# What Is the Size and Shape of a Wave Packet?

## Abstract

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## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## Appendix D

## References

- Schulman, L.S. Evolution of wave-packet spread under sequential scattering of particles of unequal mass. Phys. Rev. Lett.
**2004**, 92, 210404. [Google Scholar] [CrossRef] [PubMed] - Gaveau, B.; Schulman, L.S. Reconciling Kinetic and Quantum Theory. Found. Phys.
**2020**, 50, 55–60. [Google Scholar] [CrossRef] [Green Version] - Schulman, L.S.; Schulman, L.J. Convergence of matrices under random conjugation: Wave packet scattering without kinematic entanglement. J. Phys. A
**2006**, 39, 1717–1728. [Google Scholar] [CrossRef] [Green Version] - Schulman, L.S.; Schulman, L.J. Wave packet scattering without kinematic entanglement: Convergence of expectation values. IEEE Trans. Nanotech.
**2004**, 4, 8–13. [Google Scholar] [CrossRef] - Ford, G.W.; O’Connell, R.F. Wave packet spreading: Temperature and squeezing ef- fects with applications to quantum measurement and decoherence. Am. J. Phys.
**2002**, 70, 319–324. [Google Scholar] [CrossRef] [Green Version] - Joos, E.; Zeh, H. The Emergence of Classical Properties Through Interaction with the Environment. Z. Phys. B
**1985**, 59, 223–243. [Google Scholar] [CrossRef] - Gaveau, B.; Schulman, L.S. Decoherence, the Density Matrix, the Thermal State and the Classical World. J. Stat. Phys.
**2017**, 169, 889–901. [Google Scholar] [CrossRef] [Green Version] - Tegmark, M.; Shapiro, H.S. Decoherence produces coherent states: An explicit proof for harmonic chains. Phys. Rev. E
**1994**, 50, 2538–2547. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zurek, W.H.; Habib, S.; Paz, J.P. Coherent States via Decoherence. Phys. Rev. Lett.
**1993**, 70, 1187–1190. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Reduction of non-Gaussian terms with unequal mass scattering in three dimensions.Because that the problem has an intrinsic nonlinearity, the initial coefficients were varied. Usually they were taken as random between 0 and 1, but on some occasions they were taken to be 20 times that. In all cases there was convergence to zero. The programs to establish this were combinations of symbolic manipulation and numerical evaluation.

**Figure 2.**As in Figure 1, except that this looks at quartic terms. As in the other example, all terms shrink to zero.

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**MDPI and ACS Style**

Schulman, L.S.
What Is the Size and Shape of a Wave Packet? *Symmetry* **2021**, *13*, 527.
https://doi.org/10.3390/sym13040527

**AMA Style**

Schulman LS.
What Is the Size and Shape of a Wave Packet? *Symmetry*. 2021; 13(4):527.
https://doi.org/10.3390/sym13040527

**Chicago/Turabian Style**

Schulman, Larry S.
2021. "What Is the Size and Shape of a Wave Packet?" *Symmetry* 13, no. 4: 527.
https://doi.org/10.3390/sym13040527