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Dynamics of Free Particle and Harmonic Oscillator Using Propagators
The time evolution of a one-dimensional quantum system from an initial state
can be represented, in terms of the propagator, by [1, 2]
.
For the free particle,
,
while for the harmonic oscillator,
.
For compactness, we use units with
. For the initial state, we consider the Gaussian wave packet and the rectangular pulse.
The plots, for selected cases, show the probability densities
and
.
Contributed by:
S. M. Blinder
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For the free particle with initial Gaussian wave packet
, we find
.
For initial rectangular pulse
,
.
For the harmonic oscillator with initial Gaussian wave packet,
.
(For
, this reduces to the time-dependent ground-state eigenfunction
.)
For initial rectangular pulse,
.
References
[1] R. P. Feynman and A. R. Hibbs,
Quantum Mechanics and Path Integrals
, New York: McGraw-Hill, 1965.
[2] Wikipedia. "Propagator." (Feb 24, 2020)
en.wikipedia.org/wiki/Propagator
.
RELATED LINKS
Spread of a Gaussian Wave Packet with Time
(
Wolfram Demonstrations Project
)
PERMANENT CITATION
S. M. Blinder
"
Dynamics of Free Particle and Harmonic Oscillator Using Propagators
"
http://demonstrations.wolfram.com/DynamicsOfFreeParticleAndHarmonicOscillatorUsingPropagators/
Wolfram Demonstrations Project
Published: February 26, 2020
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