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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 ...
One particle: N particles: One dimension ^ = ^ + = + ^ = = ^ + (,,) = = + (,,) where the position of particle n is x n. = + = = +. (,) = /.There is a further restriction — the solution must not grow at infinity, so that it has either a finite L 2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): [1] ‖ ‖ = | |.
Quantum state tomography is a process by which, given a set of data representing the results of quantum measurements, a quantum state consistent with those measurement results is computed. [50] It is named by analogy with tomography , the reconstruction of three-dimensional images from slices taken through them, as in a CT scan .
Expected values can also be used to compute the variance, by means of the computational formula for the variance = [] ( []). A very important application of the expectation value is in the field of quantum mechanics.
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 ...
Consider a quantum system described by the (time independent) Hamiltonian . The expectation value of a physical quantity at equilibrium temperature T {\displaystyle T} , described by the operator A ^ {\displaystyle {\hat {A}}} , can be evaluated as:
The Ehrenfest theorem, named after Austrian theoretical physicist Paul Ehrenfest, relates the time derivative of the expectation values of the position and momentum operators x and p to the expectation value of the force = ′ on a massive particle moving in a scalar potential (), [1]
The expectation value of the total Hamiltonian H (including the term V ee) in the state described by ψ 0 will be an upper bound for its ground state energy. V ee is −5 E 1 /2 = 34 eV , so H is 8 E 1 − 5 E 1 /2 = −75 eV .