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Nevertheless, as explained in the introduction, for states that are highly localized in space, the expected position and momentum will approximately follow classical trajectories, which may be understood as an instance of the correspondence principle. Similarly, we can obtain the instantaneous change in the position expectation value.
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 angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum problems involving rotational symmetry. Being an observable, its eigenfunctions represent the distinguishable physical states of a system's angular momentum, and the corresponding eigenvalues the observable experimental values.
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:
The expectation value of the momentum p for the complex plane wave is ... The expectation value of the energy E is
The expectation value for the momentum is then calculated to be zero, and the variance in the momentum is calculated to be: = ) The ...
The expectation value ... if ψ is an eigenfunction of ^, then the momentum eigenvalue p is the value of the particle's momentum, found by: =. For three dimensions ...
In such a case the expectation value of neither l 1 nor l 2 is a constant of motion in general, but the expectation value of the total orbital angular momentum operator L = l 1 + l 2 is. Given the eigenstates of l 1 and l 2, the construction of eigenstates of L (which still is conserved) is the coupling of the angular momenta of electrons 1 and 2.