# Evolution of a Gaussian Wave Packet

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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.

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Contributed by: S. M. Blinder (March 2011)

With corrections by: Stefano Rigolin and Michael Trott

Open content licensed under CC BY-NC-SA

## Snapshots

## 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