Wave Packet - The Airy Wave Train

The Airy Wave Train

In contrast to the above Gaussian wavepacket, it has been observed that a particular wavefunction based on Airy functions, propagates freely without dispersion, maintaining its shape. It accelerates undistorted in the absence of a force field: ψ=Ai(B(xB ³ t ²)) exp(iB ³ t (x−2B ³ t ²/3)). (For simplicity, ħ=1, m=1/2, and B is a constant.)

Nevertheless, Ehrenfest's theorem is still valid in this force-free situation, because the state is both non-normalizable and has an undefined (infinite) ⟨x⟩ for all times. (To the extent that it can be defined, ⟨p⟩ =0 for all times, despite the apparent acceleration of the front.)

In phase space, this is evident in the pure state Wigner quasiprobability distribution of this wavetrain, whose shape in x and p is invariant as time progresses, but whose features accelerate to the right, in accelerating parabolas B(xB ³ t ²) + (p/BtB ²)² = 0,

Note the momentum distribution obtained by integrating over all x is constant.

Read more about this topic:  Wave Packet

Famous quotes containing the words airy, wave and/or train:

    But this rough magic
    I here abjure, and when I have required
    Some heavenly music—which even now I do—
    To work mine end upon their senses that
    This airy charm is for, I’ll break my staff,
    Bury it certain fathoms in the earth,
    And deeper than did ever plummet sound
    I’ll drown my book.
    William Shakespeare (1564–1616)

    Down the blue night the unending columns press
    In noiseless tumult, break and wave and flow,
    Rupert Brooke (1887–1915)

    Constant revolutionizing of production ... distinguish the bourgeois epoch from all earlier ones. All fixed, fast-frozen relations, with their train of ancient and venerable prejudices are swept away, all new-formed ones become antiquated before they can ossify. All that is solid melts into air, all that is holy is profaned, and man is at last compelled to face with sober senses, his real conditions of life, and his relations with his kind.
    Karl Marx (1818–1883)