Quantum States As Differential Forms
Main article: Wavefunction See also: Differential formsDifferential 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.
Read more about this topic: Quantum Geometry
Famous quotes containing the words quantum, states, differential and/or forms:
“A personality is an indefinite quantum of traits which is subject to constant flux, change, and growth from the birth of the individual in the world to his death. A character, on the other hand, is a fixed and definite quantum of traits which, though it may be interpreted with slight differences from age to age and actor to actor, is nevertheless in its essentials forever fixed.”
—Hubert C. Heffner (1901–1985)
“The people of the United States have been fortunate in many things. One of the things in which we have been most fortunate has been that so far, due perhaps to certain basic virtues in our traditional ways of doing things, we have managed to keep the crisis of western civilization, which has devastated the rest of the world and in which we are as much involved as anybody, more or less at arm’s length.”
—John Dos Passos (1896–1970)
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)
“The moment a person forms a theory, his imagination sees in every object only the tracts which favor that theory.”
—Thomas Jefferson (1743–1826)