Probability Current - Motivation - Continuity Equation For Quantum Mechanics

Continuity Equation For Quantum Mechanics

The definition of probability current and Schrödinger's equation can be used to derive the continuity equation, which has exactly the same forms as those for hydrodynamics and electromagnetism:

where the probability density is defined as


If one were to integrate both sides of the continuity equation with respect to volume, so that

then the divergence theorem implies the continuity equation is equivalent to the integral equation

where the V is any volume and S is the boundary of V. This is the conservation law for probability in quantum mechanics.

In particular, if Ψ is a wavefunction describing a single particle, the integral in the first term of the preceding equation (without the time derivative) is the probability of obtaining a value within V when the position of the particle is measured. The second term is then the rate at which probability is flowing out of the volume V. Altogether the equation states that the time derivative of the chance of the probability of the particle being measured in V is equal to the rate at which probability flows into V.

Read more about this topic:  Probability Current, Motivation

Famous quotes containing the words mechanics, quantum, continuity and/or equation:

    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)

    The receipt to make a speaker, and an applauded one too, is short and easy.—Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.
    Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    There is never a beginning, there is never an end, to the inexplicable continuity of this web of God, but always circular power returning into itself.
    Ralph Waldo Emerson (1803–1882)

    Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.
    Anna Quindlen (b. 1952)