Quantum Orbits of a Particle in Spherical Coordinates in a Three-Dimensional Harmonic Oscillator Potential
Quantum Orbits of a Particle in Spherical Coordinates in a Three-Dimensional Harmonic Oscillator Potential
The isotropic three-dimensional harmonic oscillator in spherical polar coordinates is described by the Schrödinger equation
-ψ+ψ=iℏψ
2
ℏ
2
m
p
2
∇
1
2
m
p
2
ω
2
r
∂
∂t
in atomic units [1], here, such that . The normalized solution is separable:
ℏ==ω=1
m
p
ψ
nlm
R
nl
Y
lm
with the radial function
R
nl
N
nl
l
r
-2
2
r
e
l+1/2
L
n
2
r
and with the normalization constant
N
nl
2n!
n+l+!
1
2
where is an associated Laguerre polynomial, is a spherical harmonic and is a normalization constant.
L
Y
N
In the de Broglie–Bohm (or causal) interpretation of quantum mechanics [2, 3], the particle position and momentum are well defined and the motion can be described by continuous evolution according to the time-dependent Schrödinger equation.
The dynamic behavior for such a system is quite complex. Some of the curves are closed and periodic, while others are quasi-periodic. Chaos is generated when an orbit approaches an unstable region in the neighborhood of a nodal point or nodal line [4] where the wavefunction vanishes and the trajectories apparently become strongly accelerated. Here, chaotic motion means the exponential divergence of initially neighboring trajectories. The initial positions have to be chosen carefully because of the singularities in the velocities and the resulting large oscillations, which can lead to very unstable trajectories. The quantum motion originates from the relative phase of the total wavefunction, which has no analog in classical particle mechanics. Further investigation to capture the full dynamics of the system is necessary.
The graphics show three-dimensional contour plots of the squared wavefunction (if enabled) and two initially neighboring possible trajectories of one particle. Black spheres mark the initial positions of the two quantum particles and green spheres the actual positions.