Probability Current - Connection With Classical Mechanics

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:

\begin{align} \bold{j} & = \frac{\hbar}{2mi}\left(\Psi^* \bold{\nabla} \Psi - \Psi \bold{\nabla}\Psi^*\right) \\ & = \frac{\hbar}{2mi}\left(R e^{-i S / \hbar } \bold{\nabla}R e^{i S / \hbar } - R e^{i S / \hbar } \bold{\nabla}R e^{-i S / \hbar }\right) \\ & = \frac{\hbar}{2mi}\left \\ \end{align}

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:

    “Morality becomes hypocrisy if it means accepting mothers’ suffering or dying in connection with unwanted pregnancies and illegal abortions and unwanted children.”
    —Gro Harlem Brundtland (b. 1939)

    “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 (1900–1980)

    “Compare the history of the novel to that of rock ‘n’ roll. Both started out a minority taste, became a mass taste, and then splintered into several subgenres. Both have been the typical cultural expressions of classes and epochs. Both started out aggressively fighting for their share of attention, novels attacking the drama, the tract, and the poem, rock attacking jazz and pop and rolling over classical music.”
    —W. T. Lhamon, U.S. educator, critic. “Material Differences,” Deliberate Speed: The Origins of a Cultural Style in the American 1950s, Smithsonian (1990)

    “the moderate Aristotelian city
    Of darning and the Eight-Fifteen, where Euclid’s geometry
    And Newton’s mechanics would account for our experience,
    And the kitchen table exists because I scrub it.”
    —W.H. (Wystan Hugh)