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In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring (e.g. measurements which ...
Dirac's rule of thumb suggests that statements in quantum mechanics which contain a commutator correspond to statements in classical mechanics where the commutator is supplanted by a Poisson bracket multiplied by iħ. This makes the operator expectation values obey corresponding classical equations of motion, provided the Hamiltonian is at most ...
The vacuum expectation value of an operator O is usually denoted by . One of the most widely used examples of an observable physical effect that results from the vacuum expectation value of an operator is the Casimir effect. This concept is important for working with correlation functions in quantum field theory. It is also important in ...
The old quantum theory is a collection of results from the years 1900–1925 [23] which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. [24] The theory is now understood as a semi-classical approximation [25] to modern quantum mechanics. [26]
In quantum mechanics, a density matrix (or density operator) is a matrix that describes an ensemble [1] of physical systems as quantum states (even if the ensemble contains only one system). It allows for the calculation of the probabilities of the outcomes of any measurements performed upon the systems of the ensemble using the Born rule .
A very important application of the expectation value is in the field of quantum mechanics. The expectation value of a quantum mechanical operator ^ operating on a quantum state vector | is written as ^ = | ^ | .
In quantum mechanics, the average, or expectation value of the position of a particle is given by = (). For the steady state particle in a box, it can be shown that the average position is always x = x c {\displaystyle \langle x\rangle =x_{c}} , regardless of the state of the particle.
In quantum mechanics, energy is defined in terms of the energy operator, ... where is the expectation value of energy. Properties. It can be shown that the ...