The Caged Anharmonic Oscillator in the Causal Interpretation of Quantum Mechanics

time steps

50

th

n

eigenstate

0

1

2

3

th

m

eigenstate

0

1

2

3

This Demonstration studies the caged anharmonic oscillator—fundamental for quantum field theory as well as many-body theory—in the causal interpretation of quantum mechanics developed by David Bohm. Among the large number of possible potentials for the Schrödinger equation, only a very few are analytically solvable. The potential

V=

V

HO

+

V

Re

is that of the harmonic oscillator

V

HO

=2k

2

x

2

ω

with rational frequency ratio, but additionally with a barrier term from inclusion of a repulsive force

V

Re

=

2k

2

x

[1].

The analytic solutions of the wavefunction are polynomials in the spatial variable

x

of rational or irrational order, depending on the parameter

k

. In the causal interpretation, every particle has a definite position and momentum at all times, but the trajectories are not measurable directly. The particles are guided by a quantum potential, which follows from the amplitude of the wavefunction. As an example, only a superposition of two eigenstates with their corresponding eigenenergies is chosen. In the graphic on the left, you can see the position of the particles, the squared wavefunction (blue), the potential (orange), the quantum potential (red), and the velocity (green). On the right, the graphic shows the squared wavefunction plus trajectories. The velocity, the quantum potential, and the classical potential are scaled to fit.