We investigate how wave functions evolve with time in the harmonic oscillator. We first review the periodicity properties over each multiple of a quarter of the classical oscillation period. Then we show that any wave function can be simply transformed so that its centroid, defined by the expectation values of position and momentum, remains at rest at the center of the oscillator. This implies that we need only consider the evolution of this restricted class of wave functions; the evolution of all others can be reduced to these. The evolution of the spread in position Δx and momentum Δp throws light on energy and uncertainty and on squeezed and coherent states. Finally, we show that any wave function can be transformed so that Δx and Δp do not change with time and that the evolution of all wave functions can easily be found from the evolution of those at rest at the origin with unchanging Δx and Δp.
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April 2016
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April 01 2016
The evolution of oscillator wave functions
Mark Andrews
Mark Andrews
a)
Department of Quantum Science,
Australian National University
, ACT2601, Australia
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a)
Electronic mail: mark.andrews@anu.edu.au
Am. J. Phys. 84, 270–278 (2016)
Article history
Received:
August 13 2015
Accepted:
January 18 2016
Citation
Mark Andrews; The evolution of oscillator wave functions. Am. J. Phys. 1 April 2016; 84 (4): 270–278. https://doi.org/10.1119/1.4941246
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