Expectation Value (quantum Mechanics) - Example in Configuration Space

Example in Configuration Space

As an example, let us consider a quantum mechanical particle in one spatial dimension, in the configuration space representation. Here the Hilbert space is, the space of square-integrable functions on the real line. Vectors are represented by functions, called wave functions. The scalar product is given by . The wave functions have a direct interpretation as a probability distribution:

gives the probability of finding the particle in an infinitesimal interval of length about some point .

As an observable, consider the position operator, which acts on wavefunctions by

The expectation value, or mean value of measurements, of performed on a very large number of identical independent systems will be given by

$\langle Q \rangle_\psi = \langle \psi | Q \psi \rangle =\int_{-\infty}^{\infty} \psi^\ast(x) \, x \, \psi(x) \, \mathrm{d}x = \int_{-\infty}^{\infty} x \, p(x) \, \mathrm{d}x$ .

The expectation value only exists if the integral converges, which is not the case for all vectors . This is because the position operator is unbounded, and has to be chosen from its domain of definition.

In general, the expectation of any observable can be calculated by replacing with the appropriate operator. For example, to calculate the average momentum, one uses the momentum operator in configuration space, . Explicitly, its expectation value is

Not all operators in general provide a measureable value. An operator that has a pure real expectation value is called an observable and its value can be directly measured in experiment.

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