User:GravitonsAndGraviolis/Expectation value (quantum mechanics)

In quantum theory, it is also possible for an operator to have a non-discrete spectrum, such as the position operator $$X$$ in quantum mechanics. This operator has a completely continuous spectrum, with eigenvalues and eigenvectors depending on a continuous parameter, $$x$$. Specifically, the operator $$X$$ acts on a spatial vector $$| x \rangle$$ as $$X | x \rangle = x |x\rangle$$. In this case, the vector $$\psi$$ can be written as a complex-valued function $$\psi(x)$$ on the spectrum of $$X$$ (usually the real line). This is formally achieved by projecting the state vector $$| \psi \rangle$$ onto the eigenvalues of the operator, as in the discrete case $ \psi(x) \equiv \langle x | \psi \rangle$. It happens that the eigenvectors of the position operator form a complete basis for the vector space of states, and therefore obey a closure relation:

$$ \int |x \rangle \langle x| dx \equiv \mathbb{I}$$

The above may be used to derive the common, integral expression for the expected value (4), by inserting identities into the vector expression of expected value, then expanding in the position basis:

Where the orthonormality relation of the position basis vectors $$\langle x | x' \rangle = \delta(x - x')$$, reduces the double integral to a single integral. The last line uses the modulus of a complex valued function to replace $$\psi^*\psi$$ with $$|\psi|^2$$, which is a common substitution in quantum-mechanical integrals.

The expectation value may then be stated, where x is unbounded, as the formula

(4)     $$ \langle X \rangle_\psi = \int_{-\infty}^{\infty} \, x \, |\psi(x)|^2 \, dx$$.

A similar formula holds for the momentum operator $$P$$, in systems where it has continuous spectrum.