Dynamics of Free Particle and Harmonic Oscillator Using Propagators
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]
ψ(x,0)
ψ(x,t)=∫ψ(x',0)K(x,x',t)dx'
For the free particle,
K
FP
1/2
m
2πiℏt
-
2
m(x-x')
2iℏt
e
while for the harmonic oscillator,
K
HO
1/2
mω
2πiℏsinωt
mω[(+)cosωt-2xx']
2
x
2
x'
2iℏsinωt
For compactness, we use units with . For the initial state, we consider the Gaussian wave packet and the rectangular pulse.
ℏ=m=ω=1
The plots, for selected cases, show the probability densities and .
2
ψ(x,0)
2
ψ(x,t)