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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 ...
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 conventional definition of the spin quantum number is s = n / 2 , where n can be any non-negative integer. Hence the allowed values of s are 0, 1 / 2 , 1, 3 / 2 , 2, etc. The value of s for an elementary particle depends only on the type of particle and cannot be altered in any known way (in contrast to the spin ...
Among numerous applications of the Kubo formula, one can calculate the charge and spin susceptibilities of systems of electrons in response to applied electric and magnetic fields. Responses to external mechanical forces and vibrations can be calculated as well.
For an observable , the expectation value given a quantum state is A = tr ( A ρ ) . {\displaystyle \langle A\rangle =\operatorname {tr} (A\rho ).} A density operator that is a rank-1 projection is known as a pure quantum state, and all quantum states that are not pure are designated mixed .
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.
There is a nonzero probability amplitude to find a significant fluctuation in the vacuum value of the field Φ(x) if one measures it locally (or, to be more precise, if one measures an operator obtained by averaging the field over a small region). Furthermore, the dynamics of the fields tend to favor spatially correlated fluctuations to some ...
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 −5E 1 /2 = 34 eV, so H is 8E 1 − 5E 1 /2 = −75 eV. A tighter upper bound can be found by using a better trial wavefunction with 'tunable' parameters.