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Quantum Mechanics: Encyclopedia - Quantum Mechanics
Quantum mechanics is a fundamental physical theory that replaces Newtonian mechanics and classical electromagnetism at the atomic and sub...
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Basics Of Quantum Mechanics: Encyclopedia - Basics Of Quantum Mechanics
Quantum mechanics is a physical science dealing with the behaviour of matter and electromagnetic waves on the scale of atoms and subatomi...
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Quantum Mechanics: Encyclopedia Ii - Quantum Mechanics - Introduction
The term quantum (Latin, "how much") refers to the discrete units that the theory assigns to certain physical quantities, such as the ene...
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Parity Physics: Encyclopedia Ii - Parity Physics - Quantum Mechanics
In quantum mechanics, spacetime transformations act on quantum states. The parity transformation, P, becomes an unitary operator in quant...
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Werner Heisenberg: Encyclopedia Ii - Werner Heisenberg - Quantum Mechanics
As a student, he met Niels Bohr in Göttingen in 1922. A fruitful collaboration developed between the two.
He invented matrix mechanics, ...
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Continuity Equation: Encyclopedia Ii - Continuity Equation - Quantum Mechanics
In quantum mechanics, the conservation of probability also yields a continuity equation. Let P(x, t) be a probability density and wr...
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Canonical Quantization: Encyclopedia Ii - Canonical Quantization - Quantum Mechanics
In the classical mechanics of a particle one has dynamical variables which are called coordinates (q) and momenta (p). These specify the ...
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John Von Neumann: Encyclopedia Ii - John Von Neumann - Quantum Mechanics
At the International Congress of Mathematicians of 1900, David Hilbert presented his famous list of twenty-three problems considered cent...
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Quantum Mechanics: Encyclopedia Ii - Quantum Mechanics - Description Of The Theory
Bra-ket notation | Commutation relation | Heisenberg picture | Schrödinger picture | Wavefunction | Measurement in quantum mechanics | S...
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Quantum: Encyclopedia - Quantum
The word quantum, pl. "quanta", comes from the Latin "quantus", for "how much". In general, it refers to an "amount of something". But, t...
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Wave Packet: Encyclopedia - Wave Packet
The wave packet is one of the most widely misunderstood and misused concepts in physics. In general, a wave packet is an envelope or pack...
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Vacuum State: Encyclopedia - Vacuum State
In quantum field theory, the vacuum state (also called the vacuum) is the quantum state with the lowest possible energy. As a result, it ...
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Classical Mechanics: Encyclopedia - Classical Mechanics
In physics, classical mechanics or Newtonian mechanics is one of the two major sub-fields of study in the science of mechanics, which is ...
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Coherent State: Encyclopedia - Coherent State
In quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely ...
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Basics Of Quantum Mechanics: Encyclopedia Ii - Basics Of Quantum Mechanics - Development Of Modern Quantum Mechanics
Basics of quantum mechanics - Full quantum mechanical theory.
Werner Heisenberg developed the full quantum mechanical theory in 1925 at...
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Interpretation Of Quantum Mechanics: Encyclopedia Ii - Interpretation Of Quantum Mechanics - Obstructions To Direct Interpretation
The perceived difficulties of interpretation reflect a number of points about the orthodox description of quantum mechanics. In this arti...
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Basics Of Quantum Mechanics: Encyclopedia Ii - Basics Of Quantum Mechanics - Background
Through over fifty years of experimentation and applied science, quantum mechanical theory has proven to be very successful and practical...
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Basics Of Quantum Mechanics: Encyclopedia Ii - Basics Of Quantum Mechanics - Old Quantum Theory
Quantum mechanics developed from the study of electromagnetic waves through spectroscopy which includes visible light seen in the colors ...
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Measurement In Quantum Mechanics: Encyclopedia Ii - Measurement In Quantum Mechanics - The Mathematical Formalism Of Measurement
Measurement in quantum mechanics - Measurable quantities observables as operators.
An observable quantity is represented mathematically...
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Measurement In Quantum Mechanics: Encyclopedia Ii - Measurement In Quantum Mechanics - Philosophical Problems Of Quantum Measurements
Measurement in quantum mechanics - What physical interaction constitutes a measurement?.
Until the advent of quantum decoherence theory...
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Hamiltonian Quantum Mechanics: Encyclopedia Ii - Hamiltonian Quantum Mechanics - The Quantum Hamiltonian
As explained in the article mathematical formulation of quantum mechanics, the physical state of a system may be characterized as a vecto...
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Interpretation Of Quantum Mechanics: Encyclopedia Ii - Interpretation Of Quantum Mechanics - Properties Of Interpretations
An interpretation can be characterized by whether it satisfies certain properties, such as:
Realism
Completeness
Local realism
Determini...
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Mathematical Formulation Of Quantum Mechanics: Encyclopedia Ii - Mathematical Formulation Of Quantum Mechanics - History Of The Formalism
Mathematical formulation of quantum mechanics - The old quantum theory and the need for new mathematics.
Main article: Old quantum theo...
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Interpretation Of Quantum Mechanics: Encyclopedia Ii - Interpretation Of Quantum Mechanics - Problematic Status Of Pictures And Interpretations
The precise ontological status, of each one of the interpreting pictures, remains a matter of philosophical argument.
In other words, if ...
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Mathematical Formulation Of Quantum Mechanics: Encyclopedia Ii - Mathematical Formulation Of Quantum Mechanics - Mathematical Structure Of Quantum Mechanics
A physical system is generally described by three basic ingredients: states; observables; and dynamics (or law of time evolution) or, mor...
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Interpretation Of Quantum Mechanics: Encyclopedia Ii - Interpretation Of Quantum Mechanics - Instrumentalist Interpretation
Any modern scientific theory requires at the very least an instrumentalist description which relates the mathematical formalism to experi...
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Interpretation Of Quantum Mechanics: Encyclopedia Ii - Interpretation Of Quantum Mechanics - Historical Background
The operational meaning of the technical terms used by researchers in quantum theory (such as wavefunctions and matrix mechanics) progres...
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Quantum Mechanics Philosophy And Controversy: Encyclopedia Ii - Quantum Mechanics Philosophy And Controversy - Consequences Of The Uncertainty Principle
The Uncertainty Principle is a main theory in the physical science of quantum mechanics that explains the universe at atomic and subatomi...
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Quantum Statistical Mechanics: Encyclopedia Ii - Quantum Statistical Mechanics - Expectation
From classical probability theory we know that the expectation of a random variable X is completely determined by its distribution DX by
...
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Mathematical Formulation Of Quantum Mechanics: Encyclopedia Ii - Mathematical Formulation Of Quantum Mechanics - The Problem Of Measurement
The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account...
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Quantum Mechanics Philosophy And Controversy: Encyclopedia Ii - Quantum Mechanics Philosophy And Controversy - Erwin Schroedinger Controversy
Later in life, the inventor of wave mechanics of quantum theory, Erwin Schrödinger began a campaign against the generally accepted quant...
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Quantum Gravity: Encyclopedia Ii - Quantum Gravity - The Incompatibility Of Quantum Mechanics And General Relativity
At present, one of the deepest problems in theoretical physics is harmonizing the theory of general relativity, which describes gravitati...
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Quantum Logic: Encyclopedia Ii - Quantum Logic - The Propositional Lattice Of A Quantum Mechanical System
This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics. This...
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Variational Method Quantum Mechanics: Encyclopedia Ii - Variational Method Quantum Mechanics - Ansatz
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 ...
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Perturbation Theory Quantum Mechanics: Encyclopedia Ii - Perturbation Theory Quantum Mechanics - Applications Of Perturbation Theory
Perturbation theory is an extremely important tool for describing real quantum systems, as it turns out to be very difficult to find exac...
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Quantum Mechanics Philosophy And Controversy: Encyclopedia Ii - Quantum Mechanics Philosophy And Controversy - Philosophical Determinism
18th century science had made rapid advances and the scientist and philosopher Laplace wrote a vision of science that was the prevailing ...
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Variational Method Quantum Mechanics: Encyclopedia Ii - Variational Method Quantum Mechanics - Introduction
Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian H. Ignoring complications about continuous s...
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Quantum Statistical Mechanics: Encyclopedia Ii - Quantum Statistical Mechanics - Von Neumann Entropy
Of particular significance for describing randomness of a state is the von Neumann entropy of S formally defined by
.
Actually the oper...
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Supersymmetry: Encyclopedia Ii - Supersymmetry - Supersymmetric Quantum Mechanics
Understanding the consequences of supersymmetry has proven mathematically daunting, and it has likewise been difficult to develop theorie...
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Supersymmetry: Encyclopedia Ii - Supersymmetry - Supersymmetric Quantum Mechanics
Understanding the consequences of supersymmetry has proven mathematically daunting, and it has likewise been difficult to develop theorie...
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Perturbation Theory Quantum Mechanics: Encyclopedia Ii - Perturbation Theory Quantum Mechanics - Time-independent Perturbation Theory
There are two categories of perturbation theory: time-independent and time-dependent. In this section, we discuss time-independent pertur...
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Momentum: Encyclopedia Ii - Momentum - Momentum In Quantum Mechanics
In quantum mechanics momentum is defined as an operator on the wave function. The Heisenberg uncertainty principle defines limits on how ...
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Interpretation Of Quantum Mechanics: Encyclopedia Ii - Interpretation Of Quantum Mechanics - Comparison
At the moment, there is no experimental evidence that would allow us to distinguish between the various interpretations listed below. To ...
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Hamiltonian Quantum Mechanics: Encyclopedia Ii - Hamiltonian Quantum Mechanics - Energy Eigenket Degeneracy Symmetry And Conservation Laws
In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstat...
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Quantum: Encyclopedia Ii - Quantum - Discovery Of Quantum Theory
Quantum theory, the branch of physics based on quantization, began in 1900 when Max Planck published his theory explaining the emission s...
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Path Integral Formulation: Encyclopedia Ii - Path Integral Formulation - Formulating Quantum Mechanics
The path integral method is an alternative formulation of quantum mechanics. The canonical approach, pioneered by Schroedinger, Heisenber...
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Loop Quantum Gravity: Encyclopedia Ii - Loop Quantum Gravity - The Incompatibility Between Quantum Mechanics And General Relativity
Main article: quantum gravity
Quantum field theory studied on curved (non-Minkowskian) backgrounds has shown that some of the core assump...
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Wave Packet: Encyclopedia Ii - Wave Packet - Quantum Mechanical Waves
A quantum mechanical wave in its most salient and simple form is a solution to a differential equation. It is a bridge that provides math...
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Gravity: Encyclopedia Ii - Gravity - Gravity And Quantum Mechanics
It is widely believed that three of the four fundamental forces (the strong nuclear force, the weak nuclear force, and the electromagneti...
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Gravity: Encyclopedia Ii - Gravity - Gravity And Quantum Mechanics
It is widely believed that three of the four fundamental forces (the strong nuclear force, the weak nuclear force, and the electromagneti...
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Vacuum: Encyclopedia Ii - Vacuum - The Quantum-mechanical Vacuum
Even an ideal vacuum, thought of as the complete absence of anything, will not in practice remain empty. One reason is that the walls of ...
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Physical Paradox: Encyclopedia Ii - Physical Paradox - Quantum Mechanical Paradoxes
A significant set of physical paradoxes are associated with the privileged position of the observer in quantum mechanics. Two of the most...
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Atomic Orbital: Encyclopedia Ii - Atomic Orbital - Quantum Mechanical Description
In quantum mechanics, the state of an atom, i.e. the eigenstates of the atomic Hamiltonian, are expanded (see configuration interaction e...
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Coherent State: Encyclopedia Ii - Coherent State - Quantum Mechanical Definition
Mathematically, the coherent state is defined to be the eigenstate of the annihilation operator a. Formally, this reads:
Note that si...
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Bell's Theorem: Encyclopedia Ii - Bell's Theorem - Comparison To Quantum Mechanical Prediction
To apply Bell's theorem we will show that quantum mechanics makes a prediction that violates a "Bell inequality" in the setup considered ...
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Quantum: Encyclopedia Ii - Quantum - Quantization In Antiquity
In a sense, it can be said that the quantization idea is very old. A string under tension, and fixed at both ends, will vibrate at certai...
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H-theorem: Encyclopedia Ii - H-theorem - Quantum Mechanical H-theorem
The following quantum-mechanical analogue of Boltzmann's H-theorem is sometimes given (e.g., Waldram (1985), p.39).
Starting from the Gib...
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Normal Mode: Encyclopedia Ii - Normal Mode - Normal Modes In Quantum Mechanics
In quantum mechanics, a state of a system is described by a wavefunction of (x, t) which solves the Schrödinger equation. The square of...
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Angular Momentum: Encyclopedia Ii - Angular Momentum - Angular Momentum In Quantum Mechanics
In quantum mechanics, angular momentum is defined like momentum - not as a quantity but as an operator on the wave function:
where r a...
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Ionization Potential: Encyclopedia Ii - Ionization Potential - Quantum-mechanical Explanation
According to the more sophisticated theory of quantum mechanics, the location of an electron is best described as a "cloud" of likely loc...
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Rydberg Atom: Encyclopedia Ii - Rydberg Atom - The Quantum Mechanical Picture
Quantum mechanically a state of high n refers to an atom in which the valence electron(s) have been excited into a formerly unpopulated e...
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Path Integral Formulation: Encyclopedia Ii - Path Integral Formulation - The Path Integral In Quantum-mechanical Interpretation
In one philosophical interpretation of quantum mechanics, the "sum over histories" interpretation, the path integral is taken to be funda...
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Atomic Orbital: Encyclopedia Ii - Atomic Orbital - Formal Quantum Mechanical Definition
In quantum mechanics, the state of an atom, i.e. the eigenstates of the atomic Hamiltonian, are expanded (see configuration interaction e...
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Heisenberg Group: Encyclopedia Ii - Heisenberg Group - Weyl's View Of Quantum Mechanics
The application that led Hermann Weyl to an explicit introduction of the Heisenberg group was the question of why the Schrödinger pictur...
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John Von Neumann: Encyclopedia Ii - John Von Neumann - Quantum Mechanics
At the International Congress of Mathematicians of 1900, David Hilbert presented his famous list of twenty-three problems considered cent...
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Identical Particles: Encyclopedia Ii - Identical Particles - Quantum Mechanical Description Of Identical Particles
Identical particles - Symmetrical and antisymmetrical states.
We will now make the above discussion concrete, using the formalism devel...
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Statistical Ensemble Mathematical Physics: Encyclopedia Ii - Statistical Ensemble Mathematical Physics - Ensembles In Quantum Statistical Mechanics
<>
See main article: Quantum statistical mechanics
Putting aside for the moment the question of how statistical ensembles are gener...
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Statistical Ensemble Mathematical Physics: Encyclopedia Ii - Statistical Ensemble Mathematical Physics - Ensembles In Quantum Statistical Mechanics
<>
See main article: Quantum statistical mechanics
Putting aside for the moment the question of how statistical ensembles are gener...
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Quantum Number: Encyclopedia Ii - Quantum Number - Single Electron In An Atom
The most widely studied set of quantum numbers is that for a single electron in an atom: not only because it is useful in chemistry, bein...
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Quantum Logic: Encyclopedia Ii - Quantum Logic - Introduction
In his classic treatise Mathematical Foundations of Quantum Mechanics, von Neumann noted that projections on a Hilbert space can be viewe...
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Hamilton's Equations: Encyclopedia Ii - Hamilton's Equations - Generalization Through Poisson Bracket
The Hamilton's equations above work perfectly for classical mechanics, but not for the quantum mechanics, since the differential equation...
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Quantum Logic: Encyclopedia Ii - Quantum Logic - The Measurement Process
Consider a quantum mechanical system with lattice Q which is in some statistical state given by a density operator S. This essentially me...
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Quantum Logic: Encyclopedia Ii - Quantum Logic - Automorphisms
An automorphism of Q is a bijective mapping α:Q → Q which preserves the orthocomplemented structure of Q, that is:
for any sequence...
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Matrix Mechanics: Encyclopedia Ii - Matrix Mechanics - Mathematical Details
In quantum mechanics in the Heisenberg picture the state vector, |ψ> does not change with time, and an observable A satisfies
In s...
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Quantum Gravity: Encyclopedia Ii - Quantum Gravity - Theories
There are a number of proposed quantum gravity theories:
String theory/superstring theory/M-theory
Supergravity
AdS/CFT
Wheeler-deWitt e...
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Quantum Logic: Encyclopedia Ii - Quantum Logic - Projections As Propositions
The so-called Hamiltonian formulations of classical mechanics have three ingredients: states, observables and dynamics. In the simplest c...
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Quantum Gravity: Encyclopedia Ii - Quantum Gravity - Historical Perspective
Historically, there have been two reactions to the apparent inconsistency of quantum theories with the necessary background independence ...
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Quantum State: Encyclopedia Ii - Quantum State - Superposition Of States
If a quantum mechanical state can be reached by more than one path, then is said to be a linear superposition of states. In the case of...
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Quantum Logic: Encyclopedia Ii - Quantum Logic - Non-relativistic Dynamics
In non-relativistic physical systems, there is no ambiguity in referring to time evolution since there is a global time parameter. Moreov...
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Quantum Logic: Encyclopedia Ii - Quantum Logic - Pure States
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 =...
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Quantum Indeterminacy: Encyclopedia Ii - Quantum Indeterminacy - Indeterminacy For Mixed States
We have described indeterminacy for a quantum system which is in a pure state. Mixed states are a more general kind of state obtained by ...
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Quantum Gravity: Encyclopedia Ii - Quantum Gravity - Overview
Much of the difficulty in merging these theories comes from the radically different assumptions that these theories make on how the unive...
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Quantum Logic: Encyclopedia Ii - Quantum Logic - Statistical Structure
Imagine a forensics lab which has some apparatus to measure the speed of a of a bullet fired from a gun. Under carefully controlled condi...
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Quantum Indeterminacy: Encyclopedia Ii - Quantum Indeterminacy - Indeterminacy And Incompleteness
Quantum indeterminacy is the assertion that the state of a system does not determine a unique collection of values for all its measurable...
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Quantum Indeterminacy: Encyclopedia Ii - Quantum Indeterminacy - Measurement
An adequate account of quantum indeterminacy requires a theory of measurement. Many theories have been proposed since the beginning of qu...
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Loop Quantum Gravity: Encyclopedia Ii - Loop Quantum Gravity - Loop Quantum Gravity In General And Its Ambitions
LQG in itself was initially less ambitious than string theory, purporting only to be a quantum theory of gravity. String theory, on the o...
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Loop Quantum Gravity: Encyclopedia Ii - Loop Quantum Gravity - Problems
As of December 2005, there is not a single experiment which verifies or refutes any aspect of LQG. This problem plagues most current theo...
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Loop Quantum Gravity: Encyclopedia Ii - Loop Quantum Gravity - The Ingredients Of Loop Quantum Gravity
Loop quantum gravity - Loop quantization.
At the core of loop quantum gravity is a framework for nonperturbative quantization of diffeo...
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Loop Quantum Gravity: Encyclopedia Ii - Loop Quantum Gravity - Loop Quantum Gravity In General And Its Ambitions
LQG in itself was initially less ambitious than string theory, claiming only to be a quantum theory of gravity. String theory, on the oth...
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Loop Quantum Gravity: Encyclopedia Ii - Loop Quantum Gravity - Problems
As of now, there is not a single experiment which verifies or refutes any aspect of LQG. This problem plagues most current theories of qu...
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Loop Quantum Gravity: Encyclopedia Ii - Loop Quantum Gravity - History Of Lqg
Main article: history of loop quantum gravity
In 1986, Abhay Ashtekar reformulated Einstein's field equations of general relativity using...
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Classical Mechanics: Encyclopedia Ii - Classical Mechanics - Limits Of Validity
Classical mechanics - The classical approximation to special relativity.
Non-relativistic classical mechanics approximates the relativi...
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Classical Mechanics: Encyclopedia Ii - Classical Mechanics - Description Of The Theory
The following introduces the basic concepts of classical mechanics. For simplicity, it uses point particles, objects with negligible size...
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Classical Mechanics: Encyclopedia Ii - Classical Mechanics - History
Main article: History of classical mechanics
The Greeks, and Aristotle in particular, were the first to propose that there are abstract p...
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Atomic Orbital: Encyclopedia Ii - Atomic Orbital - Qualitative Characterization
Atomic orbital - Limitations on the quantum numbers.
An atomic orbital is uniquely identified by the values of the three quantum number...
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Coherent State: Encyclopedia Ii - Coherent State - Coherent States Of Bose-einstein Condensates
A Bose-Einstein condensate (BEC) is a collection of boson atoms that are all in the same quantum state. An approximate theoretical desc...
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Quantum Mechanics: Encyclopedia Ii - Quantum Mechanics - History
In 1900, Max Planck introduced the idea that energy is quantized, in order to derive a formula for the observed frequency dependence of t...
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Quantum Mechanics: Encyclopedia Ii - Quantum Mechanics - Applications Of Quantum Theory
Quantum mechanics has had enormous success in explaining many of the features of our world. The individual behavior of the microscopic pa...
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Quantum Mechanics: Encyclopedia Ii - Quantum Mechanics - Applications Of Quantum Theory
Quantum mechanics has had enormous success in explaining many of the features of our world. The individual behavior of the subatomic part...
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Quantum Mechanics: Encyclopedia Ii - Quantum Mechanics - Philosophical Consequences
Since its inception, the many counter-intuitive results of quantum mechanics have provoked strong philosophical debate and many interpret...
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