GaussianWellenpaket

A Gaussian wave packet is a fundamental concept in quantum mechanics representing a localized wave. It is formed by the superposition of many plane waves, each with a slightly different wave vector. The key characteristic of a Gaussian wave packet is that its amplitude is described by a Gaussian function in both position and momentum space. This property is a direct consequence of the Heisenberg uncertainty principle, which states that a particle cannot simultaneously have a precisely defined position and momentum.

Mathematically, the wave function of a Gaussian wave packet at time t=0 can be represented by a Gaussian function. As time evolves, the packet spreads out due to the different phase velocities of its constituent plane waves. This spreading is also governed by the initial distribution in momentum. A narrower initial packet in position space implies a wider distribution in momentum space, leading to faster spreading. Conversely, a wider initial packet in position space, with a narrower momentum distribution, will spread more slowly.

The Gaussian wave packet is a useful theoretical tool because it is exactly solvable and provides a good approximation for many physical situations. For instance, it is often used to model the behavior of a free particle or a particle in a potential. Its predictable spreading and well-defined properties make it a cornerstone for understanding wave-particle duality and the dynamics of quantum systems. The concept is also relevant in fields beyond quantum mechanics, such as signal processing and optics, wherever localized wave phenomena are studied.