Connection With Classical Mechanics
The wave function can also be written in the complex exponential (polar) form:
where R and S are real functions of r and t.
Written this way, the probability density is
and the probability current is:
The exponentials and R∇R terms cancel:
Finally, combining and cancelling the constants, and replacing R2 with ρ,
If we take the familiar formula for the current:
- ,
where v is the velocity of the particle (also the group velocity of the wave), we can associate the velocity with ∇S/m, which is the same as equating ∇S with the classical momentum p = mv. This interpretation fits with Hamilton-Jacobi theory, in which
in Cartesian coordinates is given by ∇S, where S is Hamilton's principal function.
Read more about this topic: Probability Current
Famous quotes containing the words connection with, connection, classical and/or mechanics:
“Self-expression is not enough; experiment is not enough; the recording of special moments or cases is not enough. All of the arts have broken faith or lost connection with their origin and function. They have ceased to be concerned with the legitimate and permanent material of art.”
—Jane Heap (c. 18801964)
“We live in a world of things, and our only connection with them is that we know how to manipulate or to consume them.”
—Erich Fromm (19001980)
“Culture is a sham if it is only a sort of Gothic front put on an iron buildinglike Tower Bridgeor a classical front put on a steel framelike the Daily Telegraph building in Fleet Street. Culture, if it is to be a real thing and a holy thing, must be the product of what we actually do for a livingnot something added, like sugar on a pill.”
—Eric Gill (18821940)
“the moderate Aristotelian city
Of darning and the Eight-Fifteen, where Euclids geometry
And Newtons mechanics would account for our experience,
And the kitchen table exists because I scrub it.”
—W.H. (Wystan Hugh)