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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 ...
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.
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.
If a translation operator ^ acts on the state | , creating a new state | then the expectation value of position for | is equal to the expectation value of position for | plus the vector . This result is consistent with what you would expect from an operation that shifts the particle by that amount.
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:
Informally, the expected value is the mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would expect to get in reality.
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 .