle localized at x, (·,·) represents the inner product, Φ_p is an eigenfunction of the momentum operator representing a particle with precise momentum p, I is the identity operator and the integrals (first and third) represent the completeness of momentum and position eigenstates, Ψ(x) is the coordinate space wave function and Ψ(p) is the wave function in momentum space. In Dirac notation, the above equation reads

$|\Psi\rangle = I|\Psi\rangle = \int |x\rangle \langle x|\Psi\rangle dx = \int \Psi(x) |x\rangle dx = \int |p\rangle \langle p|\Psi\rangle dp = \int \Psi(p) |p\rangle dp.$

The description is not yet complete. There is a further technical requirement on the function space, that of completeness, that allows one to take limits of sequences in the function space, and be ensured that, if the limit exists, it is an element of the function space. A complete inner product space is called a Hilbert space. The property of completeness is crucial in advanced treatments and applications of quantum mechanics. It will not be very important in the subsequent discussion of wave functions, and technical details and links may be found in footnotes like the one that follows. The space L² is a Hilbert space, with inner product presented later. The function space of the example of the figure is a subspace of L². A subspace of a Hilbert space is a Hilbert space if it is closed. It is here that the topology of the function space enters into its description.

It is also important to note, in order to avoid confusion, that not all functions to be discussed are elements of some Hilbert space, say L². The most glaring example is the set of functions e^{​2πipx⁄_h}. These are solutions of the Schrödinger equation for a free particle, but are not normalizable, hence not in L². But they are nonetheless fundamental for the description. One can, using them, express functions that are normalizable using wave packets. They are, in a sense to be made precise later, a basis (but not a Hilbert space basis) in which wave functions of interest can be expressed. There is also the artifact "normalization to a delta function" that is frequently employed for notational convenience, see further down. The delta functions themselves aren't square integrable either.

Physical 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 above description of the function space containing the wave functions is mostly mathematically motivated. The function spaces are, due to completeness, very large in a certain sense. Not all functions are realistic descriptions of any physical system. For instance, in the function space L² one can find the function that takes on the value 0 for all rational numbers and -i for the irrationals in the interval [0, 1]. This is square integrable, but can hardly represent a physical state.

The following constraints on the wave function are sometimes explicitly formulated for the calculations and physical interpretation to make sense:

• The wave function must be square integrable. This is motivated by the Copenhagen interpretation of the wave function as a probability amplitude.
• It must everywhere be everywhere continuous and everywhere continuously differentiable. This is motivated by the appearance of the Schrödinger equation.

It is possible to relax these conditions somewhat for special purposes. If these requirements are not met, it is not possible to interpret the wave function as a probability amplitude.

This does not alter the structure of the Hilbert space that these particular wave functions inhabit, but it should be pointed out that the subspace of the square-integrable functions L², which is a Hilbert space, satisfying the second requirement is not closed in L², hence not a Hilbert space in itself. The functions that does not meet the requirements are still needed for both technical and practical reasons.

Definition (one spinless particle in 1d)

Standing waves for a particle in a box, examples of stationary states.

Travelling waves of a free particle.

The real parts of position wave function Ψ(x) and momentum wave function 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 colour opacity of the particles corresponds to the probability density (not the wave function) 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 wave function

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 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, the positive real number

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

is interpreted as the probability density that the particle is at x. The asterisk indicates the complex conjugate. If the particle's position is measured, its location cannot be determined from the wave function, but is described by a probability distribution. The probability that its position x will be in the interval a ≤ x ≤ b is the integral of the density over this interval:

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

where t is the time at which the particle was measured. 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.

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

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

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.

Momentum-space wave function

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.

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:

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

and the normalization condition is:

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

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,

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

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.

Definitions (other cases)

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:

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

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:

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

where r_i 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.

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. This translates to a requirement on the wave function for a system of identical particles:

$\Psi \left ( \ldots \mathbf{r}_a, \ldots , \mathbf{r}_b, \ldots \right ) = \pm \Psi \left ( \ldots \mathbf{r}_b, \ldots , \mathbf{r}_a, \ldots \right )$

where the + sign occurs if the particles are all bosons and − sign if they are all fermions. In other words, the wave function is either totally symmetric in the positions of bosons, or totally antisymmetric in the positions of fermions. 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, i.e. no two having the same set of quantum numbers), there is no requirement for the wave function to be either symmetric or antisymmetric.

For a collection of particles, some identical with coordinates 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 r_i only:

$\Psi \left ( \ldots \mathbf{r}_a, \ldots , \mathbf{r}_b, \ldots , \mathbf{x}_1, \mathbf{x}_2, \ldots \right ) = \pm \Psi \left ( \ldots \mathbf{r}_b, \ldots , \mathbf{r}_a, \ldots , \mathbf{x}_1, \mathbf{x}_2, \ldots \right )$

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

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

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

which is a complex-valued function of position r in three-dimensional space, time t, and s_z, 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 s_z parameter, unlike r and t, is a discrete variable. For example, for a spin-1/2 particle, s_z can only be +1/2 or −1/2, and not any other value. (In general, for spin s, s_z can be s, s − 1, ... , −s + 1, −s.)

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 s_z. 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:

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

The wave function for N particles each with spin is the complex-valued function:

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

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

$\rho\left(\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) = \left | \Psi\left (\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) \right |^2$

and the probability that particle 1 is in region R₁ with spin s_z1 = m₁ and particle 2 is in region R₂ with spin s_z2 = m₂ etc. at time t is the integral of the probability density over these regions and spins:

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

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

Decompositions into products

For systems in time-independent potentials, the wave function can always be written as a function of the degrees of freedom multiplied by a time-dependent phase factor, the form of which is given by the Schrödinger equation. For the case of N particles position-spin space,

$\Psi(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N,t,s_{z1},s_{z2},\ldots,s_{zN}) = e^{-i Et/\hbar} \psi(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N,s_{z1},s_{z2},\ldots,s_{zN})\,,$

where E is the energy eigenvalue of the system corresponding to the eigenstate Ψ. Wave functions of this form are called stationary states.

In some situations, the wave function for a particle with spin factors into a product of a space function ψ and a spin function ξ, where each are complex-valued functions, and the time dependence can be placed in either function:

$\Psi(\mathbf{r},t,s_z) = \psi(\mathbf{r},t)\xi(s_z) = \phi(\mathbf{r})\zeta(s_z, t)\,.$

The dynamics of each factor can be studied in isolation. This factorization is always possible when the orbital and spin angular momenta of the particle are separable in the Hamiltonian operator, that is, the Hamiltonian can be split into an orbital term and a spin term. It is not possible for those interactions where an external field or any space-dependent quantity couples to the spin; examples include a particle in a magnetic field, and spin-orbit coupling. For the time-independent case this reduces to

$\Psi(\mathbf{r},t,s_z) = e^{-iEt/\hbar} \psi(\mathbf{r})\xi(s_z) \,,$

where again E is the energy eigenvalue of the system corresponding to the eigenstate Ψ. This extends to the case of N particles:

$\Psi(\mathbf{r},t,s_z) = \psi(\mathbf{r}_1, \mathbf{r}_2,\ldots, \mathbf{r}_N,t)\xi(s_{z1}, s_{z2},\ldots,s_{zN}) = \phi(\mathbf{r}_1, \mathbf{r}_2,\ldots, \mathbf{r}_N)\zeta(s_{z1}, s_{z2},\ldots,s_{zN},t)\,.$

and for the case of identical particles, each factor has to have the correct antisymmetry or symmetry, to make the overall wave function antisymmetric for fermions or symmetric for bosons.

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)

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

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:

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

while for many spinless particles in 3d:

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

(altogether, this is N three-dimensional volume integrals with differential volume elements d³r_i, also written "dV_i" or "dx_i dy_i dz_i"). For one particle with spin in 3d:

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

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

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

(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

$\left|\left\langle\Psi_1,\Psi_2\right\rangle\right|^2 = P\left(\Psi_2 \rightarrow \Psi_1\right) \,,$

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,

$\langle\Psi,\Psi\rangle = \|\Psi\|^2 \,,$

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:

$\left\langle \Psi, \Psi \right\rangle = 1 \,.$

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:

$\left\langle \Psi_1, \Psi_2 \right\rangle = 0 \,.$

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

$\langle \Psi_m , \Psi_n \rangle = \delta_{mn} \,,$

where m and n each take values 1, 2, ..., and δ_mn is the Kronecker delta (+1 for m = n and 0 for m ≠ n). 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 Ψ_m and Ψ_n is more complicated than the mere δ_mn).

Returning to the superposition above:

$\Psi = \sum_n a_n \psi_n$

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

$a_n = \langle \psi_n , \Psi \rangle$

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:

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

and similarly for more particles in higher dimensions.

One particular solution to the time-independent Schrödinger equation is

$\Psi_p(x) = e^{ipx/\hbar},$

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

$\{\Psi_p(x, t), -\infty \le p \le \infty\}$

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,

$\langle\Psi_{p},\Psi_{p'}\rangle = \delta(p - p').$

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...dⁿr_N 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.

More on wave functions and abstract state space

Main article: Quantum state

As has been demonstrated, the set of all possible normalizable wave functions for a system with a particular choice of basis constitute a Hilbert space. This vector space is in general infinite-dimensional. Due to the multiple possible choices of basis, these Hilbert spaces are not unique. One therefore talks about an abstract Hilbert space, state space, where the choice of basis is left undetermined. The choice of basis corresponds to a choice of a maximal set of quantum numbers, each quantum number corresponding to an observable. Two observables corresponding to quantum numbers in the maximal set must commute, therefore, the basis isn't entirely arbitrary, but nonetheless, there are always several choices.

Specifically, each state is represented as an abstract vector in state space

$|\Psi\rangle$

where |Ψ⟩ is a "ket" (a vector) written in Dirac's bra–ket notation. Kets that differ by multiplication by a scalar represent the same state. A ray in Hilbert space is a set of normalized vectors differing by a complex number of modulus 1. If |ψ⟩ and |ϕ⟩ are two states in the vector space, and a and b are two complex numbers, then the linear combination

$|\Psi \rangle = a|\psi\rangle + b|\phi\rangle$

is also in the same vector space. The state space is postulated to have an inner product, denoted by

$\langle \Psi_1 | \Psi_2 \rangle,$

that is (usually, this differs) linear in the first argument and antilinear in the second argument. The dual vectors are denoted as "bras", ⟨Ψ|. These are linear functionals, elements of the dual space to the state space. The inner product, once chosen, can be used to define a unique map from state space to its dual, see Riesz representation theorem. this map is antilinear. One has

$\langle \Psi | = a^{*} \langle \psi | + b^{*} \langle \phi | \leftrightarrow a|\psi\rangle + b|\phi\rangle = |\Psi\rangle,$

where the asterisk denotes the complex conjugate. For this reason one has under this map

$\langle \Phi|\Psi\rangle = \langle \Phi| (|\Psi\rangle),$

and one may, as a practical consequence, at least notation-wise in this formalism, ignore that bra's are dual vectors.

The state vector for the system evolves in time according to the Schrödinger equation, or other dynamical pictures of quantum mechanics- In bra-ket notation this reads,

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

Abstract state space is also, by definition, required to be a Hilbert space. The only requirement missing for this in the description so far is completeness. See the quantum state article for more explanation of the Hilbert space formalism and its consequences to quantum physics.

The connection to the Hilbert spaces of wave functions is made as follows. If (a, b, … l, m, …) is a maximal set of quantum numbers, denote the state corresponding to fixed choices of these quantum numbers by

$|a, b, \ldots, l, m, \ldots\rangle.$

The wave function corresponding to an arbitrary state |Ψ⟩ is denoted

$\langle a, b, \ldots, l, m, \ldots|\Psi\rangle,$

for a concrete example,

$\Psi(x) = \langle x|\Psi\rangle.$

There are several advantages to understanding wave functions as representing 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

Discrete components A_k of a complex vector |A⟩ = ∑_k A_k|e_k⟩, which belongs to a countably infinite-dimensional Hilbert space; there are countably infinitely many k values and basis vectors |e_k⟩.

Continuous components ψ(x) of a complex vector |ψ⟩ = ∫dx ψ(x)|x⟩, 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 |ε_i⟩ 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:

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

where the complex numbers

$c_i = \langle \varepsilon_i | \Psi \rangle$

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 |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 |Ψ_ε⟩. A particular basis ket may be subscripted |ε₀⟩ ≡ |Ψ_ε0⟩ or primed |ε′⟩ ≡ |Ψ_ε′⟩, 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 |Ψ⟩ in the continuous basis of the Hilbert space, with the corresponding bra in the dual space, are respectively given by:

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

where the components are the complex-valued functions

$\Psi(\varepsilon) = \langle \varepsilon | \Psi \rangle$

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 discrete and continuous orthonormal bases, respectively. An orthonormal set of kets form bases if and only if they satisfy these relations. 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 |Ψ₁⟩ with a second |Ψ₂⟩ can also be obtained by multiplying |Ψ₁⟩ on the left and |Ψ₂⟩ 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 |Ψ₁⟩ is a state in the above basis with components c₁, c₂, ..., c_n and |Ψ₂⟩ is another state in the same basis with components z₁, z₂, ..., z_n, the inner product is the complex number:

$\langle \Psi_1 | \Psi_2 \rangle = \left(\sum_i z^*_i \langle\varepsilon_i | \right)\left(\sum_j c_j |\varepsilon_j\rangle \right) =\sum_{ij} z^*_i c_j \langle \varepsilon_i|\varepsilon_j\rangle = \sum_i z^*_i c_i \,.$

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

$\langle \Psi_1 | \Psi_2 \rangle = \left( \int d\varepsilon' {\Psi_1(\varepsilon')}^{*} \langle \varepsilon' | \right)\left(\int d\varepsilon \Psi_2(\varepsilon) |\varepsilon \rangle \right) = \int d\varepsilon' \int d\varepsilon {\Psi_1(\varepsilon')}^{*} \Psi_2(\varepsilon) \langle\varepsilon' | \varepsilon \rangle = \int d\varepsilon {\Psi_1(\varepsilon)}^{*} \Psi_2(\varepsilon) \,.$

where the integrals are taken over all ε and ε′.

The square of the norm (magnitude) of the state vector |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 each separately equal to 1. If the state is not normalized, then dividing by its magnitude normalizes the state:

$| \Psi_N \rangle = \frac{1}{\| \Psi \|} | \Psi \rangle$

Normalized components and probabilities

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

For the discrete basis, projecting the normalized state |Ψ_N⟩ onto a particular state the system may collapse to, |ε_q⟩, 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 |ε_q⟩ 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 a ≤ ε′ ≤ 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 |ε_i⟩ has distinct and definite values, λ_i, and a measurement of that variable is performed on a system in the state |Ψ⟩ then the probability of measuring λ_i is | ⟨ε_i|Ψ⟩ |². If the measurement yields λ_i, the system "collapses" to the state |ε_i⟩ irreversibly and instantaneously.

Time dependence

Main article: Dynamical pictures (quantum mechanics)

In the Schrödinger picture, the states evolve in time, so the time dependence is placed in |Ψ⟩ according to

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

for discrete bases, or

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

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)⟩. The Heisenberg picture wave function is a snapshot of a Schrödinger picture wave function, representing the whole spacetime history of the system. In the interaction picture (also called Dirac picture), the time dependence is placed in both the states and operators, the subdivision depending on the interaction term in the Hamiltonian, and can be viewed as intermediate between the Heisenberg and Schrödinger pictures. It is useful primarily in computing S-matrix elements.

Tensor product

Further information: Bra-ket notation § Composite bras and kets

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 |Ψ⟩ and |Φ⟩, the tensor product of the states forms the composite state denoted by |Ψ⟩⊗|Φ⟩ or simply without any operation symbol |Ψ⟩|Φ⟩, and the new system includes both of the original systems together. The tensor product state |Ψ⟩|Φ⟩ 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 |Ψ⟩|Φ⟩ ≠ |Φ⟩|Ψ⟩. If |Ψ⟩ has components c_i and |Φ⟩ has components z_j, each in a discrete orthonormal basis |ε_k⟩, then:

$|\Psi\rangle|\Phi\rangle = \left(\sum_i c_i |\varepsilon_i\rangle \right) \left(\sum_j z_j |\varepsilon_j\rangle \right) = \sum_{i,j}c_iz_j|\varepsilon_i\rangle|\varepsilon_j\rangle$

and the notation can be simplified by abbreviating |A⟩ = |Ψ⟩|Φ⟩, A_ij = c_iz_j, and |E_ij⟩ = |ε_i⟩|ε_j⟩, so that

$|A\rangle = \sum_{i,j}A_{ij} |E_{ij}\rangle$

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

This section applies mostly to non-relativistic quantum mechanics. In relativistic quantum mechanics, eigenstates of the position operator are problematic due to a relativistic extension of Heisenberg's uncertainty principle. In relativistic quantum field theory, they are not used at all to label physical states. Associated to a particle perfectly localized to a point in space is an infinite uncertainty in energy. This leads to pair production in the relativistic regime. Thus such a particle automatically has companions, leading to a breakdown of the description.

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 basis states; |x⟩, also written |Ψ_x⟩, corresponding to the set of all position coordinates x. The completeness condition for this basis is

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

and the orthogonality relation is

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

The state |Ψ⟩ is expressed by:

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

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

$\Psi (x) = \langle x | \Psi \rangle$

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

$\langle \Psi_1 | \Psi_2 \rangle = \langle \Psi_1 |\left(\int\limits_{-\infty}^{\infty} d x \, | x \rangle \langle x | \right)| \Psi_2 \rangle = \int\limits_{-\infty}^{\infty} d x \, \langle \Psi_1 | x \rangle \langle x | \Psi_2 \rangle = \int\limits _{-\infty}^{\infty} d x \, \Psi_1(x)^{*} \Psi_2(x) \,.$

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 |r⟩ and any state vector |Ψ⟩ in this space is expressed in terms of the basis vectors as |r⟩:

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

with components:

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

For N spinless particles in 3d, the basis states are |r₁, ..., r_N⟩. This is the tensor product of the one-particle position bases |r₁⟩, |r₂⟩, ..., |r_N⟩, each of which spans the separate one-particle Hilbert spaces, so |r₁, ..., r_N⟩ 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 |Ψ⟩ in this space is

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

with components:

$\langle \mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N | \Psi \rangle = \Psi(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N)$

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

$|\Psi\rangle = \sum_{s_z} \int\limits_\mathrm{all\,space} d^3\mathbf{r} |\mathbf{r},s_z\rangle \langle \mathbf{r},s_z | \Psi\rangle$

with components:

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

For N particles with spin in 3d, the basis states are |r₁, ..., r_N, s_{z 1}, ..., s_{z N}⟩, the tensor product of the position basis |r₁, ..., r_N⟩ and spin basis |s_{z 1}, ..., s_{z N}⟩, which exists in a new space from the spin space and position space alone. Any state in this space is:

$|\Psi\rangle = \sum_{s_{z\,1} , \ldots , s_{z\,N}}\int\limits\limits_\mathrm{all\,space} d^3\mathbf{r}_N \cdots \int\limits\limits_\mathrm{all\,space} d^3\mathbf{r}_1 \, | \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} \rangle \langle \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} | \Psi\rangle$

with components:

$\langle \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} | \Psi\rangle = \Psi(\mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} )$

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:

$| \Psi \rangle = \sum_{s_{z\,1} , \ldots , s_{z\,N}}\int\limits\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits\limits_{R_1} d^3\mathbf{r}_1 \, \Psi ( \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} ) | \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} \rangle$

The orthogonality relation for this basis is:

$\langle \mathbf{x}_1, \ldots, \mathbf{x}_{N}, m_1, \ldots, m_N | \mathbf{r}_1,\ldots,\mathbf{r}_N, s_{z\,1},\ldots,s_{z\,N} \rangle = \delta_{m_1\,s_{z\,1}}\cdots\delta_{m_N\,s_{z\,N}}\delta( \mathbf{x}_1 - \mathbf{r}_{1})\cdots\delta( \mathbf{x}_{N} - \mathbf{r}_{N})$

and the inner product of |Ψ₁⟩ and |Ψ₂⟩ is:

$\langle \Psi_1 | \Psi_2 \rangle = \sum_{s_{z\,1} , \ldots , s_{z\,N}}\int\limits\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits\limits_{R_1} d^3\mathbf{r}_1 \Psi_1(\mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} )^{*} \Psi_2(\mathbf{r}_1, \ldots, \mathbf{r}_N, s_{z\,1} , \ldots , s_{z\,N}) \,.$

Momentum space wave functions are similar, using the momentum vectors of the particles as continuous bases, namely |p⟩, |p₁, p₂, ..., p_N⟩, etc.

Ontology

Main article: Interpretations of quantum mechanics

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 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, Bohm and Everett and others, argued that the wave function must have an objective, physical existence. Einstein thought that a complete description of physical reality should refer directly to physical space and time, as distinct from the wave function, which refers to an abstract mathematical space.

Examples

Free particle

Main article: Free particle

A free particle in 3d with wave vector k and angular frequency ω has a wave function

$\Psi (\mathbf{r},t) = A e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}\,.$

Particle in a box

Main article: Particle in a box

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-dimensional quantum tunnelling

Main articles: Finite potential barrier and 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

$V(x)=\begin{cases}V_0 & |x|<a \\ 0 & \text{otherwise,}\end{cases}$

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 A_r = 1 corresponds to firing particles singly; the terms containing A_r and C_r signify motion to the right, while A_l and C_l – to the left. Under this beam interpretation, put C_l = 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.

Quantum Dots

3D confined electron wave functions in a quantum dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are more s-type and p-type. However, in a triangular dot the wave functions are mixed due to confinement symmetry. (Click for animation)

In a semiconductor crystallite whose radius is smaller than the size of its exciton Bohr radius, the excitons are squeezed, leading to quantum confinement. The energy levels can then be modeled using the particle in a box model in which the energy of different states is dependent on the length of the box.

Other

Some examples of wave functions for specific applications include:

• Finite square well
• Delta potential
• Quantum harmonic oscillator
• Hydrogen atom and Hydrogen-like atom

See also

• Boson
• de Broglie–Bohm theory
• Double-slit experiment
• Faraday wave
• Fermion
• Schrödinger equation
• Wave function collapse
• Wave packet
• Phase space formulation of quantum mechanics, wave functions are replaced by quasi-probability distributions that place the position and momenta variables on equal footing.

References

• Online copy (French) Online copy (English)

• Online copy

External links

• [1], [2], [3], [4]
• [5] Normalization.
• [6] Quantum Mechanics and Quantum Computation at BerkeleyX
• Einstein, The quantum theory of radiation