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wave function
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File:QuantumHarmonicOscillatorAnimation.gif|thumb|erect=1.5|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; here, the vertical axis shows the real part (blue) and imaginary part (red) of the wave function.Panels (C–F) show four different standing-wave solutions of the Schrödinger equationSchrödinger equationFile:Hydrogen Density Plots.png|thumb|upright=1.4|Wavefunctions of the electron of a hydrogen atom at different energies. The brightness at each point represents the probability of observing the electronelectronA wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters {{math|ψ}} or {{math|Ψ}} (lower-case and capital psi, respectively).The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state.For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, 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. Some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also 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 {{math|2 × 1}} column vector for a non-relativistic electron with spin {{math|{{frac|1|2}}}}).According to the superposition principle of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a Hilbert space. The inner product between two wave functions is a measure of the overlap between the corresponding physical states, and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The Schrödinger equation determines how wave functions evolve over time, and 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. However, the wave function in quantum mechanics describes a kind of physical phenomenon, still open to different interpretations, which fundamentally differs from that of classic mechanical waves.{{harvnb|Born|1927|pp=354–357}}{{harvnb|Heisenberg|1958|p=143}}Heisenberg, W. (1927/1985/2009). Heisenberg is translated by {{harvnb|Camilleri|2009|p=71}}, (from {{harvnb|Bohr|1985|p=142}}).{{harvnb|Murdoch|1987|p=43}}{{harvnb|de Broglie|1960|p=48}}{{harvnb|Landau|Lifshitz|p=6}}{{harvnb|Newton|2002|pp=19–21}}In Born's statistical interpretation in non-relativistic quantum mechanics,Born, M. (1954). the squared modulus of the wave function, {{math|{{abs|ψ}}2}}, is a real number interpreted as the probability density of measuring a particle's being detected at a given place – or having a given momentum – at a given time, 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 that 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 {{math|ψ}} and calculate the statistical distributions for measurable quantities.

Historical background

{{Quantum mechanics|cTopic=Fundamental concepts}}In 1905, Einstein postulated the proportionality between the frequency f of a photon and its energy E, E = hf, {{harvnb|Einstein|1905|pp=132–148}} (in German), {{harvnb|Arons|Peppard|1965|p=367}} (in English) and in 1916 the corresponding relation between photon's momentum p and wavelength lambda, lambda = dfrac{h}{p} {{harvnb|Einstein|1916|pp=47–62}}, and a nearly identical version {{harvnb|Einstein|1917|pp=121–128}} translated in {{harvnb|ter Haar|1967|pages=167–183}}., where h is the Planck constant. In 1923, De Broglie was the first to suggest that the relation lambda = frac{h}{p}, now called the De Broglie relation, holds for massive particles, the chief clue being Lorentz invariance,{{harvnb|de Broglie|1923|pp=507–510,548,630}} 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.{{harvnb|Hanle|1977|pp=606–609}}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.{{harvnb|Schrödinger|1926|pages=1049–1070}} However, no one was clear on how to interpret it.{{harvnb|Tipler|Mosca|Freeman|2008}} 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.{{harvnb|Weinberg|2013}} This was shown to be incompatible with the elastic scattering of a wave packet (representing a particle) off a target; it spreads out in all directions.{{harvnb|Born|1926a}}, translated in {{harvnb|Wheeler|Zurek|1983}} at pages 52–55. 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.{{harvnb|Born|1926b}}, translated in {{harvnb|Ludwig|1968|pages=206–225}}. Also here.{{harvnb|Young|Freedman|2008|page=1333}} 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.{{harvnb|Atkins|1974}} 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.{{harvnb|Martin|Shaw|2008}}In 1927, Pauli phenomenologically found a non-relativistic equation to describe spin-1/2 particles in electromagnetic fields, now called the Pauli equation.{{harvnb|Pauli|1927|pages=601–623.}} 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 counterparts.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.{{harvtxt|Weinberg|2002}} takes the standpoint that quantum field theory appears the way it does because it is the only way to reconcile quantum mechanics with special relativity. 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 {{math|0}}) and the Dirac equation (spin {{math|{{frac|1|2}}}}) in this guise remain in the theory. Higher spin analogues include the Proca equation (spin {{math|1}}), Rarita–Schwinger equation (spin {{math|{{frac|3|2}}}}), and, more generally, the Bargmann–Wigner equations. For massless free fields two examples are the free field Maxwell equation (spin {{math|1}}) and the free field Einstein equation (spin {{math|2}}) for the field operators.{{harvtxt|Weinberg|2002}} See especially chapter 5, where some of these results are derived. 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,{{harvnb|Weinberg|2002}} Chapter 4. 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. If a Lagrangian density (including interactions) is available, then the Lagrangian formalism will yield an equation of motion at the classical level. This equation may be very complex and not amenable to solution. Any solution would refer to a fixed number of particles and would not account for the term "interaction" as referred to in these theories, which involves the creation and annihilation of particles and not external potentials as in ordinary "first quantized" quantum theory.In string theory, the situation remains analogous. For instance, a wave function in momentum space has the role of Fourier expansion coefficient in a general state of a particle (string) with momentum that is not sharply defined.{{harvnb|Zwiebach|2009}}

Definition (one spinless particle in one dimension)

{{multiple image
| align = right
| direction = vertical
| width = 402
| footer = The real parts of position wave function {{math|Ψ(x)}} and momentum wave function {{math|Φ(p)}}, and corresponding probability densities {{math|{{!}}Ψ(x){{!}}2}} and {{math|{{!}}Φ(p){{!}}2}}, for one spin-0 particle in one {{math|x}} or {{math|p}} dimension. The colour opacity of the particles corresponds to the probability density (not the wave function) of finding the particle at position {{math|x}} or momentum {{math|p}}.
| image1 = Quantum mechanics standing wavefunctions.svg
| caption1 = Standing waves for a particle in a box, examples of stationary states.
| image2 = Quantum mechanics travelling wavefunctions.svg
| caption2 = Travelling waves of a free particle. }}
For now, consider the simple case of a non-relativistic single particle, without spin, in one spatial dimension. More general cases are discussed below.

Position-space wave functions

The state of such a particle is completely described by its wave function,
Psi(x,t),,
where {{math|x}} is position and {{math|t}} is time. This is a complex-valued function of two real variables {{math|x}} and {{math|t}}.For one spinless particle in 1d, if the wave function is 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 {{math|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 {{math|x}} will be in the interval {{math|a ≤ x ≤ b}} is the integral of the density over this interval:
P_{ale xle b} (t) = intlimits_a^b d x,|Psi(x,t)|^2
where {{math|t}} is the time at which the particle was measured. This leads to the normalization condition:
intlimits_{-infty}^infty d x , |Psi(x,t)|^2 = 1,,
because if the particle is measured, there is 100% probability that it will be somewhere.For a given system, the set of all possible normalizable wave functions (at any given time) forms an abstract mathematical vector space, meaning that it is possible to add together different wave functions, and multiply wave functions by complex numbers (see vector space for details). Technically, because of the normalization condition, wave functions form a projective space rather than an ordinary vector space. This vector space is infinite-dimensional, because there is no finite set of functions which can be added together in various combinations to create every possible function. Also, it is a Hilbert space, because the inner product of two wave functions {{math|Ψ1}} and {{math|Ψ2}} can be defined as the complex number (at time {{math|t}})The functions are here assumed to be elements of {{math|L2}}, the space of square integrable functions. The elements of this space are more precisely equivalence classes of square integrable functions, two functions declared equivalent if they differ on a set of Lebesgue measure {{math|0}}. This is necessary to obtain an inner product (that is, {{math|(Ψ, Ψ) {{=}} 0 ⇒ Ψ ≡ 0}}) as opposed to a semi-inner product. The integral is taken to be the Lebesque integral. This is essential for completeness of the space, thus yielding a complete inner product space = Hilbert space.
( Psi_1 , Psi_2 ) = intlimits_{-infty}^infty d x , Psi_1^*(x, t)Psi_2(x, t).
More details are given below. Although the inner product of two wave functions is a complex number, the inner product of a wave function {{math|Ψ}} with itself,
(Psi,Psi) = |Psi|^2 ,,
is always a positive real number. The number {{math|{{!}}{{!}}Ψ{{!}}{{!}}}} (not {{math|{{!}}{{!}}Ψ{{!}}{{!}}2}}) is called the norm of the wave function {{math|Ψ}}.If {{math|(Ψ, Ψ) {{=}} 1}}, then {{math|Ψ}} is normalized. If {{math|Ψ}} is not normalized, then dividing by its norm gives the normalized function {{math|Ψ/{{!}}{{!}}Ψ{{!}}{{!}}}}. Two wave functions {{math|Ψ1}} and {{math|Ψ2}} are orthogonal if {{math|(Ψ1, Ψ2) {{=}} 0}}. If they are normalized and orthogonal, they are orthonormal. Orthogonality (hence also orthonormality) of wave functions is not a necessary condition wave functions must satisfy, but is instructive to consider since this guarantees linear independence of the functions. In a linear combination of orthogonal wave functions {{math|Ψn}} we have,
Psi = sum_n a_n Psi_n ,,quad a_n = frac{( Psi_n , Psi )}{( Psi_n , Psi_n )}
If the wave functions {{math|Ψn}} were nonorthogonal, the coefficients would be less simple to obtain.In the Copenhagen interpretation, the modulus squared of the inner product (a complex number) gives a real number
left|(Psi_1,Psi_2)right|^2 = Pleft(Psi_2 rightarrow Psi_1right) ,,
which, assuming both wave functions are normalized, is interpreted as the probability of the wave function {{math|Ψ2}} "collapsing" to the new wave function {{math|Ψ1}} upon measurement of an observable, whose eigenvalues are the possible results of the measurement, with {{math|Ψ1}} being an eigenvector of the resulting eigenvalue. This is the Born rule, and is one of the fundamental postulates of quantum mechanics.At a particular instant of time, all values of the wave function {{math|Ψ(x, t)}} are components of a vector. There are uncountably infinitely many of them and integration is used in place of summation. In Bra–ket notation, this vector is written
|Psi(t)rangle = int dx Psi(x,t) |xrangle
and is referred to as a "quantum state vector", or simply "quantum state". 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 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.
The time parameter is often suppressed, and will be in the following. The {{math|x}} coordinate is a continuous index. The {{math|{{ket|x}}}} are the basis vectors, which are orthonormal so their inner product is a delta function;
langle x' | x rangle = delta(x' - x)
thus
langle x' |Psirangle = int dx Psi(x) langle x'|xrangle = Psi(x')
and
|Psirangle = int dx |xrangle langle x |Psirangle = left( int dx |xrangle langle x |right) |Psirangle
which illuminates the identity operator
I = int dx |xrangle langle x | ,.
Finding the identity operator in a basis allows the abstract state to be expressed explicitly in a basis, and more (the inner product between two state vectors, and other operators for observables, can be expressed in the basis).

Momentum-space wave functions

The particle also has a wave function in momentum space:
Phi(p,t)
where {{math|p}} is the momentum in one dimension, which can be any value from {{math|−∞}} to {{math|+∞}}, and {{math|t}} is time.Analogous to the position case, the inner product of two wave functions {{math|Φ1(p, t)}} and {{math|Φ2(p, t)}} can be defined as:
(Phi_1 , Phi_2 ) = intlimits_{-infty}^infty d p , Phi_1^*(p, t)Phi_2(p, t) ,.
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 {{math|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,
(Psi_{p},Psi_{p'}) = 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 next.

Relations between position and momentum representations

The {{math|x}} and {{math|p}} representations are
begin{align}
Psirangle &= int Psirangle dx = int Psi(x) |xrangle dx,
Psirangle &= int Psirangle dp = int Phi(p) |prangle dp.end{align}Now take the projection of the state {{math|Ψ}} onto eigenfunctions of momentum using the last expression in the two equations,{{harvnb|Shankar|1994|loc=Ch. 1}}
int Psi(x) langle p|xrangle dx = int Phi(p') langle p|p'rangle dp' = int Phi(p') delta(p-p') dp' = Phi(p).
Then utilizing the known expression for suitably normalized eigenstates of momentum in the position representation solutions of the free Schrödinger equation
langle x | p rangle = p(x) = frac{1}{sqrt{2pihbar}}e^{frac{i}{hbar}px} Rightarrow langle p | x rangle = frac{1}{sqrt{2pihbar}}e^{-frac{i}{hbar}px},
one obtains
Phi(p) = frac{1}{sqrt{2pihbar}}int Psi(x)e^{-frac{i}{hbar}px}dx,.
Likewise, using eigenfunctions of position,
Psi(x) = frac{1}{sqrt{2pihbar}}int Phi(p)e^{frac{i}{hbar}px}dp,.
The position-space and momentum-space wave functions are thus found to be Fourier transforms of each other.{{harvnb|Griffiths|2004}} The two wave functions contain the same information, and either one alone is sufficient to calculate any property of the particle. As representatives of elements of abstract physical Hilbert space, whose elements are the possible states of the system under consideration, they represent the same state vector, hence identical physical states, but they are not generally equal when viewed as square-integrable functions.In practice, the position-space wave function is used much more often than the momentum-space wave function. The potential entering the relevant equation (Schrödinger, Dirac, etc.) determines in which basis the description is easiest. For the harmonic oscillator, {{math|x}} and {{math|p}} enter symmetrically, so there it doesn't matter which description one uses. The same equation (modulo constants) results. From this follows, with a little bit of afterthought, a factoid: The solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in {{math|L2}}.The Fourier transform viewed as a unitary operator on the space {{math|L2}} has eigenvalues {{math|±1, ±i}}. The eigenvectors are "Hermite functions", i.e. Hermite polynomials multiplied by a Gaussian function. See {{harvtxt|Byron|Fuller|1992}} for a description of the Fourier transform as a unitary transformation. For eigenvalues and eigenvalues, refer to Problem 27 Ch. 9.

Definitions (other cases)

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

One-particle states in 3d position space

The position-space wave function of a single particle without spin in three spatial dimensions is similar to the case of one spatial dimension above:
Psi(mathbf{r},t)
where {{math|r}} is the position vector in three-dimensional space, and {{math|t}} is time. As always {{math|Ψ(r, t)}} is a complex-valued function of real variables. As a single vector in Dirac notation
|Psi(t)rangle = int d^3! mathbf{r}, Psi(mathbf{r},t) ,|mathbf{r}rangle
All the previous remarks on inner products, momentum space wave functions, Fourier transforms, and so on extend to higher dimensions.For a particle with spin, ignoring the position degrees of freedom, the wave function is a function of spin only (time is a parameter);
xi(s_z,t)
where {{math|sz}} is the spin projection quantum number along the {{math|z}} axis. (The {{math|z}} axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) The {{math|sz}} parameter, unlike {{math|r}} and {{math|t}}, is a discrete variable. For example, for a spin-1/2 particle, {{math|sz}} can only be {{math|+1/2}} or {{math|−1/2}}, and not any other value. (In general, for spin {{math|s}}, {{math|sz}} can be {{math|s, s − 1, ... , −s + 1, −s}}). Inserting each quantum number gives a complex valued function of space and time, there are {{math|2s + 1}} of them. These can be arranged into a column vectorColumn vectors can be motivated by the convenience of expressing the spin operator for a given spin as a matrix, for the z-component spin operator (divided by hbar to nondimensionalize)
frac{1}{hbar}hat{S}_z = begin{bmatrix} s & 0 & cdots & 0 & 0 0 & s-1 & cdots & 0 & 0 vdots & vdots & ddots & vdots & vdots 0 & 0 & cdots & -(s-1) & 0 0 & 0 & cdots & 0 & -s end{bmatrix}
The eigenvectors of this matrix are the above column vectors, with eigenvalues being the corresponding spin quantum numbers.
{ xi = begin{bmatrix} xi(s,t) xi(s-1,t) vdots xi(-(s-1),t) xi(-s,t) end{bmatrix} = xi(s,t) begin{bmatrix} 1 0 vdots 0 0 end{bmatrix} + xi(s-1,t)begin{bmatrix} 0 1 vdots 0 0 end{bmatrix} + cdots + xi(-(s-1),t) begin{bmatrix} 0 0 vdots 1 0 end{bmatrix} + xi(-s,t) begin{bmatrix} 0 0 vdots 0 1 end{bmatrix} }
In bra-ket notation, these easily arrange into the components of a vectorEach {{math|{{ket|sz}}}} is usually identified as a column vector
|srangle leftrightarrow begin{bmatrix} 1 0 vdots 0 0 end{bmatrix} ,, quad |s-1rangle leftrightarrow begin{bmatrix} 0 1 vdots 0 0 end{bmatrix} ,, ldots ,, quad |-(s-1)rangle leftrightarrow begin{bmatrix} 0 0 vdots 1 0 end{bmatrix} ,,quad |-srangle leftrightarrow begin{bmatrix} 0 0 vdots 0 1 end{bmatrix}
but it is a common abuse of notation to write
|srangle = begin{bmatrix} 1 0 vdots 0 0 end{bmatrix} , ldots ,,
because the kets {{math|{{ket|sz}}}} are not synonymous or equal to the column vectors. Column vectors simply provide a convenient way to express the spin components.
|xi (t)rangle = sum_{s_z=-s}^s xi(s_z,t) ,| s_z rangle
The entire vector {{math|ξ}} is a solution of the Schrödinger equation (with a suitable Hamiltonian), which unfolds to a coupled system of {{math|2s + 1}} ordinary differential equations with solutions {{math|ξ(s, t), ξ(s − 1, t), ..., ξ(−s, t)}}. The term "spin function" instead of "wave function" is used by some authors. This contrasts the solutions to position space wave functions, the position coordinates being continuous degrees of freedom, because then the Schrödinger equation does take the form of a wave equation.More generally, for a particle in 3d with any spin, the wave function can be written in "position–spin space" as:
Psi(mathbf{r},s_z,t)
and these can also be arranged into a column vector
Psi(mathbf{r},t) = begin{bmatrix} Psi(mathbf{r},s,t) Psi(mathbf{r},s-1,t) vdots Psi(mathbf{r},-(s-1),t) Psi(mathbf{r},-s,t) end{bmatrix}
in which the spin dependence is placed in indexing the entries, and the wave function is a complex vector-valued function of space and time only.All values of the wave function, not only for discrete but continuous variables also, collect into a single vector
|Psi(t)rangle

sum_{s_z}int d^3!mathbf{r} ,Psi(mathbf{r},s_z,t), |mathbf{r}, s_zrangle For a single particle, the tensor product {{math|⊗}} of its position state vector {{math|{{ket|ψ}}}} and spin state vector {{math|{{ket|ξ}}}} gives the composite position-spin state vector
|psi(t)rangle! otimes! |xi(t)rangle

sum_{s_z}int d^3! mathbf{r}, psi(mathbf{r},t),xi(s_z,t) ,|mathbf{r}rangle !otimes! |s_zrangle with the identifications
|Psi (t)rangle = |psi(t)rangle
!otimes!
|xi(t)rangle


Psi(mathbf{r},s_z,t) = psi(mathbf{r},t),xi(s_z,t) |mathbf{r},s_z rangle= |mathbf{r}rangle !otimes! |s_zrangle
The tensor product factorization is only possible if the orbital and spin angular momenta of the particle are separable in the Hamiltonian operator underlying the system's dynamics (in other words, the Hamiltonian can be split into the sum of orbital and spin terms{{harvnb|Shankar|1994|page=378–379}}). The time dependence can be placed in either factor, and time evolution of each can be studied separately. The factorization 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.The preceding discussion is not limited to spin as a discrete variable, the total angular momentum J may also be used.{{harvnb|Landau|Lifshitz|1977}} Other discrete degrees of freedom, like isospin, can expressed similarly to the case of spin above.

Many particle states in 3d position space

(File:Two particle wavefunction.svg|right|402px|thumb|Traveling waves of two free particles, with two of three dimensions suppressed. Top is position space wave function, bottom is momentum space wave function, with corresponding probability densities.)If there are many particles, in general there is only one wave function, not a separate wave function for each particle. The fact that one wave function describes many particles is what makes quantum entanglement and the EPR paradox possible. The position-space wave function for {{math|N}} particles is written:
Psi(mathbf{r}_1,mathbf{r}_2 cdots mathbf{r}_N,t)
where {{math|ri}} is the position of the {{math|i}}th particle in three-dimensional space, and {{math|t}} is time. Altogether, this is a complex-valued function of {{math|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 {{math|+}} sign occurs if the particles are all bosons and {{math|−}} sign if they are all fermions. In other words, the wave function is either totally symmetric in the positions of bosons, or totally antisymmetric in the positions of fermions.{{harvnb|Zettili|2009|p=463}} The physical interchange of particles corresponds to mathematically switching arguments in the wave function. The antisymmetry feature of fermionic wave functions leads to the Pauli principle. Generally, bosonic and fermionic symmetry requirements are the manifestation of particle statistics and are present in other quantum state formalisms.For {{math|N}} distinguishable particles (no two being identical, i.e. no two having the same set of quantum numbers), there is no requirement for the wave function to be either symmetric or antisymmetric.For a collection of particles, some identical with coordinates {{math|r1, r2, ...}} and others distinguishable {{math|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 {{math|ri}} 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 {{math|xi}}.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)
Accumulating all these components into a single vector,underbrace{| Psi rangle}_{text{state vector (ket)}} = underbrace{overbrace{sum_{s_{z,1},ldots,s_{z,N}}}^{text{discrete labels}} overbrace{intlimitslimits_{R_N} d^3mathbf{r}_N cdots intlimitslimits_{R_1} d^3mathbf{r}_1}^{text{continuous labels}}}_{text{adding up}} , underbrace{{Psi}( mathbf{r}_1, ldots, mathbf{r}_N , s_{z,1} , ldots , s_{z,N} )}_{text{wavefunction (component of state vector along basis state)}} underbrace{| mathbf{r}_1, ldots, mathbf{r}_N , s_{z,1} , ldots , s_{z,N} rangle }_{text{basis state (basis ket)}},.For identical particles, symmetry requirements apply to both position and spin arguments of the wave function so it has the overall correct symmetry.The formulae for the inner products are integrals over all coordinates or momenta and sums over all spin quantum numbers. For the general case of {{math|N}} particles with spin in 3d,
( Psi_1 , Psi_2 ) = sum_{s_{z,N}} cdots sum_{s_{z,2}} sum_{s_{z,1}} intlimits_{mathrm{ all , space}} d ^3mathbf{r}_1 intlimits_{mathrm{ all , space}} d ^3mathbf{r}_2cdots intlimits_{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 )
this is altogether {{math|N}} three-dimensional volume integrals and {{math|N}} sums over the spins. The differential volume elements {{math|d3ri}} are also written "{{math|dVi}}" or "{{math|dxi dyi dzi}}".The multidimensional Fourier transforms of the position or position–spin space wave functions yields momentum or momentum–spin space wave functions.

Probability interpretation

For the general case of {{math|N}} particles with spin in 3d, if {{math|Ψ}} is interpreted as a probability amplitude, the probability density is
rholeft(mathbf{r}_1 cdots mathbf{r}_N,s_{z,1}cdots s_{z,N},t right ) = left | Psileft (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 {{math|R1}} with spin {{math|s'z1 {{=}} m1}} and particle 2 is in region {{math|R2}} with spin {{math|s'z2 {{=}} m2}} etc. at time {{math|t}} is the integral of the probability density over these regions and evaluated at these spin numbers:
P_{mathbf{r}_1in R_1,s_{z,1} = m_1, ldots, mathbf{r}_Nin R_N,s_{z,N} = m_N} (t) = intlimits_{R_1} d ^3mathbf{r}_1 intlimits_{R_2} d ^3mathbf{r}_2cdots intlimits_{R_N} d ^3mathbf{r}_N left | Psileft (mathbf{r}_1 cdots mathbf{r}_N,m_1cdots m_N,t right ) right |^2

Time dependence

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 {{math|N}} particles, considering their positions only and suppressing other degrees of freedom,
Psi(mathbf{r}_1,mathbf{r}_2,ldots,mathbf{r}_N,t) = e^{-i Et/hbar} ,psi(mathbf{r}_1,mathbf{r}_2,ldots,mathbf{r}_N),,
where {{math|E}} is the energy eigenvalue of the system corresponding to the eigenstate {{math|Ψ}}. Wave functions of this form are called stationary states.The time dependence of the quantum state and the operators can be placed according to unitary transformations on the operators and states. For any quantum state {{math|{{ket|Ψ}}}} and operator {{math|O}}, in the Schrödinger picture {{math|{{ket|Ψ(t)}}}} changes with time according to the Schrödinger equation while {{math|O}} is constant. In the Heisenberg picture it is the other way round, {{math|{{ket|Ψ}}}} is constant while {{math|O(t)}} evolves with time according to the Heisenberg equation of motion. The Dirac (or interaction) picture is intermediate, time dependence is places in both operators and states which evolve according to equations of motion. It is useful primarily in computing S-matrix elements.{{Harvnb|Weinberg|2002}} Chapter 3, Scattering matrix.

Non-relativistic examples

The following are solutions to the Schrödinger equation for one nonrelativistic spinless particle.

Finite potential barrier

missing image!
- Finitepot.png -
right|Scattering at a finite potential barrier of height {{math|V0}}. 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. {{math|E > V0}} 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. A common model is the "potential barrier", the one-dimensional case has the potential
V(x)=begin{cases}V_0 & |x|


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