What is Expectation Value in quantum mechanics?

Answer 1

In probability theory, an "expectation value" is the average of all values of a measurable quantity that one would expect, if a measurement was repeated a large number of times on a given system. For example, for an unbiased coin, the expectation value for "heads" is half of all tosses.

Each measurable quantity of a quantum system has an operator that, when mathematically applied to the system, gives a value of that quantity for that system. The expectation value for that quantity, for a given quantum system, is the product of that operator on a given state of the system, times the probability of the system being in that state, integrated over all possible states of the system. A more formally stated example:

For a quantum state Ψ(x), where 'x' can vary from -∞ to ∞, and for which Q(x) is a measurable quantity, then the expectation value of Q(x) would be equal to

∫Ψ*(x)Ψ(x)Q(x)dx

integrated from x = -∞ to x = ∞

As an example, suppose we wanted the expectation value for the radial position of an electron in its '1S' state within a hydrogen atom. When doing the formal math, we find that this value exactly equals the Bohr Radius. In contrast to the Bohr Model of an atom, this expectration value does NOT state that this electron IS at this radius, only that an AVERAGE of all radial measurements of such an electron would be the Bohr Radius.

Answer 2

In quantum mechanics, the expectation value is the average value of a physical quantity obtained from a wave function. It represents the most probable outcome of measuring that quantity for a given quantum state. Mathematically, it is calculated by taking the integral of the observable operator over the wave function squared.