Quantum mechanics is a fundamental physical theory that replaces Newtonian mechanics and classical electromagnetism at the atomic and subatomic levels and is the underlying framework of many fields of physics and chemistry, including condensed matter physics, quantum chemistry, and particle physics. Along with general relativity, it is one of the pillars of modern physics. Quantum mechanics - Introduction. The term quantum (Latin, "how much") refers to the discrete units that the theory assign ...

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Quantum mechanics - Introduction Quantum mechanics - Description of the theory
Quantum mechanics - Quantum mechanical effects Quantum mechanics - Mathematical formulation Quantum mechanics - Interactions with other scientific theories
Quantum mechanics - Applications of quantum theory Quantum mechanics - Philosophical consequences Quantum mechanics - History
Quantum mechanics - Founding experiments
Quantum mechanics - Notes

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Quantum mechanics, Quantum mechanics - Introduction, Quantum mechanics - Description of the theory, Quantum mechanics - Quantum mechanical effects, Quantum mechanics - Mathematical formulation, Quantum mechanics - Interactions with other scientific theories, Quantum mechanics - Applications of quantum theory, Quantum mechanics - Philosophical consequences, Quantum mechanics - History, Quantum mechanics - Founding experiments, Quantum mechanics - Notes

Read more here: » Quantum mechanics: Encyclopedia II - Quantum mechanics - Introduction

Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian H. Ignoring complications about continuous spectra, we look at the discrete spectrum of H and the corresponding eigenspaces of each eigenvalue λ (see spectral theorem for Hermitian operators for the mathematical background): with and . Physical states are normalized, meaning that their norm is equal to 1. Once again ignoring complica ...

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Variational method quantum mechanics, Variational method quantum mechanics - Introduction, Variational method quantum mechanics - Ansatz

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In physics, momentum is the product of the mass and velocity of an object. Momentum - Introduction - Momentum in Classical mechanics. If an object is moving in any reference frame, then it has momentum in that frame. The amount of momentum that an object has depends on two variables: the mass and the velocity of the moving object in the frame of reference. This can be written as: momentum = mass × velocity In physics, the symbol for momentum is a small p, so the above equation can be r ...

Including:

Momentum - Introduction - Momentum in Classical mechanics Momentum - Origin of momentum Momentum - Conservation of momentum
Momentum - Conservation of momentum and collisions
Momentum - Changes in momentum Momentum - Momentum in relativistic mechanics Momentum - Momentum in quantum mechanics Momentum - Figurative use

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In physics, the WKB (Wentzel-Kramers-Brillouin) approximation, also known as WKBJ approximation, is the most familiar example of a semiclassical calculation in quantum mechanics in which the wavefunction is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is taken to be slowly changing. This method is named after physicists Wentzel, Kramers, and Brillouin, who all developed it in 1926. In 1923, mathematician Harold Jeffreys had developed a general method of approxima ...

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WKB approximation - Derivation

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Computational chemistry is a branch of theoretical chemistry whose major goals are to create efficient mathematical approximations and computer programs that calculate the properties of molecules (such as total energy, dipole and quadrupole moment, vibrational frequencies, reactivity and other diverse spectroscopic quantitities and cross sections for collision of molecules with diverse atomic or subatomic projectiles) and to apply these programs to concrete chemical objects. The term is also sometimes used to cover the areas of overla ...

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Computational chemistry - Introduction Computational chemistry - Ab initio methods
Computational chemistry - Electronic structure Computational chemistry - Chemical dynamics
Computational chemistry - Semiempirical methods
Computational chemistry - Electronic structure
Computational chemistry - Molecular mechanics Computational chemistry - Software packages

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Alfred Clebsch (1832-1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. His collaboration with Paul Gordan led to the introduction of Clebsch-Gordan coefficients for spherical harmonics, which are now widely used in quantum mechanics. Together with Carl Neumann he founded the mathematical research journal Mathematische Annalen. Alfred Clebsch - External link. Biography at the MacTutor archive ...

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Alfred Clebsch - External link

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In his classic treatise Mathematical Foundations of Quantum Mechanics, von Neumann noted that projections on a Hilbert space can be viewed as propositions about physical observables. The set of principles for manipulating these quantum propositions was called quantum logic by von Neumann and Birkhoff. In his book (also called Mathematical Foundations of Quantum Mechanics) Mackey attempted to provide a set of axioms for this propositional system as an orthocomplemented partially ordered set. Mackey viewed elements of this ...

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Quantum logic, Quantum logic - Introduction, Quantum logic - Projections as propositions, Quantum logic - The propositional lattice of a quantum mechanical system, Quantum logic - Statistical structure, Quantum logic - Automorphisms, Quantum logic - Non-relativistic dynamics, Quantum logic - Pure states, Quantum logic - The measurement process, Quantum logic - Limitations of quantum logic

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An automorphism of Q is a bijective mapping α:Q → Q which preserves the orthocomplemented structure of Q, that is: for any sequence {Ei}i of pairwise orthogonal self-adjoint projections. Note that this property implies monotonicity of α. If P is a quantum probability measure on Q, then E → α(E) is also a quantum probability measure on Q. By the Gleason theorem characterizing quantum probability measures quoted above, any automorphism α induces a mapping α* on the density operators ...

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Quantum logic, Quantum logic - Introduction, Quantum logic - Projections as propositions, Quantum logic - The propositional lattice of a quantum mechanical system, Quantum logic - Statistical structure, Quantum logic - Automorphisms, Quantum logic - Non-relativistic dynamics, Quantum logic - Pure states, Quantum logic - The measurement process, Quantum logic - Limitations of quantum logic

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Consider a quantum mechanical system with lattice Q which is in some statistical state given by a density operator S. This essentially means an ensemble of systems specified by a repeatable lab preparation process. The result of a cluster of measurements intended to determine the truth value of proposition E, is just as in the classical case, a probability distribution of truth values T and F. Say the probabilities are p for T and q = 1 - p for F. By the previous section p = Tr(S < ...

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Quantum logic, Quantum logic - Introduction, Quantum logic - Projections as propositions, Quantum logic - The propositional lattice of a quantum mechanical system, Quantum logic - Statistical structure, Quantum logic - Automorphisms, Quantum logic - Non-relativistic dynamics, Quantum logic - Pure states, Quantum logic - The measurement process, Quantum logic - Limitations of quantum logic

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The so-called Hamiltonian formulations of classical mechanics have three ingredients: states, observables and dynamics. In the simplest case of a single particle moving in R3, the state space is the position-momentum space R6. We will merely note here that an observable is some real-valued function f on the state space. Examples of observables are position, momentum or energy of a particle. For classical systems, the value f(x), that is the value of f ...

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Quantum logic, Quantum logic - Introduction, Quantum logic - Projections as propositions, Quantum logic - The propositional lattice of a quantum mechanical system, Quantum logic - Statistical structure, Quantum logic - Automorphisms, Quantum logic - Non-relativistic dynamics, Quantum logic - Pure states, Quantum logic - The measurement process, Quantum logic - Limitations of quantum logic

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In non-relativistic physical systems, there is no ambiguity in referring to time evolution since there is a global time parameter. Moreover an isolated quantum system evolves in a deterministic way: if the system is in a state S at time t then at time s> t, the system is in a state Fs,t(S). Moreover, we assume The dependence is reversible: The operators Fs,t are bijective. The dependence is homogeneous: Fs,t ...

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Quantum logic, Quantum logic - Introduction, Quantum logic - Projections as propositions, Quantum logic - The propositional lattice of a quantum mechanical system, Quantum logic - Statistical structure, Quantum logic - Automorphisms, Quantum logic - Non-relativistic dynamics, Quantum logic - Pure states, Quantum logic - The measurement process, Quantum logic - Limitations of quantum logic

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A convex combinations of statistical states S1 and S2 is a state of the form S = p2 S1 +p2 S2 where p1, p2 are non-negative and p1 + p2 =1. Considering the statistical state of system as specified by lab conditions used for its preparation, the convex combination S can be regarded as the state formed in the following way: toss a biased coin with outco ...

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Quantum logic, Quantum logic - Introduction, Quantum logic - Projections as propositions, Quantum logic - The propositional lattice of a quantum mechanical system, Quantum logic - Statistical structure, Quantum logic - Automorphisms, Quantum logic - Non-relativistic dynamics, Quantum logic - Pure states, Quantum logic - The measurement process, Quantum logic - Limitations of quantum logic

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Imagine a forensics lab which has some apparatus to measure the speed of a of a bullet fired from a gun. Under carefully controlled conditions of temperature, humidity, pressure and so on the same gun is fired repeatedly and speed measurements taken. This produces some distribution of speeds. Though we will not get exactly the same value for each individual measurement, for each cluster of measurements, we would expect the experiment to lead to the same distribution of speeds. In particular, we can expect to assign probability distributions to ...

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Quantum logic, Quantum logic - Introduction, Quantum logic - Projections as propositions, Quantum logic - The propositional lattice of a quantum mechanical system, Quantum logic - Statistical structure, Quantum logic - Automorphisms, Quantum logic - Non-relativistic dynamics, Quantum logic - Pure states, Quantum logic - The measurement process, Quantum logic - Limitations of quantum logic

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Obviously, if we were to vary over all possible states with norm 1 trying to minimize the expectation value of H, the lowest value would be E0 and the corresponding state would be an eigenstate of E0. Varying over the entire Hilbert space is usually too complicated for physical calculations, and a subspace of the entire Hilbert space is chosen, parametrized by some (real) differentiable parameters αi (i=1,2..,N). The choice of the subspace is called the ansatz. Some choices of ansatzes lead to better approxima ...

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Variational method quantum mechanics, Variational method quantum mechanics - Introduction, Variational method quantum mechanics - Ansatz

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In quantum mechanics momentum is defined as an operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are conjugate variables. For a single particle with no electric charge and no spin, the momentum operator can be written in the position basis as where is the gradient operator. This is a commonly encountered form ...

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Momentum, Momentum - Introduction - Momentum in Classical mechanics, Momentum - Origin of momentum, Momentum - Conservation of momentum, Momentum - Conservation of momentum and collisions, Momentum - Changes in momentum, Momentum - Momentum in relativistic mechanics, Momentum - Momentum in quantum mechanics, Momentum - Figurative use

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Relativistic momentum as proposed by Albert Einstein arises from the invariance of four-vectors under Lorentzian translation. These four-vectors appear spontaneously in the Green's function from quantum field theory. A vector, called the Four-momentum is defined as: [E/c p] where E is the total energy of the system, and p is called the "relativistic momentum" defined thus: where where . Setting velocity to zero, one derives that t ...

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Momentum, Momentum - Introduction - Momentum in Classical mechanics, Momentum - Origin of momentum, Momentum - Conservation of momentum, Momentum - Conservation of momentum and collisions, Momentum - Changes in momentum, Momentum - Momentum in relativistic mechanics, Momentum - Momentum in quantum mechanics, Momentum - Figurative use

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Although momentum is conserved within a closed system, individual parts of a system can undergo changes in momentum. In classical mechanics, an impulse changes the momentum of a body, and has the same units and dimensions as momentum. The SI unit of impulse is the same as for momentum (kg m/s). An impulse is calculated as the integral of force with respect to time. where I is the impulse, measured in kilogram metres per second F is the force, measured in newtons t ...

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Momentum, Momentum - Introduction - Momentum in Classical mechanics, Momentum - Origin of momentum, Momentum - Conservation of momentum, Momentum - Conservation of momentum and collisions, Momentum - Changes in momentum, Momentum - Momentum in relativistic mechanics, Momentum - Momentum in quantum mechanics, Momentum - Figurative use

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If an object is moving in any reference frame, then it has momentum in that frame. The amount of momentum that an object has depends on two variables: the mass and the velocity of the moving object in the frame of reference. This can be written as: momentum = mass × velocity In physics, the symbol for momentum is a small p, so the above equation can be rewritten as: where m is the mass and v th ...

See also:

Momentum, Momentum - Introduction - Momentum in Classical mechanics, Momentum - Origin of momentum, Momentum - Conservation of momentum, Momentum - Conservation of momentum and collisions, Momentum - Changes in momentum, Momentum - Momentum in relativistic mechanics, Momentum - Momentum in quantum mechanics, Momentum - Figurative use

Read more here: » Momentum: Encyclopedia II - Momentum - Introduction - Momentum in Classical mechanics

Because of the way it is defined, momentum is always conserved. In the absence of external forces, a system will have constant total momentum: a property that is identical to Newton's law of inertia, his first law of motion. Newton's third law of motion, the law of reciprocal actions, dictates that the forces acting between systems are equal, which is equivalent to a statement of the conservation of momentum. Momen ...

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Momentum, Momentum - Introduction - Momentum in Classical mechanics, Momentum - Origin of momentum, Momentum - Conservation of momentum, Momentum - Conservation of momentum and collisions, Momentum - Changes in momentum, Momentum - Momentum in relativistic mechanics, Momentum - Momentum in quantum mechanics, Momentum - Figurative use

Read more here: » Momentum: Encyclopedia II - Momentum - Conservation of momentum