An * operator* in a vector space maps a set of vectors to another set of vectors (or transforms one function into another function), where . For example, the operator transform the function into :

* Linear operators* have the following properties:

Two operators commonly encountered in quantum mechanics are the ** position** and

**operators. To construct these operators, we refer to probability theory, where the expectation value of the position of a particle in a 1-D box of length is**

*linear momentum*where is the probability of observing the particle at a particular position between and , and is the particle’s wavefunction, which is assumed to be real.

Comparing the above equation with the expression of the expectation value of a quantum-mechanical operator, .

One may infer that the linear momentum operator is . However, we must find a form of that is a function of so that we can compute . If we compare the time-independent Schrodinger equation with the total energy equation , we have .

To test the validity of and , we compute and using the 1-D box wavefunction of and check if the results are reasonable with respect to classical mechanics.

Integrating by parts, we have . In classical mechanics, the particle can be anywhere in the 1-D box with equal probability. Therefore, the average position of is reasonable.

For the linear momentum operator, we have . Since , we expect . Therefore, and are reasonable assignments of the position and linear momentum operators respectively. In 3-D, the position and linear momentum operators are:

To see the proof that the position and linear momentum operators are Hermitian, read this article.