Wave function

Last updated
Comparison of classical and quantum harmonic oscillator conceptions for a single spinless particle. The two processes differ greatly. The classical process (A-B) is represented as the motion of a particle along a trajectory. The quantum process (C-H) has no such trajectory. Rather, it is represented as a wave; here, the vertical axis shows the real part (blue) and imaginary part (red) of the wave function. Panels (C-F) show four different standing-wave solutions of the Schrodinger equation. Panels (G-H) further show two different wave functions that are solutions of the Schrodinger equation but not standing waves. QuantumHarmonicOscillatorAnimation.gif
Comparison of classical and quantum harmonic oscillator conceptions for a single spinless particle. The two processes differ greatly. The classical process (A–B) is represented as the motion of a particle along a trajectory. The quantum process (C–H) has no such trajectory. Rather, it is represented as a wave; here, the vertical axis shows the real part (blue) and imaginary part (red) of the wave function. Panels (C–F) show four different standing-wave solutions of the Schrödinger equation. Panels (G–H) further show two different wave functions that are solutions of the Schrödinger equation but not standing waves.

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi, respectively).

In quantum physics, a quantum state is the state of an isolated quantum system. A quantum state provides a probability distribution for the value of each observable, i.e. for the outcome of each possible measurement on the system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior.

A quantum system is a portion of the whole Universe which is taken under consideration to make analysis or to study for quantum mechanics pertaining to the wave-particle duality in that system. Everything outside this system is studied only to observe its effects on the system. A quantum system involves the wave function and its constituents, such as the momentum and wavelength of the wave for which wave function is being defined.

Complex number Element of a number system in which –1 has a square root

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

Contents

The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state.

Function (mathematics) Mapping that associates a single output value to each input

In mathematics, a function is a relation between sets that associates to every element of a first set exactly one element of the second set. Typical examples are functions from integers to integers or from the real numbers to real numbers.

In physics, the degree of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration. It is the number of parameters that determine the state of a physical system and is important to the analysis of systems of bodies in mechanical engineering, aeronautical engineering, robotics, and structural engineering.

For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier transform. Some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as isospin. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for each possible value of the discrete degrees of freedom (e.g., z-component of spin) – these values are often displayed in a column matrix (e.g., a 2 × 1 column vector for a non-relativistic electron with spin 12).

Domain of a function mathematical concept

In mathematics, the domain of definition of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain. Conversely, the set of values the function takes on as output is termed the image of the function, which is sometimes also referred to as the range of the function.

Fourier transform mathematical transform that expresses a mathematical function of time as a function of frequency

The Fourier transform (FT) decomposes a function of time into its constituent frequencies. This is similar to the way a musical chord can be expressed in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform of a function of time is itself a complex-valued function of frequency, whose magnitude (modulus) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation.

Electron subatomic particle with negative electric charge

The electron is a subatomic particle, symbol
e
or
β
, whose electric charge is negative one elementary charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron has a mass that is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value, expressed in units of the reduced Planck constant, ħ. Being fermions, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: they can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer de Broglie wavelength for a given energy.

According to the superposition principle of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a Hilbert space. The inner product between two wave functions is a measure of the overlap between the corresponding physical states, and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality. However, the wave function in quantum mechanics describes a kind of physical phenomenon, still open to different interpretations, which fundamentally differs from that of classic mechanical waves. [1] [2] [3] [4] [5] [6] [7]

Superposition principle fundamental physics principle stating that physical solutions of linear systems are linear

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 that if input A produces response X and input B produces response Y then input produces response.

Hilbert space inner product space that is metrically complete; a Banach space whose norm induces an inner product (follows the parallelogram identity)

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.

The Born rule, formulated by German physicist Max Born in 1926, is a physical law of quantum mechanics giving the probability that a measurement on a quantum system will yield a given result. In its simplest form it states that the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's wavefunction at that point. The Born rule is one of the key principles of quantum mechanics.

In Born's statistical interpretation in non-relativistic quantum mechanics, [8] [9] [10] the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle's being detected at a given place – or having a given momentum – at a given time, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation. This general requirement 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 ψ and calculate the statistical distributions for measurable quantities.

Max Born physicist

Max Born was a German-Jewish 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 number of notable physicists in the 1920s and 1930s. Born won the 1954 Nobel Prize in Physics for his "fundamental research in quantum mechanics, especially in the statistical interpretation of the wave function".

Real number Number representing a continuous quantity

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as 2. Included within the irrationals are the transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.

Probability density function Function whose integral over a region describes the probability of an event occurring in that region

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.

Historical background

In 1905, Einstein postulated the proportionality between the frequency of a photon and its energy , , [11] and in 1916 the corresponding relation between photon's momentum and wavelength , [12] , 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, [13] 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.

Momentum conserved physical quantity related to the motion of a body

In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction in three-dimensional space. If m is an object's mass and v is the velocity, then the momentum is

Wavelength spatial period of the wave—the distance over which the waves shape repeats, and thus the inverse of the spatial frequency

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

Planck constant physical constant representing the quantum of action

The Planck constant, or Planck's constant, denoted is a physical constant that is the quantum of electromagnetic action, which relates the energy carried by a photon to its frequency. A photon's energy is equal to its frequency multiplied by the Planck constant. The Planck constant is of fundamental importance in quantum mechanics, and in metrology it is the basis for the definition of the kilogram.

In the 1920s and 1930s, quantum mechanics was developed using calculus and linear algebra. Those who used the techniques of calculus included Louis de Broglie, Erwin Schrödinger, and others, developing "wave mechanics". Those who applied the methods of linear algebra included Werner Heisenberg, Max Born, and others, developing "matrix mechanics". Schrödinger subsequently showed that the two approaches were equivalent. [14]

In 1926, Schrödinger published the famous wave equation now named after him, the Schrödinger equation. This equation was based on classical conservation of energy using quantum operators and the de Broglie relations, and the solutions of the equation are the wave functions for the quantum system. [15] However, no one was clear on how to interpret it. [16] 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. [17] 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. [8] 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. [8] [9] [18] 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. [19] The Slater determinant and permanent (of a matrix) was part of the method, provided by John C. Slater.

Schrödinger did encounter an equation for the wave function that satisfied relativistic energy conservation before he published the non-relativistic one, but discarded it as it predicted negative probabilities and negative energies. In 1927, Klein, Gordon and Fock also found it, but incorporated the electromagnetic interaction and proved that it was Lorentz invariant. De Broglie also arrived at the same equation in 1928. This relativistic wave equation is now most commonly known as the Klein–Gordon equation. [20]

In 1927, Pauli phenomenologically found a non-relativistic equation to describe spin-1/2 particles in electromagnetic fields, now called the Pauli equation. [21] 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: [19] two for the electron and two for the electron's antiparticle, the positron. In the non-relativistic limit, the Dirac wave function resembles the Pauli wave function for the electron. Later, other relativistic wave equations were found.

Wave functions and wave equations in modern theories

All these wave equations are of enduring importance. The Schrödinger equation and the Pauli equation are under many circumstances excellent approximations of the relativistic variants. They are considerably easier to solve in practical problems than the relativistic counterparts.

The Klein–Gordon equation and the Dirac equation, while being relativistic, do not represent full reconciliation of quantum mechanics and special relativity. The branch of quantum mechanics where these equations are studied the same way as the Schrödinger equation, often called relativistic quantum mechanics, while very successful, has its limitations (see e.g. Lamb shift) and conceptual problems (see e.g. Dirac sea).

Relativity makes it inevitable that the number of particles in a system is not constant. For full reconciliation, quantum field theory is needed. [22] In this theory, the wave equations and the wave functions have their place, but in a somewhat different guise. The main objects of interest are not the wave functions, but rather operators, so called field operators (or just fields where "operator" is understood) on the Hilbert space of states (to be described next section). It turns out that the original relativistic wave equations and their solutions are still needed to build the Hilbert space. Moreover, the free fields operators, i.e. when interactions are assumed not to exist, turn out to (formally) satisfy the same equation as do the fields (wave functions) in many cases.

Thus the Klein–Gordon equation (spin 0) and the Dirac equation (spin 12) in this guise remain in the theory. Higher spin analogues include the Proca equation (spin 1), Rarita–Schwinger equation (spin 32), and, more generally, the Bargmann–Wigner equations. For massless free fields two examples are the free field Maxwell equation (spin 1) and the free field Einstein equation (spin 2) for the field operators. [23] All of them are essentially a direct consequence of the requirement of Lorentz invariance. Their solutions must transform under Lorentz transformation in a prescribed way, i.e. under a particular representation of the Lorentz group and that together with few other reasonable demands, e.g. the cluster decomposition principle, [24] with implications for causality is enough to fix the equations.

It should be emphasized that this applies to free field equations; interactions are not included. If a Lagrangian density (including interactions) is available, then the Lagrangian formalism will yield an equation of motion at the classical level. This equation may be very complex and not amenable to solution. Any solution would refer to a fixed number of particles and would not account for the term "interaction" as referred to in these theories, which involves the creation and annihilation of particles and not external potentials as in ordinary "first quantized" quantum theory.

In string theory, the situation remains analogous. For instance, a wave function in momentum space has the role of Fourier expansion coefficient in a general state of a particle (string) with momentum that is not sharply defined. [25]

Definition (one spinless particle in one dimension)

Quantum mechanics travelling wavefunctions.svg
Travelling waves of a free particle.
The real parts of position wave function Ψ(x) and momentum wave function Φ(p), and corresponding probability densities |Ψ(x)|2 and |Φ(p)|2, for one spin-0 particle in one x or p dimension. The colour opacity of the particles corresponds to the probability density (not the wave function) of finding the particle at position x or momentum p.

For now, consider the simple case of a non-relativistic single particle, without spin, in one spatial dimension. More general cases are discussed below.

Position-space wave functions

The state of such a particle is completely described by its wave function,

where x is position and t is time. This is a complex-valued function of two real variables x and t.

For one spinless particle in 1d, if the wave function is interpreted as a probability amplitude, the square modulus of the wave function, the positive real number

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

where t 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 rather than an ordinary vector space. This vector space is infinite-dimensional, because there is no finite set of functions which can be added together in various combinations to create every possible function. Also, it is a Hilbert space, because the inner product of two wave functions Ψ1 and Ψ2 can be defined as the complex number (at time t) [nb 1]

More details are given below. Although the inner product of two wave functions is a complex number, the inner product of a wave function Ψ with itself,

is always a positive real number. The number ||Ψ|| (not ||Ψ||2) is called the norm of the wave function Ψ.

If (Ψ, Ψ) = 1, then Ψ is normalized. If Ψ is not normalized, then dividing by its norm gives the normalized function Ψ/||Ψ||. Two wave functions Ψ1 and Ψ2 are orthogonal if 1, Ψ2) = 0. If they are normalized and orthogonal, they are orthonormal. Orthogonality (hence also orthonormality) of wave functions is not a necessary condition wave functions must satisfy, but is instructive to consider since this guarantees linear independence of the functions. In a linear combination of orthogonal wave functions Ψn we have,

If the wave functions Ψn were nonorthogonal, the coefficients would be less simple to obtain.

In the Copenhagen interpretation, the modulus squared of the inner product (a complex number) gives a real number

which, assuming both wave functions are normalized, is interpreted as the probability of the wave function Ψ2 "collapsing" to the new wave function Ψ1 upon measurement of an observable, whose eigenvalues are the possible results of the measurement, with Ψ1 being an eigenvector of the resulting eigenvalue. This is the Born rule, [8] and is one of the fundamental postulates of quantum mechanics.

At a particular instant of time, all values of the wave function Ψ(x, t) are components of a vector. There are uncountably infinitely many of them and integration is used in place of summation. In Bra–ket notation, this vector is written

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:

The time parameter is often suppressed, and will be in the following. The x coordinate is a continuous index. The |x 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 p is the momentum in one dimension, which can be any value from −∞ to +∞, and t is time.

Analogous to the position case, the inner product of two wave functions Φ1(p, t) and Φ2(p, t) 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 p, since it is an eigenfunction of the momentum operator. These functions are not normalizable to unity (they aren't square-integrable), so they are not really elements of physical Hilbert space. The set

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,

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 x and p representations are

Now take the projection of the state Ψ onto eigenfunctions of momentum using the last expression in the two equations, [26]

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 transforms of each other. [27] 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, x and p enter symmetrically, so there it doesn't matter which description one uses. The same equation (modulo constants) results. From this follows, with a little bit of afterthought, a factoid: The solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in L2. [nb 2]

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 r is the position vector in three-dimensional space, and t is time. As always Ψ(r, t) 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, ignoring the position degrees of freedom, the wave function is a function of spin only (time is a parameter);

where sz is the spin projection quantum number along the z axis. (The z axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) The sz parameter, unlike r and t, is a discrete variable. For example, for a spin-1/2 particle, sz can only be +1/2 or −1/2, and not any other value. (In general, for spin s, sz can be s, s − 1, ... , −s + 1, −s). Inserting each quantum number gives a complex valued function of space and time, there are 2s + 1 of them. These can be arranged into a column vector [nb 3]

In bra–ket notation, these easily arrange into the components of a vector [nb 4]

The entire vector ξ is a solution of the Schrödinger equation (with a suitable Hamiltonian), which unfolds to a coupled system of 2s + 1 ordinary differential equations with solutions ξ(s, t), ξ(s − 1, t), ..., ξ(−s, t). The term "spin function" instead of "wave function" is used by some authors. This contrasts the solutions to position space wave functions, the position coordinates being continuous degrees of freedom, because then the Schrödinger equation does take the form of a wave equation.

More generally, for a particle in 3d with any spin, the wave function can be written in "position–spin space" as:

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 |ψ and spin state vector |ξ 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 [28] ). The time dependence can be placed in either factor, and time evolution of each can be studied separately. The factorization is not possible for those interactions where an external field or any space-dependent quantity couples to the spin; examples include a particle in a magnetic field, and spin–orbit coupling.

The preceding discussion is not limited to spin as a discrete variable, the total angular momentum J may also be used. [29] Other discrete degrees of freedom, like isospin, can expressed similarly to the case of spin above.

Many-particle states in 3d position space

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. Two particle wavefunction.svg
Traveling waves of two free particles, with two of three dimensions suppressed. Top is position-space wave function, bottom is momentum-space wave function, with corresponding probability densities.

If there are many particles, in general there is only one wave function, not a separate wave function for each particle. The fact that one wave function describes many particles is what makes quantum entanglement and the EPR paradox possible. The position-space wave function for N particles is written: [19]

where ri is the position of the ith particle in three-dimensional space, and t is time. Altogether, this is a complex-valued function of 3N + 1 real variables.

In quantum mechanics there is a fundamental distinction between identical particles and distinguishable particles. For example, any two electrons are identical and fundamentally indistinguishable from each other; the laws of physics make it impossible to "stamp an identification number" on a certain electron to keep track of it. [27] 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. [30] 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 Ndistinguishable particles (no two being identical, i.e. no two having the same set of quantum numbers), there is no requirement for the wave function to be either symmetric or antisymmetric.

For a collection of particles, some identical with coordinates r1, r2, ... and others distinguishable x1, x2, ... (not identical with each other, and not identical to the aforementioned identical particles), the wave function is symmetric or antisymmetric in the identical particle coordinates ri only:

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

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 N particles with spin in 3d,

this is altogether N three-dimensional volume integrals and N sums over the spins. The differential volume elements d3ri 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.

Probability interpretation

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

and the probability that particle 1 is in region R1 with spin sz1 = m1and particle 2 is in region R2 with spin sz2 = m2 etc. at time t 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 N particles, considering their positions only and suppressing other degrees of freedom,

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

The time dependence of the quantum state and the operators can be placed according to unitary transformations on the operators and states. For any quantum state |Ψ and operator O, in the Schrödinger picture |Ψ(t) changes with time according to the Schrödinger equation while O is constant. In the Heisenberg picture it is the other way round, |Ψ is constant while O(t) evolves with time according to the Heisenberg equation of motion. The Dirac (or interaction) picture is intermediate, time dependence is places in both operators and states which evolve according to equations of motion. It is useful primarily in computing S-matrix elements. [31]

Non-relativistic examples

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

Finite potential barrier

Scattering at a finite potential barrier of height V0. The amplitudes and direction of left and right moving waves are indicated. In red, those waves used for the derivation of the reflection and transmission amplitude. E > V0 for this illustration. Finitepot.png
Scattering at a finite potential barrier of height V0. The amplitudes and direction of left and right moving waves are indicated. In red, those waves used for the derivation of the reflection and transmission amplitude. E > V0 for this illustration.

One of most prominent features of the wave mechanics is a possibility for a particle to reach a location with a prohibitive (in classical mechanics) force potential. A common model is the "potential barrier", the one-dimensional case has the potential

and the steady-state solutions to the wave equation have the form (for some constants k, κ)

Note that these wave functions are not normalized; see scattering theory for discussion.

The standard interpretation of this is as a stream of particles being fired at the step from the left (the direction of negative x): setting Ar = 1 corresponds to firing particles singly; the terms containing Ar and Cr signify motion to the right, while Al and Cl – to the left. Under this beam interpretation, put Cl = 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 more s-type and p-type. However, in a triangular dot the wave functions are mixed due to confinement symmetry. (Click for animation) Quantum dot.png
3D confined electron wave functions in a quantum dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are more s-type and p-type. However, in a triangular dot the wave functions are mixed due to confinement symmetry. (Click for animation)

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

Quantum harmonic oscillator

The wave functions for the quantum harmonic oscillator can be expressed in terms of Hermite polynomials Hn, they are

where n = 0,1,2,....

The electron probability density for the first few hydrogen atom electron orbitals shown as cross-sections. These orbitals form an orthonormal basis for the wave function of the electron. Different orbitals are depicted with different scale. Hydrogen Density Plots.png
The electron probability density for the first few hydrogen atom electron orbitals shown as cross-sections. These orbitals form an orthonormal basis for the wave function of the electron. Different orbitals are depicted with different scale.

Hydrogen atom

The wave functions of an electron in a Hydrogen atom are expressed in terms of spherical harmonics and generalized Laguerre polynomials (these are defined differently by different authors—see main article on them and the hydrogen atom).

It is convenient to use spherical coordinates, and the wave function can be separated into functions of each coordinate, [32]

where R are radial functions and Ym
(θ, φ)
are spherical harmonics of degree and order m. 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: [33]

where a0 = 4πε0ħ2/mee2 is the Bohr radius, L2 + 1
n − 1
are the generalized Laguerre polynomials of degree n − 1, n = 1, 2, ... is the principal quantum number, = 0, 1, ... n − 1 the azimuthal quantum number, m = −, − + 1, ..., − 1, the magnetic 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.

Wave functions and function spaces

The concept of function spaces enters naturally in the discussion about wave functions. A function space is a set of functions, usually with some defining requirements on the functions (in the present case that they are square integrable), sometimes with an algebraic structure on the set (in the present case a vector space structure with an inner product), together with a topology 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 be closed. It will be concluded below that the function space of wave functions is a Hilbert space. This observation is the foundation of the predominant mathematical formulation of quantum mechanics.

Vector space structure

A wave function is an element of a function space partly characterized by the following concrete and abstract descriptions.

This similarity is of course not accidental. There are also a distinctions between the spaces to keep in mind.

Representations

Basic states are characterized by a set of quantum numbers. This is a set of eigenvalues of a maximal set of commuting observables. 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 of representation.

The abstract states are "abstract" only in that an arbitrary choice necessary for a particular explicit description of it is not given. This is the same as saying that no choice of maximal set of commuting observables has been given. This is analogous to a vector space without a specified basis. 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) and Ψ(p, Sy), both describing the same state.

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.

Inner product

There is an additional algebraic structure on the vector spaces of wave functions and the abstract state space.

where m, n are (sets of) indices (quantum numbers) labeling different solutions, the strictly positive function w is called a weight function, and δmn is the Kronecker delta. The integration is taken over all of the relevant space.

This motivates the introduction of an inner 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 the Bra–ket notation Ψ|Φ. It yields a complex number. With the inner product, the function space is an inner 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 the Born rule. It states that the probability p of finding upon measurement the state Φ given the system is in the state Ψ is

where Φ and Ψ are assumed normalized. Consider a scattering experiment. In quantum field theory, if Φout describes a state in the "distant future" (an "out state") after interactions between scattering particles have ceased, and Ψin an "in state" in the "distant past", then the quantities out, Ψin), with Φout and Ψin varying over a complete set of in states and out states respectively, is called the S-matrix or scattering matrix. Knowledge of it is, effectively, having solved the theory at hand, at least as far as predictions go. Measurable quantities such as decay rates and scattering cross sections are calculable from the S-matrix. [35]

Hilbert space

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 of completeness, that allows one to take limits of sequences in the function space, and be ensured that, if the limit exists, it is an element of the function space. A complete inner product space is called a Hilbert space. The property of completeness is crucial in advanced treatments and applications of quantum mechanics. For instance, the existence of projection operators or orthogonal projections relies on the completeness of the space. [36] These projection operators, in turn, are essential for the statement and proof of many useful theorems, e.g. the spectral 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 7] The space L2 is a Hilbert space, with inner product presented later. The function space of the example of the figure is a subspace of L2. A subspace of a Hilbert space is a Hilbert space if it is closed.

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, say L2. The most glaring example is the set of functions e2πip · xh. These are plane wave solutions of the Schrödinger equation for a free particle, but are not normalizable, hence not in L2. But they are nonetheless fundamental for the description. One can, using them, express functions that are normalizable using wave packets. They are, in a sense, a basis (but not a Hilbert space basis, nor a Hamel basis) in which wave functions of interest can be expressed. There is also the artifact "normalization to a delta function" that is frequently employed for notational convenience, see further down. The delta functions themselves aren't square integrable either.

The above description of the function space containing the wave functions is mostly mathematically motivated. The function spaces are, due to completeness, very large in a certain sense. Not all functions are realistic descriptions of any physical system. For instance, in the function space L2 one can find the function that takes on the value 0 for all rational numbers and -i for the irrationals in the interval [0, 1]. This is square integrable, [nb 8] but can hardly represent a physical state.

Common Hilbert spaces

While the space of solutions as a whole is a Hilbert space there are many other Hilbert spaces that commonly occur as ingredients.

More generally, one may consider a unified treatment of all second order polynomial solutions to the Sturm–Liouville equations in the setting of Hilbert space. These include the Legendre and Laguerre polynomials as well as Chebyshev polynomials, Jacobi polynomials and Hermite polynomials. All of these actually appear in physical problems, the latter ones in the harmonic oscillator, and what is otherwise a bewildering maze of properties of special functions becomes an organized body of facts. For this, see Byron & Fuller (1992 , Chapter 5).

There occurs also finite-dimensional Hilbert spaces. The space n is a Hilbert space of dimension n. The inner product is the standard inner product on these spaces. In it, the "spin part" of a single particle wave function resides.

With more particles, the situations is more complicated. One has to employ tensor products and use representation theory of the symmetry groups involved (the rotation group and the Lorentz 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. [37] See the Bethe–Salpeter equation.) Corresponding remarks apply to the concept of isospin, for which the symmetry group is SU(2). The models of the nuclear forces of the sixties (still useful today, see nuclear force) used the symmetry group SU(3). In this case, as well, the part of the wave functions corresponding to the inner symmetries reside in some n or subspaces of tensor products of such spaces.

Due to the infinite-dimensional nature of the system, the appropriate mathematical tools are objects of study in functional analysis.

Simplified description

Continuity of the wave function and its first spatial derivative (in the x direction, y and z coordinates not shown), at some time t. Wavefunction continuity space.svg
Continuity of the wave function and its first spatial derivative (in the x direction, y and z coordinates not shown), at some time t.

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: [38] [39]

It is possible to relax these conditions somewhat for special purposes. [nb 10] If these requirements are not met, it is not possible to interpret the wave function as a probability amplitude. [40]

This does not alter the structure of the Hilbert space that these particular wave functions inhabit, but it should be pointed out that the subspace of the square-integrable functions L2, which is a Hilbert space, satisfying the second requirement is not closed in L2, hence not a Hilbert space in itself. [nb 11] The functions that does not meet the requirements are still needed for both technical and practical reasons. [nb 12] [nb 13]

More on wave functions and abstract state space

As has been demonstrated, the set of all possible wave functions in some representation for a system constitute an in general infinite-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. [41] A quantum state |Ψ in any representation is generally expressed as a vector

where

These quantum numbers index the components of the state vector. More, all α are in an n-dimensional set A = A1 × A2 × ... An where each Ai is the set of allowed values for αi; all ω are in an m-dimensional "volume" Ω ⊆ ℝm where Ω = Ω1 × Ω2 × ... Ωm and each Ωi ⊆ ℝ is the set of allowed values for ωi, a subset of the real numbers . For generality n and m are not necessarily equal.

Example:

(a) For a single particle in 3d with spin s, neglecting other degrees of freedom, using Cartesian coordinates, we could take α = (sz) for the spin quantum number of the particle along the z direction, and ω = (x, y, z) for the particle's position coordinates. Here A = {−s, −s + 1, ..., s − 1, s} is the set of allowed spin quantum numbers and Ω = ℝ3 is the set of all possible particle positions throughout 3d position space.

(b) An alternative choice is α = (sy) for the spin quantum number along the y direction and ω = (px, py, pz) for the particle's momentum components. In this case A and Ω are the same as before.

The probability density of finding the system at time at state |α, ω is

The probability of finding system with α in some or all possible discrete-variable configurations, DA, and ω in some or all possible continuous-variable configurations, C ⊆ Ω, is the sum and integral over the density, [nb 14]

Since the sum of all probabilities must be 1, the normalization condition

must hold at all times during the evolution of the system.

The normalization condition requires ρ dmω to be dimensionless, by dimensional analysis Ψ must have the same units as (ω1ω2...ωm)−1/2.

Ontology

Whether the wave function really exists, and what it represents, are major questions in the interpretation of quantum mechanics. Many famous physicists of a previous generation puzzled over this problem, such as Schrödinger, Einstein and Bohr. Some advocate formulations or variants of the Copenhagen interpretation (e.g. Bohr, Wigner and von Neumann) while others, such as Wheeler or Jaynes, take the more classical approach [42] and regard the wave function as representing information in the mind of the observer, i.e. a measure of our knowledge of reality. Some, including Schrödinger, Bohm and Everett and others, argued that the wave function must have an objective, physical existence. Einstein thought that a complete description of physical reality should refer directly to physical space and time, as distinct from the wave function, which refers to an abstract mathematical space. [43]

See also

Remarks

  1. The functions are here assumed to be elements of L2, the space of square integrable functions. The elements of this space are more precisely equivalence classes of square integrable functions, two functions declared equivalent if they differ on a set of Lebesgue measure 0. This is necessary to obtain an inner product (that is, (Ψ, Ψ) = 0 ⇒ Ψ ≡ 0) as opposed to a semi-inner product. The integral is taken to be the Lebesque integral. This is essential for completeness of the space, thus yielding a complete inner product space = Hilbert space.
  2. The Fourier transform viewed as a unitary operator on the space L2 has eigenvalues ±1, ±i. The eigenvectors are "Hermite functions", i.e. Hermite polynomials multiplied by a Gaussian function. See Byron & Fuller (1992) for a description of the Fourier transform as a unitary transformation. For eigenvalues and eigenvalues, refer to Problem 27 Ch. 9.
  3. Column vectors can be motivated by the convenience of expressing the spin operator for a given spin as a matrix, for the z-component spin operator (divided by hbar to nondimensionalize)
    The eigenvectors of this matrix are the above column vectors, with eigenvalues being the corresponding spin quantum numbers.
  4. Each |sz is usually identified as a column vector
    but it is a common abuse of notation to write
    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.
  5. 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 is not a generator of any symmetry in nature. On the other hand, the total momentum is a generator of a symmetry in nature; the translational symmetry.
  6. 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 of wave packets or by enclosing the system in a "box". See further remarks below.
  7. In technical terms, this is formulated the following way. The inner product yields a norm. This norm, in turn, induces a metric. If this metric is complete, then the aforementioned limits will be in the function space. The inner product space is then called complete. A complete inner product space is a Hilbert 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.
  8. As is explained in a later footnote, the integral must be taken to be the Lebesgue integral, the Riemann integral is not sufficient.
  9. 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 the category of Hilbert spaces.
  10. One such relaxation is that the wave function must belong to the Sobolev space W1,2. It means that it is differentiable in the sense of distributions, and its gradient is square-integrable. This relaxation is necessary for potentials that are not functions but are distributions, such as the Dirac delta function.
  11. It is easy to visualize a sequence of functions meeting the requirement that converges to a discontinuous function. For this, modify an example given in Inner product space#Examples. This element though is an element of L2.
  12. For instance, in perturbation 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.
  13. 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.
  14. Here
    is a multiple sum.

Notes

  1. Born 1927 , pp. 354–357
  2. Heisenberg 1958 , p. 143
  3. Heisenberg, W. (1927/1985/2009). Heisenberg is translated by Camilleri 2009 , p. 71, (from Bohr 1985 , p. 142).
  4. Murdoch 1987 , p. 43
  5. de Broglie 1960 , p. 48
  6. Landau & Lifshitz , p. 6
  7. Newton 2002 , pp. 19–21
  8. 1 2 3 4 Born 1926a, translated in Wheeler & Zurek 1983 at pages 52–55.
  9. 1 2 Born 1926b, translated in Ludwig 1968 , pp. 206–225. Also here.
  10. Born, M. (1954).
  11. Einstein 1905 , pp. 132–148 (in German), Arons & Peppard 1965 , p. 367 (in English)
  12. Einstein 1916 , pp. 47–62, and a nearly identical version Einstein 1917 , pp. 121–128 translated in ter Haar 1967 , pp. 167–183.
  13. de Broglie 1923 , pp. 507–510,548,630
  14. Hanle 1977 , pp. 606–609
  15. Schrödinger 1926 , pp. 1049–1070
  16. Tipler, Mosca & Freeman 2008
  17. 1 2 3 Weinberg 2013
  18. Young & Freedman 2008 , p. 1333
  19. 1 2 3 Atkins 1974
  20. Martin & Shaw 2008
  21. Pauli 1927 , pp. 601–623.
  22. Weinberg (2002) takes the standpoint that quantum field theory appears the way it does because it is the only way to reconcile quantum mechanics with special relativity.
  23. Weinberg (2002) See especially chapter 5, where some of these results are derived.
  24. Weinberg 2002 Chapter 4.
  25. Zwiebach 2009
  26. Shankar 1994 , Ch. 1
  27. 1 2 Griffiths 2004
  28. Shankar 1994 , p. 378–379
  29. Landau & Lifshitz 1977
  30. Zettili 2009 , p. 463
  31. Weinberg 2002 Chapter 3, Scattering matrix.
  32. Physics for Scientists and Engineers – with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN   0-7167-8964-7
  33. David Griffiths (2008). Introduction to elementary particles. Wiley-VCH. pp. 162–. ISBN   978-3-527-40601-2 . Retrieved 27 June 2011.
  34. Weinberg 2002
  35. Weinberg 2002 , Chapter 3
  36. Conway 1990
  37. Greiner & Reinhardt 2008
  38. Eisberg & Resnick 1985
  39. Rae 2008
  40. Atkins 1974 , p. 258
  41. Dirac 1982
  42. Jaynes 2003
  43. Einstein 1998 , p. 682

Related Research Articles

In quantum mechanics, bra–ket notation is a common notation for quantum states i.e. vectors in a complex Hilbert space on which an algebra of observables act. More generally the notation uses the angle brackets and a vertical bar, for a ket like to denote a vector in an abstract vector space and a bra, like to denote a linear functional on , i.e. a co-vector, an element of the dual vector space . The natural pairing of a linear functional with a vector is then written as . On Hilbert spaces, the scalar product gives an (anti-linear) identification of a vector ket with a linear functional bra . Using this notation, the scalar product . For the vector space , kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and operators are interpreted using matrix multiplication. If has the standard hermitian inner product , under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the hermitian conjugate .

In quantum mechanics, a Hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system. It is usually denoted by , but also or to highlight its function as an operator. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces and operators on these spaces. Many of these structures are drawn from functional analysis, a research area within pure mathematics that was influenced in part by the needs of quantum mechanics. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.

Dirac equation Relativistic quantum mechanical wave equation

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.

Schrödinger equation Linear partial differential equation whose solution describes the quantum-mechanical system.

The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

A density matrix is a matrix that describes the statistical state of a system in quantum mechanics. The probability for any outcome of any well-defined measurement upon a system can be calculated from the density matrix for that system. The extreme points in the set of density matrices are the pure states, which can also be written as state vectors or wavefunctions. Density matrices that are not pure states are mixed states. Any mixed state can be represented as a convex combination of pure states, and so density matrices are helpful for dealing with statistical ensembles of different possible preparations of a quantum system, or situations where a precise preparation is not known, as in quantum statistical mechanics.

Quantum superposition Principle of quantum mechanics

Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in classical physics, any two quantum states can be added together ("superposed") and the result will be another valid quantum state; and conversely, that every quantum state can be represented as a sum of two or more other distinct states. Mathematically, it refers to a property of solutions to the Schrödinger equation; since the Schrödinger equation is linear, any linear combination of solutions will also be a solution.

The Klein–Gordon equation is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pions are unstable and also experience the strong interaction the practical utility is limited.

Probability amplitude complex number whose squared absolute value is a probability

In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability or probability density.

Canonical quantization Process of converting a classical physical theory into one compatible with quantum mechanics

In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible.

In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. In quantum mechanics, it means a region of uniform potential, usually set to zero in the region of interest since potential can be arbitrarily set to zero at any point in space.

Two-state quantum system Quantum system that can be measured as one of two values; sought for "quantum bits" in quantum computing

In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

In quantum mechanics, the momentum operator is the operator associated with the measurement of linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one dimension, the definition is:

In quantum mechanics, the probability current is a mathematical quantity describing the flow of probability in terms of probability per unit time per unit area. Specifically, if one describes the probability density as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. This is analogous to mass currents in hydrodynamics and electric currents in electromagnetism. It is a real vector, like electric current density. The concept of a probability current is a useful formalism in quantum mechanics.

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

The Koopman–von Neumann mechanics is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932.

References

Further reading