Quantum Geometry - Quantum States As Differential Forms

Quantum States As Differential Forms

Differential forms are used to express quantum states, using the wedge product:

where the position vector is

the differential volume element is

and x1, x2, x3 are an arbitrary set of coordinates, the upper indices indicate contravariance, lower indices indicate covariance, so explicitly the quantum state in differential form is:

The overlap integral is given by:

in differential form this is

The probability of finding the particle in some region of space R is given by the integral over that region:

provided the wave function is normalized. When R is all of 3d position space, the integral must be 1 if the particle exists.

Differential forms are an approach for describing the geometry of curves and surfaces in a coordinate independent way. In quantum mechanics, idealized situations occur in rectangular Cartesian coordinates, such as the potential well, particle in a box, quantum harmonic oscillator, and more realistic approximations in spherical polar coordinates such as electrons in atoms and molecules. For generality, a formalism which can be used in any coordinate system is useful.