background preloader

Quantum harmonic oscillator

Quantum harmonic oscillator
Some trajectories of a harmonic oscillator according to Newton's laws of classical mechanics (A-B), and according to the Schrödinger equation of quantum mechanics (C-H). In A-B, the particle (represented as a ball attached to a spring) oscillates back and forth. In C-H, some solutions to the Schrödinger Equation are shown, where the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. One-dimensional harmonic oscillator[edit] Hamiltonian and energy eigenstates[edit] Wavefunction representations for the first eight bound eigenstates, n = 0 to 7. Corresponding probability densities. where m is the particle's mass, ω is the angular frequency of the oscillator, is the position operator, and is the momentum operator, given by One may write the time-independent Schrödinger equation, The functions Hn are the Hermite polynomials, and Related:  QUANTUM PHYSICS 2Physics

Schrödinger's cat Schrödinger's cat: a cat, a flask of poison, and a radioactive source are placed in a sealed box. If an internal monitor detects radioactivity (i.e. a single atom decaying), the flask is shattered, releasing the poison that kills the cat. The Copenhagen interpretation of quantum mechanics implies that after a while, the cat is simultaneously alive and dead. Yet, when one looks in the box, one sees the cat either alive or dead, not both alive and dead. Schrödinger's cat is a thought experiment, sometimes described as a paradox, devised by Austrian physicist Erwin Schrödinger in 1935.[1] It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics applied to everyday objects. Origin and motivation[edit] Real-size cat figure in the garden of Huttenstrasse 9, Zurich, where Erwin Schrödinger lived 1921 – 1926. The thought experiment[edit] Schrödinger wrote:[1][10] One can even set up quite ridiculous cases. Interpretations of the experiment[edit]

Finite potential well The finite potential well (also known as the finite square well) is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a box, but one which has finite potential walls. Unlike the infinite potential well, there is a probability associated with the particle being found outside the box. Particle in a 1-dimensional box[edit] For the 1-dimensional case on the x-axis, the time-independent Schrödinger equation can be written as: where is Planck's constant, is the mass of the particle, is the (complex valued) wavefunction that we want to find, is a function describing the potential energy at each point x, and is the energy, a real number, sometimes called eigenenergy. For the case of the particle in a 1-dimensional box of length L, the potential is zero inside the box, but rises abruptly to a value at x = -L/2 and x = L/2. Inside the box[edit] For the region inside the box V(x) = 0 and Equation 1 reduces to Letting the equation becomes .

Quantum reflection Quantum reflection is a physical phenomenon involving the reflection of a matter wave from an attractive potential. In classical mechanics, such a phenomenon is not possible; for instance when one magnet is pulled toward another, the observer does not expect one of the magnets to suddenly (i.e. before the magnets 'touch') turn around and retreat in the opposite direction. Definition[edit] Quantum reflection became an important branch of physics in the 21st century. In a workshop about quantum reflection,[1] the following definition of quantum reflection was suggested: Quantum reflection is a classically counterintuitive phenomenon whereby the motion of particles is reverted "against the force" acting on them. Observation of quantum reflection has become possible thanks to recent advances in trapping and cooling atoms. Reflection of slow atoms[edit] Single-dimensional approximation[edit] and ), such that only a single coordinate (say ) is important. axis, where is the atomic mass, Fig. . , where .

Mehler kernel In mathematics and physics, the Mehler kernel is the fundamental solution, or non-relativistic propagator of the Hamiltonian for the quantum harmonic oscillator. Mehler (1866) gave an explicit formula for it, called Mehler's formula. The Mehler kernel provides the fundamental solution for the most general solution φ(x, y; t) to Specifically, Mehler's kernel is By a simple transformation, this is, apart from a multiplying factor, the bivariate Gaussian probability density given by When t=0 variables x and y coincide resulting in the formula The bivariate probability density can be written as an infinite series involving the one-dimensional probability densities and Hermite polynomials of x and y.

Consciousness and the Prospects of Physicalism In recent decades a number of arguments have emerged designed to show that physicalist theories of mind cannot do justice to the nature of consciousness. These arguments have resulted in a huge literature with a dialectic that has led to increasingly interesting and sophisticated positions on both sides of the issue. In particular, some physicalists have lately been exploring new and out-of-the-mainstream ways of answering the anti-physicalist arguments. The book divides into three parts: in each of chapters 1-4 and 5-6 Pereboom considers a possible physicalist response to the anti-physicalist arguments. The anti-physicalist arguments all take as their point of departure a first-person or introspective perspective. Pereboom may seem to be proposing that introspective representations are just flat out illusory, but it's a little more complicated than that. The point of all this, of course, is to bring the introspective representations in line with physicalism.

Particle in a box In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example a ball trapped inside a large box, the particle can move at any speed within the box and it is no more likely to be found at one position than another. The particle in a box model provides one of the very few problems in quantum mechanics which can be solved analytically, without approximations. One-dimensional solution[edit] The barriers outside a one-dimensional box have infinitely large potential, while the interior of the box has a constant, zero potential. The simplest form of the particle in a box model considers a one-dimensional system. where is the length of the box and is the position of the particle within the box. and . .

Fresnel diffraction In optics, the Fresnel diffraction equation for near-field diffraction, is an approximation of Kirchhoff-Fresnel diffraction that can be applied to the propagation of waves in the near field.[1] It is used to calculate the diffraction pattern created by waves passing through an aperture or around an object, when viewed from relatively close to the object. In contrast the diffraction pattern in the far field region is given by the Fraunhofer diffraction equation. The near field can be specified by the Fresnel number, F of the optical arrangement. When the diffracted wave is considered to be in the near field. However, the validity of the Fresnel diffraction integral is deduced by the approximations derived below. where is the maximal angle described by , a and L the same as in the definition of the Fresnel number. The multiple Fresnel diffraction at nearly placed periodical ridges (ridged mirror) causes the specular reflection; this effect can be used for atomic mirrors.[2] is the aperture, .

Mock modular form In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. History[edit] "Suppose there is a function in the Eulerian form and suppose that all or an infinity of points are exponential singularities, and also suppose that at these points the asymptotic form closes as neatly as in the cases of (A) and (B). Ramanujan's original definition of a mock theta function, from (Ramanujan 2000, Appendix II) Ramanujan's 12 January 1920 letter to Hardy, reprinted in (Ramanujan 2000, Appendix II), listed 17 examples of functions that he called mock theta functions, and his lost notebook (Ramanujan 1988) contained several more examples. Ramanujan associated an order to his mock theta functions, which was not clearly defined. Definition[edit] Fix a weight k, usually with 2k integral.

Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (/ˈʃroʊdɪŋər/; German: [ˈɛʁviːn ˈʃʁøːdɪŋɐ]; 12 August 1887 – 4 January 1961), a Nobel Prize-winning Austrian physicist who developed a number of fundamental results in the field of quantum theory, which formed the basis of wave mechanics: he formulated the wave equation (stationary and time-dependent Schrödinger equation) and revealed the identity of his development of the formalism and matrix mechanics. Schrödinger proposed an original interpretation of the physical meaning of the wave function and in subsequent years repeatedly criticized the conventional Copenhagen interpretation of quantum mechanics (using e.g. the paradox of Schrödinger's cat). In addition, he was the author of many works in various fields of physics: statistical mechanics and thermodynamics, physics of dielectrics, color theory, electrodynamics, general relativity, and cosmology, and he made several attempts to construct a unified field theory. In his book What Is Life?

Density matrix Explicitly, suppose a quantum system may be found in state with probability p1, or it may be found in state with probability p2, or it may be found in state with probability p3, and so on. By choosing a basis (which need not be orthogonal), one may resolve the density operator into the density matrix, whose elements are[1] For an operator (which describes an observable is given by[1] In words, the expectation value of A for the mixed state is the sum of the expectation values of A for each of the pure states Mixed states arise in situations where the experimenter does not know which particular states are being manipulated. Pure and mixed states[edit] In quantum mechanics, a quantum system is represented by a state vector (or ket) . is called a pure state. and a 50% chance that the state vector is . A mixed state is different from a quantum superposition. Example: Light polarization[edit] The incandescent light bulb (1) emits completely random polarized photons (2) with mixed state density matrix .

Doughnut theory of the universe Bloom Toroidal Model of the Universe The doughnut theory of the universe is an informal description of the theory that the shape of the universe is a three-dimensional torus. The name comes from the shape of a doughnut, whose surface has the topology of a two-dimensional torus. The foundation for the doughnut theory started with Bell Lab’s discovery of cosmic microwave background (CMB). Supporting evidence[edit] Dr. Cosmic Background Explorer (COBE)[edit] The Cosmic Background Explorer was an explorer satellite launched in 1989 by NASA that used a Far Infrared Absolute Spectrometer (FIRAS) to measure the radiation of the universe.[2] Led by researchers John C. Wilkinson Microwave Anisotropy Probe (WMAP)[edit] WMAP cosmic microwave background map The Wilkinson Microwave Anisotropy Probe (WMAP) was launched in 2001 as NASA’s second explorer satellite intended to map the precise distribution of CMB across the universe. Notes and references[edit] Notes References External links[edit]

Harmonic series (mathematics) In mathematics, the harmonic series is the divergent infinite series: The fact that the harmonic series diverges was first proven in the 14th century by Nicole Oresme,[1] but this achievement fell into obscurity. Proofs were given in the 17th century by Pietro Mengoli,[2] Johann Bernoulli,[3] and Jacob Bernoulli.[4] The harmonic series is counterintuitive to students first encountering it, because it is a divergent series though the limit of the nth term as n goes to infinity is zero. Because the series gets arbitrarily large as n becomes larger, eventually this ratio must exceed 1, which implies that the worm reaches the end of the rubber band. Another example is: given a collection of identical dominoes, it is clearly possible to stack them at the edge of a table so that they hang over the edge of the table without falling. There are several well-known proofs of the divergence of the harmonic series. for every positive integer k. The harmonic series diverges very slowly. where The series

Qualia In philosophy, qualia (/ˈkwɑːliə/ or /ˈkweɪliə/; singular form: quale) are what some consider to be individual instances of subjective, conscious experience. The term "qualia" derives from the Latin neuter plural form (qualia) of the Latin adjective quālis (Latin pronunciation: [ˈkʷaːlɪs]) meaning "of what sort" or "of what kind"). Examples of qualia include the pain of a headache, the taste of wine, or the perceived redness of an evening sky. As qualitative characters of sensation, qualia stand in contrast to "propositional attitudes".[1] Daniel Dennett (b. 1942), American philosopher and cognitive scientist, regards qualia as "an unfamiliar term for something that could not be more familiar to each of us: the ways things seem to us".[2] Erwin Schrödinger (1887–1961), the famous physicist, had this counter-materialist take: The sensation of color cannot be accounted for by the physicist's objective picture of light-waves. Definitions[edit] Arguments for the existence of qualia[edit] E. J.