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WOLFRAM|DEMONSTRATIONS PROJECT

Bohm Trajectories for Quantum Particles in a Time-Dependent Linear Potential

​
time steps
5
field strength
-1
B
0
1
C
0
1
initial density
0.2
The time-dependent probability density associated with this Gaussian wave packet is Gaussian for all times. The de Broglie–Bohm formulation of quantum mechanics, also known as the quantum theory of motion, is a single-valued theory in configuration space with one possible velocity
v=
p
m
(
p
is momentum) for a given position
x
at time
t
. A point particle follows a trajectory given by the equation of motion. In this Demonstration the time-dependent potential is assumed to be
V=f(t)x=Ktcos(t)x
. Therefore,
ψ(x,t)
describes a normalized Gaussian wave packet that is initially centered at
-
C
0
B
0
. The shape of the wave packet is not changed by the external force. The particle trajectories are obtained via the velocity field
v=
1
m
∂S
∂x
(adopting
ℏ=m=
A
0
=1
), where
S
is the phase of the wave function in the eikonal form
ψ=R
S
ℏ
e
. Their time evolution can be given in closed form:
x
QM
(t)=-
C
0
B
0
+2Ksin(t)-Ktcos(t)+
c
1
4
2
a
2
B
0
2
(1-
B
0
t)
+1
, where
a
,
B
0
, and
C
0
are arbitrary real constants, and where
c
1
, the integration constant, is used to estimated the positions of the initial particles. For the classical case the equation of motion can be solved analytically from Newton's second law
F=
∇
x
V=m
2
∂
x
Cl
∂
2
t
:
x
Cl
(t)=
x
0
+2Ksin(t)-Ktcos(t)+vt
, which is considered with the quantum motion. The quantum particle dynamics is a sum of classical terms (with
x
0
=-
C
0
B
0
, but without the
vt
-term) and a term due to the spreading of the packet. When the particle lies initially at the center of the squared wavefunction (
c
1
=0
) the motion becomes classical. You can calculate the wavefunction, the gradient of the phase (
S
x
=
∂S
∂x
), and the analytic solution of the quantum motion for any arbitrary time-dependent function
f(t)
, if the terms are integrable. The graphics show the squared wavefunction and the trajectories on the right, and the position of the particles, the squared wavefunction (blue), and the quantum potential (red) on the left.
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