![How does a Bohm Particle Localise?
Next Steps
Introduction
The De Broglie-Bohm or‘Pilot wave’interpretation of quantum mechanics [1] has long been completely overshadowed by
the standard‘Copenhagen’interpretation of quantum mechanics. In this project, a program designed to re-interpret the
one dimensional‘Anderson’model [2] and‘Aubry-Andre’model [3] within the context of de Broglie-Bohm has been mostly
fixed, and initial steps have been made to create new methods based on the de Broglie-Bohm interpretation to interpret
these systems.
The Current Stage
The program appears to correctly produce trajectories
and wavefunctions, from testing the program with
simple systems such as the free Gaussian wave packet
case. However, the methods used appear to have a very
very high numerical inaccuracy, where the error
introduced is also asymmetric.
Figure 1 above shows some of the current plots of
trajectories (with space on the y axis, and time on the x
axis). Plots to the left have lower disorder, and plots to
the right have higher disorder.
Once the numerical inaccuracy problems with the program
are fixed, corrected trajectory plots can be produced and
analysed. Then phase diagrams and graphs of mean kinetic
energy can be produced.
It should prove interesting to see what insights these
diagrams can give to a quantum mechanical system in
general, or if any interesting features about the Anderson
model and Aubry-Andre model can be found through these
methods.
Chris Davis, Prof. Rudolf Roemer, Edoardo Carnio
Warwick Department of Physics and Centre for Scientific Computing
The de Broglie-Bohm interpretation
Figure 2 : Trajectories for the double slit experiment
under the de Broglie-Bohm interpretation. Trajectories
are calculated from v in equation (1).
The Simulation
Chebyshev propagation was used to numerically calculate
wavefunctions from the time dependent Schrodinger
equation, and finite difference methods were used to
numerically calculate the trajectories from the
wavefunctions.
When finished, the program will provide further insight
into both de Broglie-Bohm mechanics and disordered
systems.
(1)
De Broglie-Bohm theory is a nonlocal hidden variable
theory, which instead of relying on Schrodinger’s
equation and the Born rule as fundamental equations,
uses an equation of motion:
where S is the action of the system, m is the mass and v
is the velocity of the particle in question.
Figure 1
References : [1] P. R. Holland, The Quantum Theory of Motion, Cambridge University Press, 1993 [2] P. W. Anderson, Phys. Rev. 124, 41 (1961) [3] Aubry S and André G 1980 Ann. Israel Phys. Soc. 3 133
Figure 3 : Trajectories for the Anderson Localisation case,
with small disorder, as calculated by the simulation. Space
is on the y-axis, and time on the x-axis. The particle’s initial
distribution is Gaussian, and is travelling in the positive
(upward) direction. The bottom diagram shows the phase
of the wavefunction plotted against trajectories.](https://image.slidesharecdn.com/6ef37e04-bf50-4d7b-8dbc-118a1b78af17-151216212652/85/chris_davis_poster-1-320.jpg)
chris_davis_poster
- 1. How does a Bohm Particle Localise? Next Steps Introduction The De Broglie-Bohm or‘Pilot wave’interpretation of quantum mechanics [1] has long been completely overshadowed by the standard‘Copenhagen’interpretation of quantum mechanics. In this project, a program designed to re-interpret the one dimensional‘Anderson’model [2] and‘Aubry-Andre’model [3] within the context of de Broglie-Bohm has been mostly fixed, and initial steps have been made to create new methods based on the de Broglie-Bohm interpretation to interpret these systems. The Current Stage The program appears to correctly produce trajectories and wavefunctions, from testing the program with simple systems such as the free Gaussian wave packet case. However, the methods used appear to have a very very high numerical inaccuracy, where the error introduced is also asymmetric. Figure 1 above shows some of the current plots of trajectories (with space on the y axis, and time on the x axis). Plots to the left have lower disorder, and plots to the right have higher disorder. Once the numerical inaccuracy problems with the program are fixed, corrected trajectory plots can be produced and analysed. Then phase diagrams and graphs of mean kinetic energy can be produced. It should prove interesting to see what insights these diagrams can give to a quantum mechanical system in general, or if any interesting features about the Anderson model and Aubry-Andre model can be found through these methods. Chris Davis, Prof. Rudolf Roemer, Edoardo Carnio Warwick Department of Physics and Centre for Scientific Computing The de Broglie-Bohm interpretation Figure 2 : Trajectories for the double slit experiment under the de Broglie-Bohm interpretation. Trajectories are calculated from v in equation (1). The Simulation Chebyshev propagation was used to numerically calculate wavefunctions from the time dependent Schrodinger equation, and finite difference methods were used to numerically calculate the trajectories from the wavefunctions. When finished, the program will provide further insight into both de Broglie-Bohm mechanics and disordered systems. (1) De Broglie-Bohm theory is a nonlocal hidden variable theory, which instead of relying on Schrodinger’s equation and the Born rule as fundamental equations, uses an equation of motion: where S is the action of the system, m is the mass and v is the velocity of the particle in question. Figure 1 References : [1] P. R. Holland, The Quantum Theory of Motion, Cambridge University Press, 1993 [2] P. W. Anderson, Phys. Rev. 124, 41 (1961) [3] Aubry S and André G 1980 Ann. Israel Phys. Soc. 3 133 Figure 3 : Trajectories for the Anderson Localisation case, with small disorder, as calculated by the simulation. Space is on the y-axis, and time on the x-axis. The particle’s initial distribution is Gaussian, and is travelling in the positive (upward) direction. The bottom diagram shows the phase of the wavefunction plotted against trajectories.