10182

# Evolution of a Gaussian Wave Packet

A Gaussian wave packet centered around at time with an average initial momentum can be represented by the wavefunction . (For convenience, we take .) The solution of the free-particle Schrödinger equation with this initial condition works out to . The probability density is then given by , where , shown as a black curve. The wave packet remains Gaussian as it spreads out, with its center moving to , thereby following the classical trajectory of the particle. The corresponding momentum probability distribution is given by , shown in red. The rms uncertainties are given by , , which is independent of . This is consistent with the fact that is a constant of the motion for a free particle.
Thus, with ℏ put back in, the uncertainty product is given by , in accord with Heisenberg's uncertainty principle. At , the Gaussian probability distribution represents a minimum uncertainty wave packet with , but the product increases when .
In this Demonstration, you can drag the time slider to simulate the simultaneous time evolution of the probability and momentum distributions. Note that the distribution broadens with time while the distribution maintains its original width. The numerical values of , , and are illustrative only and have no absolute significance.

### DETAILS

Snapshot 1-3: the position probability distribution broadens with increasing , while the momentum distribution moves with but retains its original width
Reference: S. M. Blinder, "Evolution of a Gaussian Wavepacket," Am J Phys 36(6), 1968 pp. 525–526.

### PERMANENT CITATION

Contributed by: S. M. Blinder
With corrections by: Stefano Rigolin and Michael Trott
 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.