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:
“We are little airy creatures,
All of different voice and features:
One of us in glass is set,
One of us you’ll find in jet,
T’other you may see in tin,
And the fourth a box within;
If the fifth you should pursue,
It can never fly from you.”
—Jonathan Swift (1667–1745)
“The city is loveliest when the sweet death racket begins. Her own life lived in defiance of nature, her electricity, her frigidaires, her soundproof walls, the glint of lacquered nails, the plumes that wave across the corrugated sky. Here in the coffin depths grow the everlasting flowers sent by telegraph.”
—Henry Miller (1891–1980)
“Perfect present has no existence in our consciousness. As I said years ago in Erewhon, it lives but upon the sufferance of past and future. We are like men standing on a narrow footbridge over a railway. We can watch the future hurrying like an express train towards us, and then hurrying into the past, but in the narrow strip of present we cannot see it. Strange that that which is the most essential to our consciousness should be exactly that of which we are least definitely conscious.”
—Samuel Butler (1835–1902)