A wave function in
quantum physics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
is a mathematical description of the
quantum state of an isolated
quantum system
Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...
. The wave function is a
complex-valued
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters and (lower-case and capital
psi, respectively).
The wave function is a
function of the
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
corresponding to some maximal set of
commuting observables
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum physi ...
. Once such a representation is chosen, the wave function can be derived from the quantum state.
For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the
domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over
momentum space; the two are related by a
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
. Some particles, like
electrons and
photons, have nonzero
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
, and the wave function for such particles include spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as
isospin. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for ''each'' possible value of the discrete degrees of freedom (e.g., z-component of spin) – these values are often displayed in a
column matrix
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, ...
(e.g., a column vector for a non-relativistic electron with spin ).
According to the
superposition principle
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. The inner product between two wave functions is a measure of the overlap between the corresponding physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the
Born rule, relating transition probabilities to inner products. The
Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other
waves, such as
water waves or waves on a string, because the Schrödinger equation is mathematically a type of
wave equation. This explains the name "wave function", and gives rise to
wave–particle duality. However, the wave function in quantum mechanics describes a kind of physical phenomenon, still open to different
interpretations, which fundamentally differs from that of
classic mechanical waves.
In
Born
Born may refer to:
* Childbirth
* Born (surname), a surname (see also for a list of people with the name)
* ''Born'' (comics), a comic book limited series
Places
* Born, Belgium, a village in the German-speaking Community of Belgium
* Born, Luxe ...
's statistical interpretation in non-relativistic quantum mechanics,
the
squared modulus
In mathematics, a square is the result of multiplication, multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as exponentiation, raising to the power 2 (number), 2, and is denoted by a ...
of the wave function, , is a
real number interpreted as the
probability density of
measuring a particle as being at a given place – or having a given momentum – at a given time, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation. This general requirement that a wave function must satisfy is called the ''normalization condition''. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured—its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables; one has to apply
quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function and calculate the statistical distributions for measurable quantities.
Historical background
In 1905,
Albert Einstein postulated the proportionality between the frequency
of a photon and its energy
and in 1916 the corresponding relation between a photon's
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
and
wavelength
where
is the
Planck constant. In 1923, De Broglie was the first to suggest that the relation 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
Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a n ...
, 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, the
Schrödinger equation. This equation was based on
classical conservation of energy
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
using
quantum operators and the de Broglie relations and 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 the elastic scattering of a wave packet (representing a particle) off a target; it spreads out in all directions.
[, translated in at pages 52–55.]
While a scattered particle may scatter in any direction, it does not break up and take off in all directions. In 1926, Born provided the perspective of
probability amplitude.
[, translated in . Als]
here
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
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
) 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
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
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
Pauli is a surname and also a Finnish male given name (variant of Paul) and may refer to:
* Arthur Pauli (born 1989), Austrian ski jumper
* Barbara Pauli (1752 or 1753 - fl. 1781), Swedish fashion trader
*Gabriele Pauli (born 1957), German politi ...
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
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collides ...
. In the non-relativistic limit, the Dirac wave function resembles the Pauli wave function for the electron. Later, other
relativistic wave equations were found.
Wave functions and wave equations in modern theories
All these wave equations are of enduring importance. The Schrödinger equation and the Pauli equation are under many circumstances excellent approximations of the relativistic variants. They are considerably easier to solve in practical problems than the relativistic counterparts.
The
Klein–Gordon equation and the
Dirac equation, while being relativistic, do not represent full reconciliation of quantum mechanics and special relativity. The branch of quantum mechanics where these equations are studied the same way as the Schrödinger equation, often called
relativistic quantum mechanics, while very successful, has its limitations (see e.g.
Lamb shift) and conceptual problems (see e.g.
Dirac sea).
Relativity makes it inevitable that the number of particles in a system is not constant. For full reconciliation,
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
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 ) and the Dirac equation (spin ) in this guise remain in the theory. Higher spin analogues include the
Proca equation
In physics, specifically field theory (physics), field theory and particle physics, the Proca action describes a massive spin (physics), spin-1 quantum field, field of mass ''m'' in Minkowski spacetime. The corresponding equation is a relativisti ...
(spin ),
Rarita–Schwinger equation (spin ), and, more generally, the
Bargmann–Wigner equations
:''This article uses the Einstein summation convention for tensor/spinor indices, and uses hats for quantum operators.
In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles with non ...
. For ''massless'' free fields two examples are the free field
Maxwell equation
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
Th ...
(spin ) and the free field
Einstein equation (spin ) 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
The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of repres ...
and that together with few other reasonable demands, e.g. the
cluster decomposition property,
with implications for
causality
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
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 a ''fixed'' number of particles and would not account for the term "interaction" as referred to in these theories, which involves the creation and annihilation of particles and not external potentials as in ordinary "first quantized" quantum theory.
In
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
, 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.
Definition (one spinless particle in one dimension)
For now, consider the simple case of a non-relativistic single particle, without
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
, in one spatial dimension. More general cases are discussed below.
Position-space wave functions
The state of such a particle is completely described by its wave function,
where is position and is time. This is a
complex-valued function of two real variables and .
For one spinless particle in one dimension, if the wave function is interpreted as a
probability amplitude, the square
modulus of the wave function, the positive real number
is interpreted as the
probability density that the particle is at . The asterisk indicates the
complex conjugate. If the particle's position is
measured
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared t ...
, its location cannot be determined from the wave function, but is described by a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
.
Normalization condition
The probability that its position will be in the interval is the integral of the density over this interval:
where is the time at which the particle was measured. This leads to the normalization condition:
because if the particle is measured, there is 100% probability that it will be ''somewhere''.
For a given system, the set of all possible normalizable wave functions (at any given time) forms an abstract mathematical
vector space, meaning that it is possible to add together different wave functions, and multiply wave functions by complex numbers (see
vector space for details). Technically, because of the normalization condition, wave functions form a
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
rather than an ordinary vector space. This vector space is infinite-
dimensional, because there is no finite set of functions which can be added together in various combinations to create every possible function. Also, it is a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, because the
inner product of two wave functions and can be defined as the complex number (at time )
[The functions are here assumed to be elements of , the space of square integrable functions. The elements of this space are more precisely equivalence classes of square integrable functions, two functions declared equivalent if they differ on a set of ]Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
. This is necessary to obtain an inner product (that is, ) as opposed to a semi-inner product. The integral is taken to be the Lebesgue integral. This is essential for completeness of the space, thus yielding a complete inner product space = Hilbert space.
More details are given
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
. Although the inner product of two wave functions is a complex number, the inner product of a wave function with itself,
is ''always'' a positive real number. The number (not ) is called the
norm of the wave function .
If , then is normalized. If is not normalized, then dividing by its norm gives the normalized function . Two wave functions and are
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
if . If they are normalized ''and'' orthogonal, they are
orthonormal. Orthogonality (hence also orthonormality) of wave functions is not a necessary condition wave functions must satisfy, but is instructive to consider since this guarantees
linear independence of the functions. In a linear combination of orthogonal wave functions we have,
If the wave functions were nonorthogonal, the coefficients would be less simple to obtain.
Quantum states as vectors
In the
Copenhagen interpretation, the modulus squared of the inner product (a complex number) gives a real number
which, assuming both wave functions are normalized, is interpreted as the probability of the wave function
"collapsing" to the new wave function upon measurement of an observable, whose eigenvalues are the possible results of the measurement, with being an eigenvector of the resulting eigenvalue. This is the
Born rule,
and is one of the fundamental postulates of quantum mechanics.
At a particular instant of time, all values of the wave function are components of a vector. There are uncountably infinitely many of them and integration is used in place of summation. In
Bra–ket notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathema ...
, this vector is written
and is referred to as a "quantum state vector", or simply "quantum state". There are several advantages to understanding wave functions as representing elements of an abstract vector space:
* All the powerful tools of
linear algebra can be used to manipulate and understand wave functions. For example:
** Linear algebra explains how a vector space can be given a
basis, and then any vector in the vector space can be expressed in this basis. This explains the relationship between a wave function in position space and a wave function in momentum space and suggests that there are other possibilities too.
**
Bra–ket notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathema ...
can be used to manipulate wave functions.
* The idea that
quantum states are vectors in an abstract vector space is completely general in all aspects of quantum mechanics and
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, whereas the idea that quantum states are complex-valued "wave" functions of space is only true in certain situations.
The time parameter is often suppressed, and will be in the following. The coordinate is a continuous index. The are the basis vectors, which are
orthonormal so their
inner product is a
delta function;
thus
and
which illuminates the
identity operator
Finding the identity operator in a basis allows the abstract state to be expressed explicitly in a basis, and more (the inner product between two state vectors, and other operators for observables, can be expressed in the basis).
Momentum-space wave functions
The particle also has a wave function in
momentum space:
where is the
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
in one dimension, which can be any value from to , and is time.
Analogous to the position case, the inner product of two wave functions and can be defined as:
One particular solution to the time-independent Schrödinger equation is
a
plane wave, which can be used in the description of a particle with momentum exactly , since it is an eigenfunction of the momentum operator. These functions are not normalizable to unity (they aren't square-integrable), so they are not really elements of physical Hilbert space. The set
forms what is called the momentum basis. This "basis" is not a basis in the usual mathematical sense. For one thing, since the functions aren't normalizable, they are instead normalized to a delta function,
[Also called "Dirac orthonormality", according to ]
For another thing, though they are linearly independent, there are too many of them (they form an uncountable set) for a basis for physical Hilbert space. They can still be used to express all functions in it using Fourier transforms as described next.
Relations between position and momentum representations
The and representations are
Now take the projection of the state 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 the free Schrödinger equation
one obtains
Likewise, using eigenfunctions of position,
The position-space and momentum-space wave functions are thus found to be
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
s 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.
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
In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'':
\v ...
, and 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 .
[The Fourier transform viewed as a unitary operator on the space has eigenvalues . The eigenvectors are "Hermite functions", i.e. ]Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
* signal processing as Hermitian wavelets for wavelet transform analysis
* probability, such as the Edgeworth series, as well a ...
multiplied by a Gaussian function. See for a description of the Fourier transform as a unitary transformation. For eigenvalues and eigenvalues, refer to Problem 27 Ch. 9.
Definitions (other cases)
Following are the general forms of the wave function for systems in higher dimensions and more particles, as well as including other degrees of freedom than position coordinates or momentum components.
One-particle states in 3d position space
The position-space wave function of a single particle without spin in three spatial dimensions is similar to the case of one spatial dimension above:
where is the
position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or s ...
in three-dimensional space, and is time. As always is a complex-valued function of real variables. As a single vector in
Dirac notation
All the previous remarks on inner products, momentum space wave functions, Fourier transforms, and so on extend to higher dimensions.
For a particle with
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
, ignoring the position degrees of freedom, the wave function is a function of spin only (time is a parameter);
where is the
spin projection quantum number along the axis. (The axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) The parameter, unlike and , is a ''discrete variable''. For example, for a
spin-1/2 particle, can only be or , and not any other value. (In general, for spin , can be ). Inserting each quantum number gives a complex valued function of space and time, there are of them. These can be arranged into a
column vector[Column vectors can be motivated by the convenience of expressing the spin operator for a given spin as a ]matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
, for the z-component spin operator (divided by hbar to nondimensionalize): The eigenvectors of this matrix are the above column vectors, with eigenvalues being the corresponding spin quantum numbers.
In
bra–ket notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathema ...
, these easily arrange into the components of a vector
[Each is usually identified as a column vector: but it is a common abuse of notation to write: because the kets are not synonymous or equal to the column vectors. Column vectors simply provide a convenient way to express the spin components.]
The entire vector is a solution of the Schrödinger equation (with a suitable Hamiltonian), which unfolds to a coupled system of ordinary differential equations with solutions . 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 vector
in which the spin dependence is placed in indexing the entries, and the wave function is a complex vector-valued function of space and time only.
All values of the wave function, not only for discrete but continuous variables also, collect into a single vector
For a single particle, the
tensor product of its position state vector and spin state vector gives the composite position-spin state vector
with the identifications
The tensor product factorization is only possible if the orbital and spin angular momenta of the particle are separable in the
Hamiltonian operator underlying the system's dynamics (in other words, the Hamiltonian can be split into the sum of orbital and spin terms). The time dependence can be placed in either factor, and time evolution of each can be studied separately. The factorization is not possible for those interactions where an external field or any space-dependent quantity couples to the spin; examples include a particle in a
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
, and
spin–orbit coupling.
The preceding discussion is not limited to spin as a discrete variable, the total
angular momentum ''J'' may also be used. Other discrete degrees of freedom, like
isospin, can expressed similarly to the case of spin above.
Many-particle states in 3d position space
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 particles is written:
where is the position of the -th particle in three-dimensional space, and is time. Altogether, this is a complex-valued function of 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:
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 ''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 and others distinguishable (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 only:
Again, there is no symmetry requirement for the distinguishable particle coordinates .
The wave function for ''N'' 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 of particles with spin in 3-d,
this is altogether three-dimensional
volume integrals and sums over the spins. The differential volume elements are also written "" or "".
The multidimensional Fourier transforms of the position or position–spin space wave functions yields momentum or momentum–spin space wave functions.
Probability interpretation
For the general case of particles with spin in 3d, if is interpreted as a probability amplitude, the probability density is
and the probability that particle 1 is in region with spin ''and'' particle 2 is in region with spin etc. at time is the integral of the probability density over these regions and evaluated at these spin numbers:
Time dependence
For systems in time-independent potentials, the wave function can always be written as a function of the degrees of freedom multiplied by a time-dependent phase factor, the form of which is given by the Schrödinger equation. For particles, considering their positions only and suppressing other degrees of freedom,
where is the energy eigenvalue of the system corresponding to the eigenstate . Wave functions of this form are called
stationary states.
The time dependence of the quantum state and the operators can be placed according to unitary transformations on the operators and states. For any quantum state and operator , in the Schrödinger picture changes with time according to the Schrödinger equation while is constant. In the Heisenberg picture it is the other way round, is constant while evolves with time according to the Heisenberg equation of motion. The Dirac (or interaction) picture is intermediate, time dependence is places in both operators and states which evolve according to equations of motion. It is useful primarily in computing
S-matrix elements.
Non-relativistic examples
The following are solutions to the Schrödinger equation for one nonrelativistic spinless particle.
Finite potential barrier
One of most prominent features of the wave mechanics is a possibility for a particle to reach a location with a prohibitive (in classical mechanics)
force potential. A common model is the "
potential barrier", the one-dimensional case has the potential