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Particle physics and representation theory. The connection between particle physics and representation theory is a natural connection, first noted in the 1930s by Eugene Wigner,[1] between the properties of elementary particles and the structure of Lie groups and Lie algebras. According to this connection, the different quantum states of an elementary particle give rise to an irreducible representation of the Poincaré group. Moreover, the properties of the various particles, including their spectra, can be related to representations of Lie algebras, corresponding to "approximate symmetries" of the universe. General picture[edit] In quantum mechanics, any particular particle (with a given momentum distribution, location distribution, spin state, etc.) is written as a vector (or "ket") in a Hilbert space H.

To help understand what types of particles can exist, it is important to classify the possibilities for H, and their properties. . Poincaré group[edit] Other symmetries[edit] Approximate symmetries[edit] Hypothetical example[edit] Wave function. However, complex numbers are not necessarily used in all treatments.

Louis de Broglie in his later years proposed a real-valued wave function connected to the complex wave function by a proportionality constant and developed the de Broglie–Bohm theory. The unit of measurement for ψ depends on the system. For one particle in three dimensions, its units are [length]−3/2. These unusual units are required so that an integral of |ψ|2 over a region of three-dimensional space is a unitless probability (the probability that the particle is in that region). For different numbers of particles and/or dimensions, the units may be different and can be found by dimensional analysis.[1] Historical background[edit] In the 1920s and 1930s, quantum mechanics was developed using calculus and linear algebra.

Wave functions and function spaces[edit] If the wave function is to change throughout space and time, one would expect the wave function to be a function of the position and time coordinates. Quantum tunnelling. Quantum mechanical phenomenon In physics, quantum tunnelling, barrier penetration, or simply tunnelling is a quantum mechanical phenomenon in which an object such as an electron or atom passes through a potential energy barrier that, according to classical mechanics, should not be passable due to the object not having sufficient energy to pass or surmount the barrier. Tunneling is a consequence of the wave nature of matter, where the quantum wave function describes the state of a particle or other physical system, and wave equations such as the Schrödinger equation describe their behavior.

The probability of transmission of a wave packet through a barrier decreases exponentially with the barrier height, the barrier width, and the tunneling particle's mass, so tunneling is seen most prominently in low-mass particles such as electrons or protons tunneling through microscopically narrow barriers. The effect was predicted in the early 20th century. Introduction to the concept[edit] or where . Quantum superposition. Quantum superposition is a fundamental principle of quantum mechanics that holds that a physical system—such as an electron—exists partly in all its particular theoretically possible states (or, configuration of its properties) simultaneously; but when measured or observed, it gives a result corresponding to only one of the possible configurations (as described in interpretation of quantum mechanics). and . Here is the Dirac notation for the quantum state that will always give the result 0 when converted to classical logic by a measurement.

Likewise is the state that will always convert to 1. Concept[edit] The principle of quantum superposition states that if a physical system may be in one of many configurations—arrangements of particles or fields—then the most general state is a combination of all of these possibilities, where the amount in each configuration is specified by a complex number. For example, if there are two configurations labelled by 0 and 1, the most general state would be . Quantum nonlocality. Quantum nonlocality is the phenomenon by which the measurements made at a microscopic level necessarily refute one or more notions (often referred to as local realism) that are regarded as intuitively true in classical mechanics. Rigorously, quantum nonlocality refers to quantum mechanical predictions of many-system measurement correlations that cannot be simulated by any local hidden variable theory.

Many entangled quantum states produce such correlations when measured, as demonstrated by Bell's theorem. Experiments have generally favoured quantum mechanics as a description of nature, over local hidden variable theories.[1][2] Any physical theory that supersedes or replaces quantum theory must make similar experimental predictions and must therefore also be nonlocal in this sense; quantum nonlocality is a property of the universe that is independent of our description of nature. Example[edit] Imagine two experimentalists, Alice and Bob, situated in separate laboratories. And P(b0|A1) = or. Complementarity (physics) In physics, complementarity is a fundamental principle of quantum mechanics, closely associated with the Copenhagen interpretation. It holds that objects governed by quantum mechanics, when measured, give results that depend inherently upon the type of measuring device used, and must necessarily be described in classical mechanical terms.

Further, a full description of a particular type of phenomenon can only be achieved through measurements made in each of the various possible bases — which are thus complementary. The complementarity principle was formulated by Niels Bohr, the developer of the Bohr model of the atom, and a leading founder of quantum mechanics.[1] Bohr summarized the principle as follows: ...however far the [quantum physical] phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. For example, the particle and wave aspects of physical objects are such complementary phenomena. Physicists F.A.M.

Dr. Quantum state. Quantum decoherence. Decoherence can be viewed as the loss of information from a system into the environment (often modeled as a heat bath),[2] since every system is loosely coupled with the energetic state of its surroundings. Viewed in isolation, the system's dynamics are non-unitary (although the combined system plus environment evolves in a unitary fashion).[3] Thus the dynamics of the system alone are irreversible.

As with any coupling, entanglements are generated between the system and environment. These have the effect of sharing quantum information with—or transferring it to—the surroundings. Decoherence does not generate actual wave function collapse. It only provides an explanation for the observance of wave function collapse, as the quantum nature of the system "leaks" into the environment. That is, components of the wavefunction are decoupled from a coherent system, and acquire phases from their immediate surroundings. Mechanisms[edit] Phase space picture[edit] . Dirac notation[edit] , where . Where.