Quantum harmonic oscillators for a single spinless particle. The oscillations have no trajectory, but are instead represented each as waves; the vertical axis shows the real part (blue) and imaginary part (red) of the wave function. Panels (A-D) show four different standing-wave solutions of theSchrödinger equation. Panels (E–F) show two different wave functions that are solutions of the Schrödinger equation but not standing waves.The wave function of an initially very localized free particle.
Inquantum physics, awave function (orwavefunction) is a mathematical description of thequantum state of an isolatedquantum system. The most common symbols for a wave function are the Greek lettersψ andΨ (lower-case and capitalpsi, respectively). Wave functions arecomplex-valued. For example, a wave function might assign a complex number to each point in a region of space. TheBorn rule[1][2][3] provides the means to turn these complexprobability amplitudes into actual probabilities. In one common form, it says that thesquared modulus of a wave function that depends upon position is theprobability density ofmeasuring a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition callednormalization. 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 applyquantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave functionψ and calculate the statistical distributions for measurable quantities.
Wave functions can befunctions of variables other than position, such asmomentum. The information represented by a wave function that is dependent upon position can be converted into a wave function dependent upon momentum and vice versa, by means of aFourier transform. Some particles, likeelectrons andphotons, have nonzerospin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such asisospin. 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 foreach possible value of the discrete degrees of freedom (e.g., z-component of spin). These values are often displayed in acolumn matrix (e.g., a2 × 1 column vector for a non-relativistic electron with spin1⁄2).
According to thesuperposition principle of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form aHilbert space. The inner product of 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, theBorn rule, relating transition probabilities to inner products. TheSchrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like otherwaves, such aswater waves or waves on a string, because the Schrödinger equation is mathematically a type ofwave equation. This explains the name "wave function", and gives rise towave–particle duality. However, the wave function in quantum mechanics describes a kind of physical phenomenon, as of 2023 still open to differentinterpretations, which fundamentally differs from that ofclassic mechanical waves.[4][5][6][7][8][9][10]
In 1900,Max Planck postulated the proportionality between the frequency of a photon and its energy,,[11][12]and in 1916 the corresponding relation between a photon'smomentum andwavelength,,[13]where is thePlanck constant. In 1923, De Broglie was the first to suggest that the relation, now called theDe Broglie relation, holds formassive particles, the chief clue beingLorentz invariance,[14] and this can be viewed as the starting point for the modern development of quantum mechanics. The equations representwave–particle duality for both massless and massive particles.
In 1926, Schrödinger published the famous wave equation now named after him, theSchrödinger equation. This equation was based onclassicalconservation of energy usingquantum operators and the de Broglie relations and the solutions of the equation are the wave functions for the quantum system.[16] However, no one was clear on how to interpret it.[17] 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.[18] 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.[1]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 ofprobability amplitude.[1][2][19] This relates calculations of quantum mechanics directly to probabilistic experimental observations. It is accepted as part of theCopenhagen interpretation of quantum mechanics. There are many otherinterpretations of quantum mechanics. In 1927,Hartree andFock made the first step in an attempt to solve theN-body wave function, and developed theself-consistency cycle: aniterativealgorithm to approximate the solution. Now it is also known as theHartree–Fock method.[20] TheSlater determinant andpermanent (of amatrix) was part of the method, provided byJohn C. Slater.
Schrödinger did encounter an equation for the wave function that satisfiedrelativistic energy conservationbefore he published the non-relativistic one, but discarded it as it predicted negativeprobabilities and negativeenergies. In 1927,Klein,Gordon and Fock also found it, but incorporated theelectromagneticinteraction and proved that it wasLorentz invariant. De Broglie also arrived at the same equation in 1928. This relativistic wave equation is now most commonly known as theKlein–Gordon equation.[21]
In 1927,Pauli phenomenologically found a non-relativistic equation to describe spin-1/2 particles in electromagnetic fields, now called thePauli equation.[22] 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 ofspecial relativity and quantum mechanics applied to theelectron, now called theDirac equation. In this, the wave function is aspinor represented by four complex-valued components:[20] two for the electron and two for the electron'santiparticle, thepositron. In the non-relativistic limit, the Dirac wave function resembles the Pauli wave function for the electron. Later, otherrelativistic 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.
TheKlein–Gordon equation and theDirac 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 calledrelativistic 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.[23]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 calledfield 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, thefree 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 (spin0) and the Dirac equation (spin1⁄2) in this guise remain in the theory. Higher spin analogues include theProca equation (spin1),Rarita–Schwinger equation (spin3⁄2), and, more generally, theBargmann–Wigner equations. Formassless free fields two examples are the free fieldMaxwell equation (spin1) and the free fieldEinstein equation (spin2) for the field operators.[24]All of them are essentially a direct consequence of the requirement ofLorentz invariance. Their solutions must transform under Lorentz transformation in a prescribed way, i.e. under a particularrepresentation of the Lorentz group and that together with few other reasonable demands, e.g. thecluster decomposition property,[25]with implications forcausality is enough to fix the equations.
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 afixed 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.
Instring 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.[26]
Definition (one spinless particle in one dimension)
Thereal parts of position wave functionΨ(x) and momentum wave functionΦ(p), and corresponding probability densities|Ψ(x)|2 and|Φ(p)|2, for one spin-0 particle in onex orp dimension. The colour opacity of the particles corresponds to the probability density (not the wave function) of finding the particle at positionx or momentump.
For now, consider the simple case of a non-relativistic single particle, withoutspin, in one spatial dimension. More general cases are discussed below.
The state of such a particle is completely described by its wave function, wherex is position andt is time. This is acomplex-valued function of two real variablesx andt.
For one spinless particle in one dimension, if the wave function is interpreted as aprobability amplitude; the squaremodulus of the wave function, the positive real numberis interpreted as theprobability density for a measurement of the particle's position at a given timet. The asterisk indicates thecomplex conjugate. If the particle's position ismeasured, its location cannot be determined from the wave function, but is described by aprobability distribution.
The probability that its positionx will be in the intervala ≤x ≤b is the integral of the density over this interval:wheret is the time at which the particle was measured. This leads to thenormalization condition:because if the particle is measured, there is 100% probability that it will besomewhere.
For a given system, the set of all possible normalizable wave functions (at any given time) forms an abstract mathematicalvector space, meaning that it is possible to add together different wave functions, and multiply wave functions by complex numbers. Technically, wave functions form aray in aprojective Hilbert space rather than an ordinary vector space.
At a particular instant of time, all values of the wave functionΨ(x,t) are components of a vector. There are uncountably infinitely many of them and integration is used in place of summation. InBra–ket notation, this vector is writtenand 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 oflinear algebra can be used to manipulate and understand wave functions. For example:
Linear algebra explains how a vector space can be given abasis, 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.
The idea thatquantum states are vectors in an abstract vector space is completely general in all aspects of quantum mechanics andquantum 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. Thex coordinate is a continuous index. The|x⟩ are calledimproper vectors which, unlikeproper vectors that are normalizable to unity, can only be normalized to a Dirac delta function.[nb 3][nb 4][29]thusandwhich illuminates theidentity operatorwhich is analogous to completeness relation of orthonormal basis in N-dimensional Hilbert space.
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).
The particle also has a wave function inmomentum space:wherep is themomentum in one dimension, which can be any value from−∞ to+∞, andt is time.
Analogous to the position case, the inner product of two wave functionsΦ1(p,t) andΦ2(p,t) can be defined as:
One particular solution to the time-independent Schrödinger equation isaplane wave, which can be used in the description of a particle with momentum exactlyp, since it is an eigenfunction of the momentum operator. These functions are not normalizable to unity (they are not square-integrable), so they are not really elements of physical Hilbert space. The setforms what is called themomentum basis. This "basis" is not a basis in the usual mathematical sense. For one thing, since the functions are not normalizable, they are insteadnormalized to a delta function,[nb 4]
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
Now take the projection of the stateΨ onto eigenfunctions of momentum using the last expression in the two equations,
Then utilizing the known expression for suitably normalized eigenstates of momentum in the position representation solutions of thefree Schrödinger equationone obtains
Likewise, using eigenfunctions of position,
The position-space and momentum-space wave functions are thus found to beFourier transforms of each other.[30] They are two representations of the same state; containing the same information, and either one is sufficient to calculate any property of the particle.
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 theharmonic oscillator,x andp enter symmetrically, so there it does not matter which description one uses. The same equation (modulo constants) results. From this, with a little bit of afterthought, it follows that solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform inL2.[nb 5]
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.
If theN-dimensional set is orthonormal, then the projection operator for the space spanned by these states is given by:
where the projection is equivalent to identity operator since spans the entire Hilbert space, thus leaving any vector from Hilbert space unchanged. This is also known as completeness relation of finite dimensional Hilbert space.
The wavefunction is instead given by:
where, is a set of complex numbers which can be used to construct a wavefunction using the above formula.
For non-degenerate of some observable, if eigenvalues have subset of eigenvectors labelled as, by thepostulates of quantum mechanics, the probability of measuring the observable to be is given by:
where is a projection operator of states to subspace spanned by. The equality follows due to orthogonal nature of.
Hence, which specify state of the quantum mechanical system, have magnitudes whose square gives the probability of measuring the respective state.
While the relative phase has observable effects in experiments, the global phase of the system is experimentally indistinguishable. For example in a particle in superposition of two states, the global phase of the particle cannot be distinguished by finding expectation value of observable or probabilities of observing different states but relative phases can affect the expectation values of observables.
While the overall phase of the system is considered to be arbitrary, the relative phase for each state of a prepared state in superposition can be determined based on physical meaning of the prepared state and its symmetry. For example, the construction of spin states along x direction as a superposition of spin states along z direction, can done by applying appropriate rotation transformation on the spin along z states which provides appropriate phase of the states relative to each other.
An example of finite dimensional Hilbert space can be constructed using spin eigenkets of-spin particles which forms a dimensionalHilbert space. However, the general wavefunction of a particle that fully describes its state, is always from an infinite dimensionalHilbert space since it involves a tensor product withHilbert space relating to the position or momentum of the particle. Nonetheless, the techniques developed for finite dimensional Hilbert space are useful since they can either be treated independently or treated in consideration of linearity of tensor product.
Since thespin operator for a given-spin particles can be represented as a finitematrix which acts on independent spin vector components, it is usually preferable to denote spin components using matrix/column/row notation as applicable.
For example, each|sz⟩ is usually identified as a column vector:
but it is a common abuse of notation, because the kets|sz⟩ are not synonymous or equal to the column vectors. Column vectors simply provide a convenient way to express the spin components.
Corresponding to the notation, the z-component spin operator can be written as:
since theeigenvectors of z-component spin operator are the above column vectors, with eigenvalues being the corresponding spin quantum numbers.
Corresponding to the notation, a vector from such a finite dimensional Hilbert space is hence represented as:
where are corresponding complex numbers.
In the following discussion involving spin, the complete wavefunction is considered as tensor product of spin states from finite dimensional Hilbert spaces and the wavefunction which was previously developed. The basis for this Hilbert space are hence considered:.
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: wherer is theposition vector in three-dimensional space, andt is time. As alwaysΨ(r, t) is a complex-valued function of real variables. As a single vector inDirac notation
All the previous remarks on inner products, momentum space wave functions, Fourier transforms, and so on extend to higher dimensions.
For a particle withspin, ignoring the position degrees of freedom, the wave function is a function of spin only (time is a parameter);wheresz is thespin projection quantum number along thez axis. (Thez axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) Thesz parameter, unliker andt, is adiscrete variable. For example, for aspin-1/2 particle,sz can only be+1/2 or−1/2, and not any other value. (In general, for spins,sz can bes,s − 1, ..., −s + 1, −s). Inserting each quantum number gives a complex valued function of space and time, there are2s + 1 of them. These can be arranged into acolumn vector
Inbra–ket notation, these easily arrange into the components of a vector:
The entire vectorξ is a solution of the Schrödinger equation (with a suitable Hamiltonian), which unfolds to a coupled system of2s + 1 ordinary differential equations with solutionsξ(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:and these can also be arranged into a column vectorin 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 butcontinuous variables also, collect into a single vector
For a single particle, thetensor product⊗ of its position state vector|ψ⟩ and spin state vector|ξ⟩ gives the composite position-spin state vectorwith the identifications
The tensor product factorization of energy eigenstates is always possible if the orbital and spin angular momenta of the particle are separable in theHamiltonian operator underlying the system's dynamics (in other words, the Hamiltonian can be split into the sum of orbital and spin terms[34]). The time dependence can be placed in either factor, and time evolution of each can be studied separately. Under such Hamiltonians, any tensor product state evolves into another tensor product state, which essentially means any unentangled state remains unentangled under time evolution. This is said to happen when there is no physical interaction between the states of the tensor products. In the case of non separable Hamiltonians, energy eigenstates are said to be some linear combination of such states, which need not be factorizable; examples include a particle in amagnetic field, andspin–orbit coupling.
The preceding discussion is not limited to spin as a discrete variable, the totalangular momentumJ may also be used.[35] Other discrete degrees of freedom, likeisospin, can expressed similarly to the case of spin above.
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 thatone wave function describesmany particles is what makesquantum entanglement and theEPR paradox possible. The position-space wave function forN particles is written:[20]whereri is the position of thei-th particle in three-dimensional space, andt is time. Altogether, this is a complex-valued function of3N + 1 real variables.
In quantum mechanics there is a fundamental distinction betweenidentical particles anddistinguishable 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.[30] This translates to a requirement on the wave function for a system of identical particles:where the+ sign occurs if the particles areall bosons and− sign if they areall fermions. In other words, the wave function is either totally symmetric in the positions of bosons, or totally antisymmetric in the positions of fermions.[36] The physical interchange of particles corresponds to mathematically switching arguments in the wave function. The antisymmetry feature of fermionic wave functions leads to thePauli principle. Generally, bosonic and fermionic symmetry requirements are the manifestation ofparticle statistics and are present in other quantum state formalisms.
ForNdistinguishable particles (no two beingidentical, 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 coordinatesr1,r2, ... and others distinguishablex1,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 coordinatesri only:
Again, there is no symmetry requirement for the distinguishable particle coordinatesxi.
The wave function forN particles each with spin is the complex-valued function
Accumulating all these components into a single vector,
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 ofN particles with spin in 3-d,this is altogetherN three-dimensionalvolume integrals andN sums over the spins. The differential volume elementsd3ri are also written "dVi" or "dxi dyi dzi".
The multidimensional Fourier transforms of the position or position–spin space wave functions yields momentum or momentum–spin space wave functions.
For the general case ofN particles with spin in 3d, ifΨ is interpreted as a probability amplitude, the probability density is
and the probability that particle 1 is in regionR1 with spinsz1 =m1and particle 2 is in regionR2 with spinsz2 =m2 etc. at timet is the integral of the probability density over these regions and evaluated at these spin numbers:
In non-relativistic quantum mechanics, it can be shown using Schrodinger's time dependent wave equation that the equation:
is satisfied, where is the probability density and, is known as theprobability flux in accordance with the continuity equation form of the above equation.
Using the following expression for wavefunction:where is the probability density and is the phase of the wavefunction, it can be shown that:
Hence the spacial variation of phase characterizes theprobability flux.
In classical analogy, for, the quantity is analogous with velocity. Note that this does not imply a literal interpretation of as velocity since velocity and position cannot be simultaneously determined as per theuncertainty principle. Substituting the form of wavefunction in Schrodinger's time dependent wave equation, and taking the classical limit,:
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. ForN particles, considering their positions only and suppressing other degrees of freedom,whereE is the energy eigenvalue of the system corresponding to the eigenstateΨ. Wave functions of this form are calledstationary 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|Ψ⟩ and operatorO, in the Schrödinger picture|Ψ(t)⟩ changes with time according to the Schrödinger equation whileO is constant. In the Heisenberg picture it is the other way round,|Ψ⟩ is constant whileO(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 computingS-matrix elements.[38]
Scattering at a finite potential barrier of heightV0. 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 >V0 for this illustration.
One of the most prominent features of wave mechanics is the 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 potentialand the steady-state solutions to the wave equation have the form (for some constantsk,κ)
Note that these wave functions are not normalized; seescattering 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 negativex): settingAr = 1 corresponds to firing particles singly; the terms containingAr andCr signify motion to the right, whileAl andCl – to the left. Under this beam interpretation, putCl = 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.
3D confined electron wave functions in a quantum dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are mores-type andp-type. However, in a triangular dot the wave functions are mixed due to confinement symmetry. (Click for animation)
In a semiconductorcrystallite whose radius is smaller than the size of itsexcitonBohr radius, the excitons are squeezed, leading toquantum confinement. The energy levels can then be modeled using theparticle in a box model in which the energy of different states is dependent on the length of the box.
The electron probability density for the first fewhydrogen atom electronorbitals shown as cross-sections. These orbitals form anorthonormal basis for the wave function of the electron. Different orbitals are depicted with different scale.
It is convenient to use spherical coordinates, and the wave function can be separated into functions of each coordinate,[39]whereR are radial functions andYm ℓ(θ,φ) arespherical harmonics of degreeℓ and orderm. This is the only atom for which the Schrödinger equation has been solved exactly. Multi-electron atoms require approximative methods. The family of solutions is:[40]wherea0 = 4πε0ħ2/mee2 is theBohr radius,L2ℓ + 1 n −ℓ − 1 are thegeneralized Laguerre polynomials of degreen −ℓ − 1,n = 1, 2, ... is theprincipal quantum number,ℓ = 0, 1, ...,n − 1 theazimuthal quantum number,m = −ℓ, −ℓ + 1, ...,ℓ − 1,ℓ themagnetic quantum number.Hydrogen-like atoms have very similar solutions.
This solution does not take into account the spin of the electron.
In the figure of the hydrogen orbitals, the 19 sub-images are images of wave functions in position space (their norm squared). The wave functions 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 denotedL2.
The displayed functions are solutions to the Schrödinger equation. Obviously, not every function inL2 satisfies the Schrödinger equation for the hydrogen atom. The function space is thus a subspace ofL2.
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 acountable basis.
The concept offunction 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 (in the present case that they aresquare integrable), sometimes with analgebraic structure on the set (in the present case avector space structure with aninner product), together with atopology on the 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 beclosed. It will be concluded below that the function space of wave functions is aHilbert space. This observation is the foundation of the predominant mathematical formulation of quantum mechanics.
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 set of solutions to the Schrödinger equation is a vector space.
The superposition principle of quantum mechanics. IfΨ andΦ are two states in the abstract space ofstates of a quantum mechanical system, anda andb are any two complex numbers, thenaΨ +bΦ is a valid state as well. (Whether thenull vector counts as a valid state ("no system present") is a matter of definition. The null vector doesnot at any rate describe thevacuum state in quantum field theory.) The set of allowable states is a vector space.
This similarity is of course not accidental. 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 amaximal set ofcommutingobservables. Physical observables are represented by linear operators, also called observables, on the vectors space. Maximality means that there can be added to the set no further algebraically independent observables that commute with the ones already present. A choice of such a set may be called a choice ofrepresentation.
It is a postulate of quantum mechanics that a physically observable quantity of a system, such as position, momentum, or spin, is represented by a linearHermitian operator on the state space. The possible outcomes of measurement of the quantity are theeigenvalues of the operator.[18] At a deeper level, most observables, perhaps all, arise as generators ofsymmetries.[18][41][nb 6]
The physical interpretation is that such a set represents what can – in theory – simultaneously be measured with arbitrary precision. TheHeisenberg uncertainty relation prohibits simultaneous exact measurements of two non-commuting observables.
The set is non-unique. It may for a one-particle system, for example, be position and spinz-projection,(x,Sz), or it may be momentum and spiny-projection,(p,Sy). In this case, the operator corresponding to position (amultiplication operator in the position representation) and the operator corresponding to momentum (adifferential operator in the position representation) do not commute.
Once a representation is chosen, there is still arbitrariness. It remains to choose a coordinate system. This may, for example, correspond to a choice ofx,y- andz-axis, or a choice ofcurvilinear coordinates as exemplified by thespherical coordinates used for the Hydrogen atomic wave functions. 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.[nb 7]
The abstract states are "abstract" only in that an arbitrary choice necessary for a particularexplicit 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. Wave functions corresponding to a state are accordingly not unique. This non-uniqueness reflects the non-uniqueness in the choice of a maximal set of commuting observables. For one spin particle in one dimension, to a particular state there corresponds two wave functions,Ψ(x,Sz) andΨ(p,Sy), both describing thesame 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.
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 theFourier transform.
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.
There is an additional algebraic structure on the vector spaces of wave functions and the abstract state space.
Physically, different wave functions are interpreted to overlap to some degree. A system in a stateΨ that doesnot overlap with a stateΦ cannot be found to be in the stateΦ upon measurement. But ifΦ1, Φ2, … overlapΨ tosome degree, there is a chance that measurement of a system described byΨ will be found in statesΦ1, Φ2, …. Alsoselection rules are observed apply. These are usually formulated in the preservation of some quantum numbers. This means that certain processes allowable from some perspectives (e.g. energy and momentum conservation) do not occur because the initial and finaltotal wave functions do not overlap.
Mathematically, it turns out that solutions to the Schrödinger equation for particular potentials areorthogonal in some manner, this is usually described by an integral wherem,n are (sets of) indices (quantum numbers) labeling different solutions, the strictly positive functionw is called a weight function, andδmn is theKronecker delta. The integration is taken over all of the relevant space.
This motivates the introduction of aninner product on the vector space of abstract quantum states, compatible with the mathematical observations above when passing to a representation. It is denoted(Ψ, Φ), or in theBra–ket notation⟨Ψ|Φ⟩. It yields a complex number. With the inner product, the function space is aninner product space. The explicit appearance of the inner product (usually an integral or a sum of integrals) depends on the choice of representation, but the complex number(Ψ, Φ) does not. Much of the physical interpretation of quantum mechanics stems from theBorn rule. It states that the probabilityp of finding upon measurement the stateΦ given the system is in the stateΨ iswhereΦ andΨ are assumed normalized. Consider ascattering experiment. In quantum field theory, ifΦout describes a state in the "distant future" (an "out state") after interactions between scattering particles have ceased, andΨin an "in state" in the "distant past", then the quantities(Φout, Ψin), withΦout andΨin varying over a complete set of in states and out states respectively, is called theS-matrix orscattering matrix. Knowledge of it is, effectively, havingsolved the theory at hand, at least as far as predictions go. Measurable quantities such asdecay rates andscattering cross sections are calculable from the S-matrix.[42]
The above observations encapsulate the essence of the function spaces of which wave functions are elements. However, the description is not yet complete. There is a further technical requirement on the function space, that ofcompleteness, 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 aHilbert space. The property of completeness is crucial in advanced treatments and applications of quantum mechanics. For instance, the existence ofprojection operators ororthogonal projections relies on the completeness of the space.[43] These projection operators, in turn, are essential for the statement and proof of many useful theorems, e.g. thespectral theorem. It is not very important in introductory quantum mechanics, and technical details and links may be found in footnotes like the one that follows.[nb 8]The spaceL2 is a Hilbert space, with inner product presented later. The function space of the example of the figure is a subspace ofL2. A subspace of a Hilbert space is a Hilbert space if it is closed.
In summary, the set of all possible normalizable wave functions for a system with a particular choice of basis, together with the null vector, constitute a Hilbert space.
Not all functions of interest are elements of some Hilbert space, sayL2. The most glaring example is the set of functionse2πip ·x⁄h. These are plane wave solutions of the Schrödinger equation for afree particle that are not normalizable, hence not inL2. But they are nonetheless fundamental for the description. One can, using them, express functions thatare normalizable usingwave packets. They are, in a sense, a basis (but not a Hilbert space basis, nor aHamel 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 are not square integrable either.
The above description of the function space containing the wave functions is mostly mathematically motivated. The function spaces are, due to completeness, verylarge in a certain sense. Not all functions are realistic descriptions of any physical system. For instance, in the function spaceL2 one can find the function that takes on the value0 for all rational numbers and-i for the irrationals in the interval[0, 1]. Thisis square integrable,[nb 9]but can hardly represent a physical state.
While the space of solutions as a whole is a Hilbert space there are many other Hilbert spaces that commonly occur as ingredients.
Square integrable complex valued functions on the interval[0, 2π]. The set{eint/2π,n ∈Z} is a Hilbert space basis, i.e. a maximal orthonormal set.
TheFourier transform takes functions in the above space to elements ofl2(Z), the space ofsquare summable functionsZ →C. The latter space is a Hilbert space and the Fourier transform is an isomorphism of Hilbert spaces.[nb 10] Its basis is{ei,i ∈Z} withei(j) =δij,i,j ∈Z.
The most basic example of spanning polynomials is in the space of square integrable functions on the interval[–1, 1] for which theLegendre polynomials is a Hilbert space basis (complete orthonormal set).
The square integrable functions on theunit sphereS2 is a Hilbert space. The basis functions in this case are thespherical harmonics. The Legendre polynomials are ingredients in the spherical harmonics. Most problems with rotational symmetry will have "the same" (known) solution with respect to that symmetry, so the original problem is reduced to a problem of lower dimensionality.
Theassociated Laguerre polynomials appear in the hydrogenic wave function problem after factoring out the spherical harmonics. These span the Hilbert space of square integrable functions on the semi-infinite interval[0, ∞).
There occurs also finite-dimensional Hilbert spaces. The spaceCn is a Hilbert space of dimensionn. The inner product is the standard inner product on these spaces. In it, the "spin part" of a single particle wave function resides.
In the non-relativistic description of an electron one hasn = 2 and the total wave function is a solution of thePauli equation.
In the corresponding relativistic treatment,n = 4 and the wave function solves theDirac equation.
With more particles, the situations is more complicated. One has to employtensor products and use representation theory of the symmetry groups involved (therotation group and theLorentz group respectively) to extract from the tensor product the spaces in which the (total) spin wave functions reside. (Further problems arise in the relativistic case unless the particles are free.[44] See theBethe–Salpeter equation.) Corresponding remarks apply to the concept ofisospin, for which the symmetry group isSU(2). The models of the nuclear forces of the sixties (still useful today, seenuclear force) used the symmetry groupSU(3). In this case, as well, the part of the wave functions corresponding to the inner symmetries reside in someCn or subspaces of tensor products of such spaces.
In quantum field theory the underlying Hilbert space isFock space. It is built from free single-particle states, i.e. wave functions when a representation is chosen, and can accommodate any finite, not necessarily constant in time, number of particles. The interesting (or rather thetractable) dynamics lies not in the wave functions but in thefield operators that are operators acting on Fock space. Thus theHeisenberg picture is the most common choice (constant states, time varying operators).
Due to the infinite-dimensional nature of the system, the appropriate mathematical tools are objects of study infunctional analysis.
Continuity of the wave function and its first spatial derivative (in thex direction,y andz coordinates not shown), at some timet.
Not all introductory textbooks take the long route and introduce the full Hilbert space machinery, but the focus is on the non-relativistic Schrödinger equation in position representation for certain standard potentials. The following constraints on the wave function are sometimes explicitly formulated for the calculations and physical interpretation to make sense:[45][46]
The wave function must besquare integrable. This is motivated by the Copenhagen interpretation of the wave function as a probability amplitude.
It must be everywherecontinuous and everywherecontinuously differentiable. This is motivated by the appearance of the Schrödinger equation for most physically reasonable potentials.
It is possible to relax these conditions somewhat for special purposes.[nb 11]If these requirements are not met, it is not possible to interpret the wave function as a probability amplitude.[47] Note that exceptions can arise to the continuity of derivatives rule at points of infinite discontinuity of potential field. For example, inparticle in a box where the derivative of wavefunction can be discontinuous at the boundary of the box where the potential is known to have infinite discontinuity.
This does not alter the structure of the Hilbert space that these particular wave functions inhabit, but the subspace of the square-integrable functionsL2, which is a Hilbert space, satisfying the second requirementis not closed inL2, hence not a Hilbert space in itself.[nb 12]The functions that does not meet the requirements are still needed for both technical and practical reasons.[nb 13][nb 14]
As has been demonstrated, the set of all possible wave functions in some representation for a system constitute an in generalinfinite-dimensional Hilbert space. Due to the multiple possible choices of representation basis, these Hilbert spaces are not unique. One therefore talks about an abstract Hilbert space,state space, where the choice of representation and basis is left undetermined. Specifically, each state is represented as an abstract vector in state space.[48] A quantum state|Ψ⟩ in any representation is generally expressed as a vector[citation needed]where
|α,ω⟩ the basis vectors of the chosen representation
These quantum numbers index the components of the state vector. More, allα are in ann-dimensionalsetA =A1 ×A2 × ... ×An where eachAi is the set of allowed values forαi; allω are in anm-dimensional "volume"Ω ⊆ ℝm whereΩ = Ω1 × Ω2 × ... × Ωm and eachΩi ⊆R is the set of allowed values forωi, asubset of thereal numbersR. For generalityn andm are not necessarily equal.
Example:
For a single particle in 3d with spins, neglecting other degrees of freedom, using Cartesian coordinates, we could takeα = (sz) for the spin quantum number of the particle along the z direction, andω = (x,y,z) for the particle's position coordinates. HereA = {−s, −s + 1, ...,s − 1,s} is the set of allowed spin quantum numbers andΩ =R3 is the set of all possible particle positions throughout 3d position space.
An alternative choice isα = (sy) for the spin quantum number along the y direction andω = (px,py,pz) for the particle's momentum components. In this caseA andΩ are the same as before.
The probability of finding system withα in some or all possible discrete-variable configurations,D ⊆A, andω in some or all possible continuous-variable configurations,C ⊆ Ω, is the sum and integral over the density,[nb 15]
Since the sum of all probabilities must be 1, the normalization conditionmust hold at all times during the evolution of the system.
The normalization condition requiresρ dmω to be dimensionless, bydimensional analysisΨ must have the same units as(ω1ω2...ωm)−1/2.
Whether the wave function exists in reality, and what it represents, are major questions in theinterpretation of quantum mechanics. Many famous physicists of a previous generation puzzled over this problem, such asErwin Schrödinger,Albert Einstein andNiels Bohr. Some advocate formulations or variants of theCopenhagen interpretation (e.g. Bohr,Eugene Wigner andJohn von Neumann) while others, such asJohn Archibald Wheeler orEdwin Thompson Jaynes, take the more classical approach[49] 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,David Bohm andHugh Everett III 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.[50]
^The functions are here assumed to be elements ofL2, 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 ofLebesgue measure0. This is necessary to obtain an inner product (that is,(Ψ, Ψ) = 0 ⇒ Ψ ≡ 0) as opposed to asemi-inner product. The integral is taken to be theLebesgue integral. This is essential for completeness of the space, thus yielding a complete inner product space = Hilbert space.
^As, technically, they are not in the Hilbert space. SeeSpectral theorem for more details.
^abAlso called "Dirac orthonormality", according toGriffiths, David J.Introduction to Quantum Mechanics (3rd ed.).
^The Fourier transform viewed as a unitary operator on the spaceL2 has eigenvalues±1, ±i. The eigenvectors are "Hermite functions", i.e.Hermite polynomials multiplied by aGaussian function. SeeByron & Fuller (1992) for a description of the Fourier transform as a unitary transformation. For eigenvalues and eigenvalues, refer to Problem 27 Ch. 9.
^For this statement to make sense, the observables need to be elements of a maximal commuting set. To see this, it is a simple matter to note that, for example, the momentum operator of the i'th particle in a n-particle system isnot a generator of any symmetry in nature. On the other hand, thetotal momentumis a generator of a symmetry in nature; the translational symmetry.
^The resulting basis may or may not technically be a basis in the mathematical sense of Hilbert spaces. For instance, states of definite position and definite momentum are not square integrable. This may be overcome with the use ofwave packets or by enclosing the system in a "box". See further remarks below.
^In technical terms, this is formulated the following way. The inner product yields anorm. This norm, in turn, induces ametric. If this metric iscomplete, then the aforementioned limits will be in the function space. The inner product space is then called complete. A complete inner product space is aHilbert space. The abstract state space is always taken as a Hilbert space. The matching requirement for the function spaces is a natural one. The Hilbert space property of the abstract state space was originally extracted from the observation that the function spaces forming normalizable solutions to the Schrödinger equation are Hilbert spaces.
^Conway 1990. This means that inner products, hence norms, are preserved and that the mapping is a bounded, hence continuous, linear bijection. The property of completeness is preserved as well. Thus this is the right concept of isomorphism in thecategory of Hilbert spaces.
^One such relaxation is that the wave function must belong to theSobolev spaceW1,2. It means that it is differentiable in the sense ofdistributions, and itsgradient issquare-integrable. This relaxation is necessary for potentials that are not functions but are distributions, such as theDirac delta function.
^It is easy to visualize a sequence of functions meeting the requirement that converges to adiscontinuous function. For this, modify an example given inInner product space#Some examples. This element thoughis an element ofL2.
^For instance, inperturbation theory one may construct a sequence of functions approximating the true wave function. This sequence will be guaranteed to converge in a larger space, but without the assumption of a full-fledged Hilbert space, it will not be guaranteed that the convergence is to a function in the relevant space and hence solving the original problem.
^Some functions not being square-integrable, like the plane-wave free particle solutions are necessary for the description as outlined in a previous note and also further below.
^Weinberg (2002) takes the standpoint that quantum field theory appears the way it does because it is theonly way to reconcile quantum mechanics with special relativity.
^Weinberg (2002) See especially chapter 5, where some of these results are derived.
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