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(x−B 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(x−B ³ t ²) + (p/B − tB 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:
“With all thy dazzling other ones,
In airy dalliance,
Precipitate in love....”
—Robert Frost (18741963)
“Justice was done, and the President of the Immortals, in Æschylean phrase, had ended his sport with Tess. And the dUrberville knights and dames slept on in their tombs unknowing. The two speechless gazers bent themselves down to the earth, as if in prayer, and remained thus a long time, absolutely motionless: the flag continued to wave silently. As soon as they had strength they arose, joined hands again, and went on.
The End”
—Thomas Hardy (18401928)
“Every philosophy is tinged with the colouring of some secret imaginative background, which never emerges explicitly into its train of reasoning.”
—Alfred North Whitehead (18611947)