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(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.