Eigenvalues and eigenvectors Concepts from linear algebra In linear algebra, it is often important to know which vectors have their directions unchanged by a linear transformation. An eigenvector ( EYE-gən-) or characteristic vector is such a vector. Thus an eigenvector of a linear transformation is scaled by a constant factor when the linear transformation is applied to it: . The eigenvectors and eigenvalues of a transformation serve to characterize it, and so they play important roles in all the areas where linear algebra is applied, from geology to quantum mechanics. Definition[edit] where λ is a scalar in F, known as the eigenvalue, characteristic value, or characteristic root associated with v. If V is finite-dimensional, the above equation is equivalent to where A is the matrix representation of T and u is the coordinate vector of v. Overview[edit] In essence, an eigenvector v of a linear transformation T is a nonzero vector that, when T is applied to it, does not change direction. History[edit] In this case, or . . . .
Wave function Mathematical description of the quantum state of a system Comparison of classical and quantum harmonic oscillator conceptions for a single spinless particle. The two processes differ greatly. 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. Historical background[edit] In 1905, Albert Einstein postulated the proportionality between the frequency of a photon and its energy ,[11] and in 1916 the corresponding relation between a photon's momentum and wavelength ,[12] where is the Planck constant. , 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. In the 1920s and 1930s, quantum mechanics was developed using calculus and linear algebra. Wave functions and wave equations in modern theories[edit] thus and one obtains
Five-dimensional space Geometric space with five dimensions Physics[edit] Much of the early work on five-dimensional space was in an attempt to develop a theory that unifies the four fundamental interactions in nature: strong and weak nuclear forces, gravity, and electromagnetism. To explain why this dimension would not be directly observable, Klein suggested that the fifth dimension would be rolled up into a tiny, compact loop on the order of 10-33 centimeters.[1] Under his reasoning, he envisioned light as a disturbance caused by rippling in the higher dimension just beyond human perception, similar to how fish in a pond can only see shadows of ripples across the surface of the water caused by raindrops.[2] While not detectable, it would indirectly imply a connection between seemingly unrelated forces. The main novelty of Einstein and Bergmann was to seriously consider the fifth dimension as a physical entity, rather than an excuse to combine the metric tensor and electromagnetic potential. Polytopes[edit] .
Oskar Klein Oskar Benjamin Klein (Swedish: [klajn]; 15 September 1894 – 5 February 1977) was a Swedish theoretical physicist.[1] Biography[edit] Klein was born in Danderyd outside Stockholm, son of the chief rabbi of Stockholm, Gottlieb Klein from Humenné in Slovakia and Antonie (Toni) Levy. He became a student of Svante Arrhenius at the Nobel Institute at a young age and was on the way to Jean-Baptiste Perrin in France when World War I broke out and he was drafted into the military. From 1917, he worked a few years with Niels Bohr in the University of Copenhagen and received his doctoral degree at the University College of Stockholm (now Stockholm University) in 1921. Klein is credited for inventing the idea, part of Kaluza–Klein theory, that extra dimensions may be physically real but curled up and very small, an idea essential to string theory / M-theory. The Oskar Klein Memorial Lecture, held annually at the University of Stockholm, has been named after him. References[edit] External links[edit]
BKS theory The Bohr–Kramers–Slater theory (BKS theory) was perhaps the final attempt at understanding the interaction of matter and electromagnetic radiation on the basis of the so-called old quantum theory, in which quantum phenomena are treated by imposing quantum restrictions on classically describable behaviour.[1][2][3][4] It was advanced in 1924, and sticks to a classical wave description of the electromagnetic field. It was perhaps more a research program than a full physical theory, the ideas that are developed not being worked out in a quantitative way.[5]: 236 The purpose of BKS Theory was to disprove Einstein's hypothesis of the light quantum.[6] Origins[edit] When Einstein introduced the light quantum (photon) in 1905, there was much resistance from the scientific community. The initial idea of the BKS theory originated with John C. This fourth point reverts back to Max Planck's original view of his quantum introduction in 1900. Development with Bohr and Kramers[edit] References[edit]
Fermi–Dirac statistics In quantum statistics, a branch of physics, Fermi–Dirac statistics describes a distribution of particles in certain systems comprising many identical particles that obey the Pauli exclusion principle. It is named after Enrico Fermi and Paul Dirac, who each discovered it independently, although Enrico Fermi defined the statistics earlier than Paul Dirac.[1][2] History[edit] Before the introduction of Fermi–Dirac statistics in 1926, understanding some aspects of electron behavior was difficult due to seemingly contradictory phenomena. The difficulty encountered by the electronic theory of metals at that time was due to considering that electrons were (according to classical statistics theory) all equivalent. Fermi–Dirac distribution[edit] For a system of identical fermions, the average number of fermions in a single-particle state , is given by the Fermi–Dirac (F–D) distribution,[9] where k is Boltzmann's constant, T is the absolute temperature, is the energy of the single-particle state When .
Pauli exclusion principle Quantum mechanics rule: identical fermions cannot occupy the same quantum state simultaneously In the case of electrons in atoms, the exclusion principle can be stated as follows: in a poly-electron atom it is impossible for any two electrons to have the same two values for each pair, respectively, at all the four-member set of their quantum numbers, which are: n, the principal quantum number; ℓ, the azimuthal quantum number; mℓ, the magnetic quantum number; and ms, the spin quantum number. Thus, if two electrons reside in the same orbital, then the two values of the pairs, respectively, for the n, ℓ, and mℓ numbers will be the same. However, the two values of the ms (spin) pair must be different, so these two electrons will present opposite half-integer spin projections, namely 1/2 and −1/2. Particles with an integer spin (bosons) are not subject to the Pauli exclusion principle. Overview[edit] Half-integer spin means that the intrinsic angular momentum value of fermions is History[edit]
Fermion Type of subatomic particle In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin 1/2, spin 3/2, etc. These particles obey the Pauli exclusion principle. Fermions include all quarks and leptons and all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. As a consequence of the Pauli exclusion principle, only one fermion can occupy a particular quantum state at a given time. Composite fermions, such as protons and neutrons, are the key building blocks of everyday matter. English theoretical physicist Paul Dirac coined the name fermion from the surname of Italian physicist Enrico Fermi.[2] Elementary fermions[edit] The Standard Model recognizes two types of elementary fermions: quarks and leptons. Composite fermions[edit] See also[edit]