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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. C,D,E,F, but not G,H, are energy eigenstates. H is a coherent state, a quantum state which approximates the classical trajectory. 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 and Proof:

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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. The quantum mechanical interpretation is unlike the classical interpretation, where if the total energy of the particle is less than potential energy barrier of the walls it cannot be found outside the box.

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] 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. This poses the question of when exactly quantum superposition ends and reality collapses into one possibility or the other.

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. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important.

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.

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. Derk Pereboom's new book is the latest contribution to this literature, and it is much to be recommended to anyone who wants to keep abreast on these matters.

Solution of Schrödinger equation for a step potential In quantum mechanics and scattering theory, the one-dimensional step potential is an idealized system used to model incident, reflected and transmitted matter waves. The problem consists of solving the time-independent Schrödinger equation for a particle with a step-like potential in one dimension. Typically, the potential is modelled as a Heaviside step function. Calculation[edit] Schrödinger equation and potential function[edit] 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).

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?

Macroscopic quantum phenomena Quantum mechanics is most often used to describe matter on the scale of molecules, atoms, or elementary particles. However some phenomena, particularly at low temperatures, show quantum behavior on a macroscopic scale. The best-known examples of macroscopic quantum phenomena are superfluidity and superconductivity; another example is the quantum Hall effect. Since 2000 there has been extensive experimental work on quantum gases, particularly Bose–Einstein Condensates.

Birefringence A calcite crystal laid upon a graph paper with blue lines showing the double refraction Doubly refracted image as seen through a calcite crystal, seen through a rotating polarizing filter illustrating the opposite polarization states of the two images. Explanation[edit] The simplest (and most common) type of birefringence is that of materials with uniaxial anisotropy. 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]

DNA molecules can 'teleport', Nobel Prize winner claims A Nobel Prize winning biologist has ignited controversy after publishing details of an experiment in which a fragment of DNA appeared to ‘teleport’ or imprint itself between test tubes. According to a team headed by Luc Montagnier, previously known for his work on HIV and AIDS, two test tubes, one of which contained a tiny piece of bacterial DNA, the other pure water, were surrounded by a weak electromagnetic field of 7Hz. Eighteen hours later, after DNA amplification using a polymerase chain reaction, as if by magic the DNA was detectable in the test tube containing pure water. Oddly, the original DNA sample had to be diluted many times over for the experiment to work, which might explain why the phenomenon has not been detected before, assuming that this is what has happened.