The Quantum Hilbert Space is the Hilbert Space used to represent quantum system states. These states are acted on by Operators.

Position and Momentum basis

  • Move the position and momentum basis into their own notes

In classical mechanics we keep track of a Particle’s position and momentum directly. In quantum mechanics Particle’s are described by their Wave function. The Wave function can be expressed in different orthonormal basis. Two commonly used basis are the position basis and the momentum basis. Both of these basis are continuous, represented by the and variables respectively, and in both cases taking values in .

Properties

Equations

The results below are frequently useful when manipulating algebraic expressions related to quantum spaces

ResultComments
Representation of a Wave function in the position basis
Representation of a Wave function in the momentum basis
Transformation between the position and momentum basis
is an eigenstate of with eigenvalue
is an eigenstate of with eigenvalue