Details

A Gaussian wave packet at time can be represented by

,

where is the initial width of the wave packet and is the momentum.

At a later time , the wave packet evolves with

,

where is a constant, is the mass of the particle, and is a parameter that determines the corresponding width of the wave packet ( is the actual width). The parameter is given by

Clearly, the width increases with time , as the wave packet spreads. For simplicity, and have been set equal to unity.

The probability amplitude is a Gaussian function centered about the point .

The wave packet maintains a Gaussian shape, with changing centroid and width.

Snapshot 1: small initial width implies faster spread

Snapshot 2: initially broad wave packet spreads out more slowly

In the limiting case, of a wave packet initially equal to a Dirac delta function, it immediately transforms into an infinite plane wave.

Reference

[1] H. C. Verma, Quantum Physics, 2nd ed., Bhopura, Ghaziabad, India: Surya Publications, 2009.