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:
“My aspens dear, whose airy cages quelled,
Quelled or quenched in leaves the leaping sun,
All felled, felled, are all felled;
Of a fresh and following folded rank
Not spared, not one
That dandled a sandalled
Shadow that swam or sank
On meadow and river and wind-wandering weed-winding bank.”
—Gerard Manley Hopkins (1844–1889)
“I sometimes compare press officers to riflemen on the Somme—mowing down wave upon wave of distortion, taking out rank upon rank of supposition, deduction and gossip.”
—Bernard Ingham (b. 1932)
“The beginning of reform is not so much to equalize property as to train the noble sort of natures not to desire more, and to prevent the lower from getting more.”
—Aristotle (384–322 B.C.)