Free-Particle Schrödinger Equation: Wave Packets

G. F. Torres del Castillo, A. S. De la Paz Cota

We show that the solutions of the Schrödinger equation for a free particle are related in a simple manner with the solutions of the Schrödinger equation for a particle in a uniform force field. Making use of this relation we readily obtain the so-called Airy wave packets and the propagator for a particle in a uniform force field.

Dieter Schuch

W. van Dijk, F. Masafumi Toyama, Sjirk Jan Prins +1 more

We show that one can obtain analytic solutions of the time-dependent Schrödinger equation that are more complex than the well-known oscillating coherent wave packet. Such Hermite-Gaussian or initially square wave packets exist for a free particle or for one subject to the harmonic oscillator potential. In either case, the Hermite-Gaussian packets retain their nodal structure even after long times. There is a single class of exact solutions for the system with oscillator constant K > 0, K = 0, or K < 0, leading to wave functions for the harmonic oscillator, the free particle, and the inverted oscillator, respectively.

Tian-Hai Zeng

Preparing a particle in a superposition or a wave packet of eigenstates of a physical quantity is to let it interact with a large object. The composite system composed of the particle and the large object evolves into an entangled state. When the state of the large object is considered to be approximately unchanged, the entangled state can be approximately considered as a product state, and then the particle is prepared in an approximate superposed state. We consider the Schrodinger equation for a composite system with interactions between subsystems as a fundamental postulate and a single particle’s Schrodinger equation must be approximately obtained from it. We argue that superposition of states exists only in composite systems. Interaction exchanging some quantities between subsystems makes conservation laws strictly hold, and no wave packet of a free particle yields. With this point, we can also understand the double-slit experiment and the tunnel phenomenon.