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NON-DISPERSIVE WAVEPACKET SOLUTIONS OF THE SCHRODINGER EQUATION

Shaun N. Mosley

Published Nov 21, 2012• 2 citations
OpenAlex

Abstract

The free Schrodinger equation has constant velocity wavepacket solutions ψv of the form ψ(r − vt). These solutions are eigenvectors of a velocity operator ˜v which is symmetric in a positive definite scalar product space. We discuss whether these ψv can act as basis states rather than the usual plane wave solutions.

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The free Schrödinger equation has localized constant velocity solutions ψ v of the form ψ = f ( r − v t ) e iλ t with λ being a constant. These solutions are eigenvectors of a momentum operator such that . The wavepacket, while not normalizable, is both localized and in a definite momentum state. The ψ v are orthogonal in the inner product space ⟨ϕ| r 2 |ψ⟩, and the operator is symmetric therein. We discuss whether these ψ v can act as basis states rather than the usual plane waves.
The spreading of a one-dimensional wavepacket solution of Schrödinger’s equation is related to the diffraction of light, as can be verified by considering the three-dimensional spreading of a wavepacket for an arbitrary dispersion relation. This investigation uncovers a special property of Schrödinger’s equation for a free particle: A wavepacket with initial spherical symmetry will preserve this symmetry in all Galilean reference frames. This property leads to a derivation of de Broglie’s postulate that wave number is proportional to momentum (or velocity). The application to non-Gaussian wavepackets and to Fraunhoffer diffraction also is discussed.
Motivated by the work of Berry and Balazs and Greenberger on the 1-D Schrödinger equation, we have investigated a class of nonspreading solutions to the 3-D Schrödinger equation involving accelerating Airy envelopes. These solutions are characterized by an asymmetric structure, in contrast to recently derived spherically symmetric packets moving with constant velocities. The field of a characteristic Airy packet extends in an oscillatory fashion behind its peak amplitude, while it quickly disappears in front of the packet’s center. A particle modeled by such a packet seems to leave a wake of its field behind as it accelerates in a certain direction. On the other hand, a wave packet moving with a constant velocity has a field which is symmetrically distributed in all directions. Our work on Airy-type solutions to the 3-D Schrödinger equation has led us also to analogous solutions for the 3-D scalar wave equation.