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:

    “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)

    “Children of the same family, the same blood, with the same first associations and habits, have some means of enjoyment in their power, which no subsequent connections can supply; and it must be by a long and unnatural estrangement, by a divorce which no subsequent connection can justify, if such precious remains of the earliest attachments are ever entirely outlived.”
    —Jane Austen (1775–1817)

    “The basic difference between classical music and jazz is that in the former the music is always greater than its performance—Beethoven’s Violin Concerto, for instance, is always greater than its performance—whereas the way jazz is performed is always more important than what is being performed.”
    —André Previn (b. 1929)

    “It is only the impossible that is possible for God. He has given over the possible to the mechanics of matter and the autonomy of his creatures.”
    —Simone Weil (1909–1943)