User:Maschen/wave function

(CRUDE) PROVISIONAL RESTRUCTURE of wave function using this version

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 is a mathematical object that represents a particular quantum state of a specific isolated system of one or more particles. A single wave function describes the entire system, covering at once all the particles in it. For the one state, however, there are many different wave functions, each giving its respective representative description. Each of the different representatives contains all the information that can be known about the state, the different versions being mutually interconvertible by one-to-one mathematical transformations. A wave function can be interpreted as a probability amplitude. All 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, including for example quantum field theory. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi).

The Schrödinger equation determines the allowed wave functions for the system and how they evolve over time. Because that equation is mathematically a type of wave equation, a wave function behaves in some respects like other waves, such as water waves or waves on a string. 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 ordinary 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.

In Born's statistical interpretation, the squared modulus of the wave function, |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 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.

Status of wavefunctions

History

[first introduction of concept, modern interpretations]

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 in nonrelativistic QM

Postulates of quantum mechanics

Some of the postulates of quantum mechanics are:

• The wavefunction Ψ contains all information about system
• It is a solution of the Schrödinger equation, a linear partial differential equation,

$i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi$

with H the Hamiltonian operator of the system.

• Each physically observable quantity of a system, such as position, momentum, or spin, is represented by a Hermitian linear operator on the state space. The possible outcomes of measurement of an observable are the eigenvalues of the corresponding operator.
• the wavefunction "collapses" instantly and irreversibly after a measurement.

(There are other postulates not directly relevant to this article).

Superposition principle

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sum_n a_n \Psi_n = a_1 \Psi_1 + a_2 \Psi_2 + \cdots }

This is the superposition principle of quantum mechanics. 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.

The concept of function spaces enter 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.

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 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.

Interpretations

Wave functions can be mathematically formulated, but it is not obvious how to physically interpret the wave function. There are a number of interpretations.

The most common one is the Copenhagen interpretation, the probability density of finding the system at a any set of observables, at time t, is

$\rho (t)=|\Psi (t)|^{2}$

Probabilities are calculated by integrating over the density.

Time dependence

Note throughout, time is just a parameter and not an observable (again, non-relativistic quantum mechanics is assumed). In the Schrödinger picture, the time dependence is placed in the wave function and the observables are constant. In the Hiesenberg picture, the time dependence is switched; the wave functions are constant while the observables change with time. The intermediate case is the interaction picture, where the wavefunction and observables change with time.

Wave particle duality

Intuitive heuristics about the De Broglie relations and wave profile (momentum, wavelength, energy, kinetic energy, curvature) for one spinless particle in 1d [move to position-momentum spinless section?]

Position and momentum representations

Single spin-0 particle

[Position and momentum space wavefunctions in 1d, 3d, Fourier transforms, introduce Dirac notation here]

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:

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

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

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

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

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:

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

and the normalization condition is:

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

The wave functions above are components of a quantum state vaector, in bra-ket notation. Considering the one-dimensional non-relativistic case for a single particle,

${\begin{aligned}|\Psi \rangle =I|\Psi \rangle &=\int |x\rangle \langle x|\Psi \rangle dx=\int \Psi (x)|x\rangle dx,\\|\Psi \rangle =I|\Psi \rangle &=\int |p\rangle \langle p|\Psi \rangle dp=\int \Psi (p)|p\rangle dp.\end{aligned}}$

Now take the projection of the state Ψ onto eigenfunctions of momentum using the last expression in the two equations,

$\int \Psi (x)\langle p|x\rangle dx=\int \Psi (p')\langle p|p'\rangle dp'=\int \Psi (p')\delta (p-p')dp'=\Psi (p).$

Then utilizing the known expression for suitably normalized eigenstates of momentum in the position representation solutions of the free Schrödinger equation

$\langle x|p\rangle =p(x)={\frac {1}{\sqrt {2\pi \hbar }}}e^{{\frac {i}{\hbar }}px}\Rightarrow \langle p|x\rangle ={\frac {1}{\sqrt {2\pi \hbar }}}e^{-{\frac {i}{\hbar }}px},$

one obtains

$\Psi (p)={\frac {1}{\sqrt {2\pi \hbar }}}\int \Psi (x)e^{-{\frac {i}{\hbar }}px}dx.$

Likewise, using eigenfunctions of position,

$\Psi (x)={\frac {1}{\sqrt {2\pi \hbar }}}\int \Psi (p)e^{{\frac {i}{\hbar }}px}dp.$

The position-space and momentum-space wave functions are thus found to be Fourier transforms of each other. 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. For one dimension,

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

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, x and 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 L²!

Many particle systems

[Symmetry (bosons), antisymmetry (fermions), nonsymmetry (distinguishable), The Pauli principle, implications from them]

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.

Spin

Main page: user:Maschen/Spin wave function

[Spin s particle in 3d, 2s+1 complex numbers, or a 2s+1 column vector, wavefunctions as spinors or tensors for particles, occurrence in relativistic QM and QFT]

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.

Other representations

Position or momentum, and spin, are the common representations to express a system, but are not the only ones. Canonical transformations on position and momentum can obtain more variables. Particles (e.g. nucleons) can have isospin. Quarks and gluons have colour charge. Examples of these in their respective areas are given below.

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).

For generality, let α₁, α₂, ... be scalars each taking discrete values, ω₁, ω₂, ... be scalars each taking continuous ranges of values, a₁, a₂, ... be vectors with their components taking discrete values, and z₁, z₂, ... be vectors with their components taking continuous ranges of values. The collection of quantities α₁, α₂, ..., ω₁, ω₂, ..., a₁, a₂, ..., z₁, z₂, ... are the eigenvalues of a complete set of commuting observables. These may be freely chosen, provided completeness and commutativity holds. Such a set may be called a representation. Such observables are Hermitian linear operators on the space of states, representing physical observables.

For the particular state in this representation, there is one wave function, a complex-valued function of the real-valued observables, that belongs to the chosen representation and coordinate system, and time.

$\Psi (\alpha _{1},\alpha _{2},\ldots ,\omega _{1},\omega _{2},\ldots ,\mathbf {a} _{1},\mathbf {a} _{2},\ldots ,\mathbf {z} _{1},\mathbf {z} _{2},\ldots ,t)$

If the operators commute, the physical interpretation is that such a set represents what can – in theory – be simultaneously be measured with arbitrary precision. If they do not, then there is an uncertainty relation between the observables, and they cannot be simultaneously measured with arbitrary precision. It makes no sense to have a function of two or more observables which cannot be measured simultaneously, because if one observable is known definitely, then at least one other observable will not be at all. Therefore, a wave function can be a function of commuting observables, but not noncommuting observables. There is at least one such complete set of observables for which the state is a simultaneous eigenstate. Completeness means that there can be added to the set no further algebraically independent linear Hermitian operator that commutes with the ones already present. The set is non-unique. We could choose a different set of complete and commuting observables for the same system, and the wavefunction will contain the same information about the same quantum state. Wave functions corresponding to a state are accordingly not unique, and depends on the observables used. This non-uniqueness reflects the non-uniqueness in the choice of a complete set of commuting observables.

Once a representation is chosen, any suitable curvilinear coordinate system may be chosen, most often Cartesian coordinates (x, y, z), or spherical coordinates (r, θ, φ) used for the atomic wave functions illustrated below. This final choice of coordinate system also fixes a basis in an abstract Hilbert space. The basic states are labeled by the quantum numbers corresponding to the complete set of commuting observables and an appropriate coordinate system.

For example, for a single particle z₁ could be the position or momentum vector of the particle in Cartesian coordinates, while α₁ could be a component of its spin along any one of the Cartesian axes. All three components of position and one component of spin can be measured with arbitrary precision. Likewise all three components of momentum and one component of spin can be measured with arbitrary precision. However, position and momentum cannot be measured simultaneously, so the wave function cannot be a function of position and momentum vectors (one could define new variables according to a canonical transformation, but this is a separate matter). Also, we cannot have (say) z₁ as the full spin vector, because the components of spin do not commute with each other. Also, the position and momentum space wave functions describe the same abstract state, and are related by a Fourier transform.

For example, for a one-particle system, the set of observables may 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.

If there are many particles, then z₁, z₂, ... are the positions of the particles, while α₁, α₂, ... spin projections for each particle along some common reference direction.

A non-example of the a vectors would be the full orbital angular momentum vector, because the components of the operator do not commute with each other.

• 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 complete 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 complete 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.

The set of all tuples (α₁, α₂, ..., ω₁, ω₂, ..., a₁, a₂, ..., z₁, z₂, ...) is the set of all possible states the system can be found to be in. All possible tuples form a state space, and the probability of finding the system in some "hypervolume" of the state space, which amounts to finding the continuous variables within some range and discrete variables at one or more allowed values, is given by multiple integral or multiple summation over those intervals or values:

$P=\sum _{\alpha _{1},\alpha _{2},\ldots ,\mathbf {a} _{1},\mathbf {a} _{2},\ldots }\int d\omega _{1}d\omega _{2}\ldots d^{3}\mathbf {z} _{1}d^{3}\mathbf {z} _{2}\ldots |\Psi (t)|^{2}$

at time t.

Integrating over all possible values of the continuous variables, and summing over all possible values of discrete variables, leads to the normalization condition

$\sum _{{\text{all }}\alpha _{1},\alpha _{2},\ldots ,\mathbf {a} _{1},\mathbf {a} _{2},\ldots }\int _{\text{all}}d\omega _{1}d\omega _{2}\ldots d^{3}\mathbf {z} _{1}d^{3}\mathbf {z} _{2}\ldots |\Psi (t)|^{2}=1$

because there is a 100% probability of finding the particle with some set of observables.

Units

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.

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.

More on 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.

To make things 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.

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 complete set of quantum numbers, each quantum number corresponding to an observable. Two observables corresponding to quantum numbers in the complete set must commute, therefore, the basis isn't entirely arbitrary, but nonetheless, there are always several choices.

Quantum state vectors

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.

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

$|\Psi \rangle$

known as a "ket" (a vector) written in Dirac's bra–ket notation.

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,

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

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. See dynamical pictures (quantum mechanics) for more on time dependence.

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.

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 complete 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,

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

The terminology associated with a wave function is as follows, for N distinguishable particles each with any spin.

$\underbrace {|\Psi \rangle } _{\text{state vector (ket)}}=\underbrace {\overbrace {\sum _{s_{z\,1},\ldots ,s_{z\,N}}^{\text{discrete labels}}} \overbrace {\int \limits \limits _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{R_{1}}d^{3}\mathbf {r} _{1}} ^{\text{continuous labels}}} _{\text{adding up}}\,\underbrace {\overbrace {\Psi } ^{\text{wave function}}\overbrace {(\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N})} ^{\text{eigenvalues of commuting observables}}} _{\text{component of state vector (complex number)}}\underbrace {|\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}\rangle } _{\text{basis state (basis ket)}}$

Sometimes "wave function" is used synonymously for "quantum state".

Examples

SUMMARIZE

Prototypical examples in physics

Some examples of wave functions for specific applications include:

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

Chemistry

Atomic and molecular orbitals

In particle physics

wave function of quarks, leptons

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.

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

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Further reading

• Yong-Ki Kim (September 2, 2000). "Practical Atomic Physics" (PDF). National Institute of Standards and Technology. Maryland: 1 (55 pages). Retrieved 2010-08-17.
• Polkinghorne, John (2002). Quantum Theory, A Very Short Introduction. Oxford University Press. ISBN 0-19-280252-6.

External links

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

Category:Quantum states Category:Waves