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Expectation value (quantum mechanics) 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 ...
The Ehrenfest theorem is a special case of a more general relation between the expectation of any quantum mechanical operator and the expectation of the commutator of that operator with the Hamiltonian of the system [2][3] where A is some quantum mechanical operator and A is its expectation value. It is most apparent in the Heisenberg picture ...
Position operator. In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. When the position operator is considered with a wide enough domain (e.g. the space of tempered distributions), its eigenvalues are the possible position vectors of the particle. [1]
Thus, A may be any complex number with absolute value √ 2/L; these different values of A yield the same physical state, so A = √ 2/L can be selected to simplify. It is expected that the eigenvalues , i.e., the energy E n {\displaystyle E_{n}} of the box should be the same regardless of its position in space, but ψ n ( x , t ...
Translation operator (quantum mechanics) In quantum mechanics, a translation operator is defined as an operator which shifts particles and fields by a certain amount in a certain direction. It is a special case of the shift operator from functional analysis. More specifically, for any displacement vector , there is a corresponding translation ...
One simple way to compare classical to quantum mechanics is to consider the time-evolution of the expected position and expected momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. [25]: 302 The quantum expectation values satisfy the Ehrenfest theorem.
The expectation values of position and momentum combined with variance of each variable can be derived from the wavefunction to understand the behavior of the energy eigenkets. They are shown to be x ^ = 0 {\textstyle \langle {\hat {x}}\rangle =0} and p ^ = 0 {\textstyle \langle {\hat {p}}\rangle =0} owing to the symmetry of the problem, whereas:
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected value is the mean of the possible values a random variable can take, weighted by the probability of those ...