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Dynamics of Free Particle and Harmonic Oscillator Using Propagators

system
free particle
harmonic oscillator
ψ(x,0)
Gaussian
rectangular pulse
parameter α
1
time t
0.1
The time evolution of a one-dimensional quantum system from an initial state
ψ(x,0)
can be represented, in terms of the propagator, by [1, 2]
ψ(x,t)=ψ(x',0)K(x,x',t)dx'
.
For the free particle,
K
FP
(x,x',t)=
1/2
m
2πit
-
2
m(x-x')
2it
e
,
while for the harmonic oscillator,
K
HO
(x,x',t)=
1/2
mω
2πisinωt
exp-
mω[(
2
x
+
2
x'
)cosωt-2xx']
2isinωt
.
For compactness, we use units with
=m=ω=1
. For the initial state, we consider the Gaussian wave packet and the rectangular pulse.
The plots, for selected cases, show the probability densities
2
ψ(x,0)
and
2
ψ(x,t)
.
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