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:
“Up the airy mountain,
Down the rushy glen,
We daren’t go a-hunting
For fear of little men.”
—William Allingham (1824–1889)
“Wind goes from farm to farm in wave on wave,
But carries no cry of what is hoped to be.
There may be little or much beyond the grave,
But the strong are saying nothing until they see.”
—Robert Frost (1874–1963)
“We teach boys to be such men as we are. We do not teach them to aspire to be all they can. We do not give them a training as if we believed in their noble nature. We scarce educate their bodies. We do not train the eye and the hand. We exercise their understandings to the apprehension and comparison of some facts, to a skill in numbers, in words; we aim to make accountants, attorneys, engineers; but not to make able, earnest, great- hearted men.”
—Ralph Waldo Emerson (1803–1882)