Ehrenfest theorem It reads[2] where A is some QM operator and is its expectation value. Ehrenfest's theorem is most apparent in the Heisenberg picture of quantum mechanics, where it is just the expectation value of the Heisenberg equation of motion. It provides mathematical support to the correspondence principle. Derivation in the Schrödinger picture[edit] Suppose some system is presently in a quantum state Φ. where we are integrating over all space. and Note Often (but not always) the operator A is time independent, so that its derivative is zero and we can ignore the last term. Derivation in the Heisenberg Picture[edit] In the Heisenberg picture, the derivation is trivial. we can derive Ehrenfest's theorem simply by projecting the Heisenberg equation onto from the right and from the left, or taking the expectation value, so We can pull the out of the first term since the state vectors are no longer time dependent in the Heisenberg Picture. General example[edit] where x is the position of the particle. , we get 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 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. Decoherence represents a challenge for the practical realization of quantum computers, since such machines are expected to rely heavily on the undisturbed evolution of quantum coherences. Mechanisms[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). Historical background[edit] In the 1920s and 1930s, quantum mechanics was developed using calculus and linear algebra. Wave functions and function spaces[edit] Functional analysis is commonly used to formulate the wave function with a necessary mathematical precision; usually they are quadratically integrable functions (at least locally) because it is compatible with the Hilbert space formalism mentioned below. Requirements[edit]
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.
Pauli exclusion principle A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles. This means that the wave function changes its sign if the space and spin co-ordinates of any two particles are interchanged. Integer spin particles, bosons, are not subject to the Pauli exclusion principle: any number of identical bosons can occupy the same quantum state, as with, for instance, photons produced by a laser and Bose–Einstein condensate. Overview[edit] "Half-integer spin" means that the intrinsic angular momentum value of fermions is (reduced Planck's constant) times a half-integer (1/2, 3/2, 5/2, etc.). History[edit] In the early 20th century it became evident that atoms and molecules with even numbers of electrons are more chemically stable than those with odd numbers of electrons. Pauli looked for an explanation for these numbers, which were at first only empirical. Connection to quantum state symmetry[edit] and the other in state
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. Let G be the symmetry group of the universe – that is, the set of symmetries under which the laws of physics are invariant. . Poincaré group[edit] Other symmetries[edit] Approximate symmetries[edit] Hypothetical example[edit]
Measurement in quantum mechanics A measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured, the eigenvalue of this eigenstate belongs to being equal to the result of the measurement— P.A.M. Dirac (1958) in "The Principles of Quantum Mechanics" p. 36 The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus. Measurement from a practical point of view[edit] Measurement plays an important role in quantum mechanics, and it is viewed in different ways among various interpretations of quantum mechanics. Qualitative overview[edit] In classical mechanics, a simple system consisting of only one single particle is fully described by the position and momentum The measurement process is often considered as random and indeterministic. Quantitative details[edit] Measurable quantities ("observables") as operators[edit]
Uncertainty principle where ħ is the reduced Planck constant. The original heuristic argument that such a limit should exist was given by Heisenberg, after whom it is sometimes named the Heisenberg principle. This ascribes the uncertainty in the measurable quantities to the jolt-like disturbance triggered by the act of observation. Though widely repeated in textbooks, this physical argument is now known to be fundamentally misleading.[4][5] While the act of measurement does lead to uncertainty, the loss of precision is less than that predicted by Heisenberg's argument; the formal mathematical result remains valid, however. Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. Certain experiments, however, may deliberately test a particular form of the uncertainty principle as part of their main research program. Introduction[edit] Click to see animation. Wave mechanics interpretation[edit] . with yields where
Wave–particle duality Origin of theory[edit] The idea of duality originated in a debate over the nature of light and matter that dates back to the 17th century, when Christiaan Huygens and Isaac Newton proposed competing theories of light: light was thought either to consist of waves (Huygens) or of particles (Newton). Through the work of Max Planck, Albert Einstein, Louis de Broglie, Arthur Compton, Niels Bohr, and many others, current scientific theory holds that all particles also have a wave nature (and vice versa).[2] This phenomenon has been verified not only for elementary particles, but also for compound particles like atoms and even molecules. For macroscopic particles, because of their extremely short wavelengths, wave properties usually cannot be detected.[3] Brief history of wave and particle viewpoints[edit] Thomas Young's sketch of two-slit diffraction of waves, 1803 Particle impacts make visible the interference pattern of waves. A quantum particle is represented by a wave packet.
Quantum entanglement Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently – instead, a quantum state may be given for the system as a whole. Such phenomena were the subject of a 1935 paper by Albert Einstein, Boris Podolsky and Nathan Rosen,[1] describing what came to be known as the EPR paradox, and several papers by Erwin Schrödinger shortly thereafter.[2][3] Einstein and others considered such behavior to be impossible, as it violated the local realist view of causality (Einstein referred to it as "spooky action at a distance"),[4] and argued that the accepted formulation of quantum mechanics must therefore be incomplete. History[edit] However, they did not coin the word entanglement, nor did they generalize the special properties of the state they considered. Concept[edit] Meaning of entanglement[edit] Apparent paradox[edit] The hidden variables theory[edit]