In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotations and translations in space and time form the inhomogeneous Galilean group. Without the translations in space and time the group is the homogeneous Galilean group; the Galilean group is the group of motions of Galilean relativity acting on the four dimensions of space and time, forming the Galilean geometry. This is the passive transformation point of view; the equations below, although obvious, are valid only at speeds much less than the speed of light. In special relativity the Galilean transformations are replaced by Poincaré transformations. Galileo formulated these concepts in his description of uniform motion; the topic was motivated by his description of the motion of a ball rolling down a ramp, by which he measured the numerical value for the acceleration of gravity near the surface of the Earth.
Though the transformations are named for Galileo, it is absolute time and space as conceived by Isaac Newton that provides their domain of definition. In essence, the Galilean transformations embody the intuitive notion of addition and subtraction of velocities as vectors; this assumption is abandoned in the Poincaré transformations. These relativistic transformations are applicable to all velocities, while the Galilean transformation can be regarded as a low-velocity approximation to the Poincaré transformation; the notation below describes the relationship under the Galilean transformation between the coordinates and of a single arbitrary event, as measured in two coordinate systems S and S', in uniform relative motion in their common x and x′ directions, with their spatial origins coinciding at time t = t′ = 0: x ′ = x − v t y ′ = y z ′ = z t ′ = t. Note that the last equation expresses the assumption of a universal time independent of the relative motion of different observers. In the language of linear algebra, this transformation is considered a shear mapping, is described with a matrix acting on a vector.
With motion parallel to the x-axis, the transformation acts on only two components: = Though matrix representations are not necessary for Galilean transformation, they provide the means for direct comparison to transformation methods in special relativity. The Galilean symmetries can be uniquely written as the composition of a rotation, a translation and a uniform motion of spacetime. Let x represent a point in three-dimensional space, t a point in one-dimensional time. A general point in spacetime is given by an ordered pair. A uniform motion, with velocity v, is given by ↦, where v ∈ ℝ3. A translation is given by ↦, where a ∈ ℝ3 and s ∈ ℝ. A rotation is given by ↦; as a Lie group, the Galilean transformations span 10 dimensions, i.e. comprise 10 generators. Two Galilean transformations G compose to form a third Galilean transformation, G G = G; the set of all Galilean transformations Gal on space forms a group with composition as the group operation. The group is sometimes represented as a matrix group with spacetime events as vectors where t is real and x ∈ ℝ3 is a position in space.
The action is given by =, where s is real and v, x, a ∈ ℝ3 and R is a rotation matrix. The composition of transformations is accomplished through matrix multiplication. Gal has named sub
Quantum field theory
In theoretical physics, quantum field theory is a theoretical framework that combines classical field theory, special relativity, quantum mechanics and is used to construct physical models of subatomic particles and quasiparticles. QFT treats particles as excited states of their underlying fields, which are—in a sense—more fundamental than the basic particles. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding fields; each interaction can be visually represented by Feynman diagrams, which are formal computational tools, in the process of relativistic perturbation theory. As a successful theoretical framework today, quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century, its development began in the 1920s with the description of interactions between light and electrons, culminating in the first quantum field theory — quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure.
A second major barrier came with QFT's apparent inability to describe the weak and strong interactions, to the point where some theorists called for the abandonment of the field theoretic approach. The development of gauge theory and the completion of the Standard Model in the 1970s led to a renaissance of quantum field theory. Quantum field theory is the result of the combination of classical field theory, quantum mechanics, special relativity. A brief overview of these theoretical precursors is in order; the earliest successful classical field theory is one that emerged from Newton's law of universal gravitation, despite the complete absence of the concept of fields from his 1687 treatise Philosophiæ Naturalis Principia Mathematica. The force of gravity as described by Newton is an "action at a distance" — its effects on faraway objects are instantaneous, no matter the distance. In an exchange of letters with Richard Bentley, Newton stated that "it is inconceivable that inanimate brute matter should, without the mediation of something else, not material, operate upon and affect other matter without mutual contact."
It was not until the 18th century that mathematical physicists discovered a convenient description of gravity based on fields — a numerical quantity assigned to every point in space indicating the action of gravity on any particle at that point. However, this was considered a mathematical trick. Fields began to take on an existence of their own with the development of electromagnetism in the 19th century. Michael Faraday coined the English term "field" in 1845, he introduced fields as properties of space having physical effects. He argued against "action at a distance", proposed that interactions between objects occur via space-filling "lines of force"; this description of fields remains to this day. The theory of classical electromagnetism was completed in 1862 with Maxwell's equations, which described the relationship between the electric field, the magnetic field, electric current, electric charge. Maxwell's equations implied the existence of electromagnetic waves, a phenomenon whereby electric and magnetic fields propagate from one spatial point to another at a finite speed, which turns out to be the speed of light.
Action-at-a-distance was thus conclusively refuted. Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in atomic spectra, nor for the distribution of blackbody radiation in different wavelengths. Max Planck's study of blackbody radiation marked the beginning of quantum mechanics, he treated atoms, which absorb and emit electromagnetic radiation, as tiny oscillators with the crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as quantum harmonic oscillators; this process of restricting energies to discrete values is called quantization. Building on this idea, Albert Einstein proposed in 1905 an explanation for the photoelectric effect, that light is composed of individual packets of energy called photons; this implied that the electromagnetic radiation, while being waves in the classical electromagnetic field exists in the form of particles. In 1913, Niels Bohr introduced the Bohr model of atomic structure, wherein electrons within atoms can only take on a series of discrete, rather than continuous, energies.
This is another example of quantization. The Bohr model explained the discrete nature of atomic spectral lines. In 1924, Louis de Broglie proposed the hypothesis of wave-particle duality, that microscopic particles exhibit both wave-like and particle-like properties under different circumstances. Uniting these scattered ideas, a coherent discipline, quantum mechanics, was formulated between 1925 and 1926, with important contributions from de Broglie, Werner Heisenberg, Max Born, Erwin Schrödinger, Paul Dirac, Wolfgang Pauli.:22-23In the same year as his paper on the photoelectric effect, Einstein published his theory of special relativity, built on Maxwell's electromagnetism. New rules, called Lorentz transformation, were given for the way time and space coordinates of an event change under changes in the observer's velocity, the distinction between time and space was blurred.:19 It was proposed that all physical laws must be the same for observers at different velocities, i.e. that physical laws be invariant under Lorentz transformations.
Two difficulties remained. Observationally, the Schrödinger equation underlying q
In mathematics and physics, a scalar field associates a scalar value to every point in a space – physical space. The scalar may either be a physical quantity. In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, spin-zero quantum fields, such as the Higgs field; these fields are the subject of scalar field theory. Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U; the region U may be a set in some Euclidean space, Minkowski space, or more a subset of a manifold, it is typical in mathematics to impose further conditions on the field, such that it be continuous or continuously differentiable to some order.
A scalar field is a tensor field of order zero, the term "scalar field" may be used to distinguish a function of this kind with a more general tensor field, density, or differential form. Physically, a scalar field is additionally distinguished by having units of measurement associated with it. In this context, a scalar field should be independent of the coordinate system used to describe the physical system—that is, any two observers using the same units must agree on the numerical value of a scalar field at any given point of physical space. Scalar fields are contrasted with other physical quantities such as vector fields, which associate a vector to every point of a region, as well as tensor fields and spinor fields. More subtly, scalar fields are contrasted with pseudoscalar fields. In physics, scalar fields describe the potential energy associated with a particular force; the force is a vector field, which can be obtained as the gradient of the potential energy scalar field. Examples include: Potential fields, such as the Newtonian gravitational potential, or the electric potential in electrostatics, are scalar fields which describe the more familiar forces.
A temperature, humidity or pressure field, such as those used in meteorology. In quantum field theory, a scalar field is associated with spin-0 particles; the scalar field may be complex valued. Complex scalar fields represent charged particles; these include the charged Higgs field of the Standard Model, as well as the charged pions mediating the strong nuclear interaction. In the Standard Model of elementary particles, a scalar Higgs field is used to give the leptons and massive vector bosons their mass, via a combination of the Yukawa interaction and the spontaneous symmetry breaking; this mechanism is known as the Higgs mechanism. A candidate for the Higgs boson was first detected at CERN in 2012. In scalar theories of gravitation scalar fields are used to describe the gravitational field. Scalar-tensor theories represent the gravitational interaction through a scalar; such attempts are for example the Jordan theory as a generalization of the Kaluza–Klein theory and the Brans–Dicke theory. Scalar fields like the Higgs field can be found within scalar-tensor theories, using as scalar field the Higgs field of the Standard Model.
This field interacts Yukawa-like with the particles that get mass through it. Scalar fields are found within superstring theories as dilaton fields, breaking the conformal symmetry of the string, though balancing the quantum anomalies of this tensor. Scalar fields are supposed to cause the accelerated expansion of the universe, helping to solve the horizon problem and giving a hypothetical reason for the non-vanishing cosmological constant of cosmology. Massless scalar fields in this context are known as inflatons. Massive scalar fields are proposed, using for example Higgs-like fields. Vector fields; some examples of vector fields include the electromagnetic field and the Newtonian gravitational field. Tensor fields, which associate a tensor to every point in space. For example, in general relativity gravitation is associated with the tensor field called Einstein tensor. In Kaluza–Klein theory, spacetime is extended to five dimensions and its Riemann curvature tensor can be separated out into ordinary four-dimensional gravitation plus an extra set, equivalent to Maxwell's equations for the electromagnetic field, plus an extra scalar field known as the "dilaton".
The dilaton scalar is found among the massless bosonic fields in string theory. Scalar field theory Vector-valued function
Quantum harmonic oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known; the Hamiltonian of the particle is: H ^ = p ^ 2 2 m + 1 2 k x ^ 2 = p ^ 2 2 m + 1 2 m ω 2 x ^ 2, where m is the particle's mass, k is the force constant, ω = k m is the angular frequency of the oscillator, x ^ is the position operator, p ^ is the momentum operator. The first term in the Hamiltonian represents the kinetic energy of the particle, the second term represents its potential energy, as in Hooke's law. One may write the time-independent Schrödinger equation, H ^ | ψ ⟩ = E | ψ ⟩, where E denotes a to-be-determined real number that will specify a time-independent energy level, or eigenvalue, the solution |ψ⟩ denotes that level's energy eigenstate.
One may solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave function ⟨x|ψ⟩ = ψ, using a spectral method. It turns out. In this basis, they amount to Hermite functions, ψ n = 1 2 n n! ⋅ 1 / 4 ⋅ e − m ω x 2 2 ℏ ⋅ H n, n = 0, 1, 2, …. The functions Hn are the physicists' Hermite polynomials, H n = n e z 2 d n d z n; the corresponding energy levels are E n = ℏ ω = ℏ 2 ω. This energy spectrum is noteworthy for three reasons. First, the energies are quantized. Second, these discrete energy levels are spaced, unlike in the Bohr model of the atom, or the particle in a box. Third, the lowest achievable energy is not equal to the minimum of the potential well, but ħω/2 above it; because of the zero-point energy, the position and momentum of the oscillator in the ground state are not fixed, but have a small range of variance, in accordance with the Heisenberg uncertainty principle. The ground state probability density is concentrated at the origin, which means the particle spends most of its time at the bottom of the potential well, as one would expect for a state with little energy.
As the energy increases, the probability density peaks at the classical "turning points", where the state's energy coincides with the potential energy. This is consistent with the classical harmonic oscillator, in which the particle spends more of its time near the turning points, where it is moving the slowest; the correspondence principle is thus satisfied. Moreover, special nondispersive wave