Arcadi Santamaria (Arcadi.Santamaria@uv.es)
This is a demonstration of the numerical computation of path integrals in quantum mechanics. It is based on the paper A Statistical Approach to Quantum Mechanics, by M. Creutz and B. Freedman, Annals of Physics 132, 427 (1981).
The path integral formula for the propagator is
Taking , and with
this can be used to compute the lowest energy levels and their wave functions
Although, in principle there is no integration on it is
convenient to include it also in the integration since, due to the boundary
conditions, all are completely equivalent. One does it by introducing
an integration over of
Once we have the wave function we can compute the expectation value of the Hamiltonian
which should give us the energy of the lowest bound state, . For
this we use the virial theorem
The applet gives you in the upper panel the normalized wave function as a histogram (to be compared with a Gaussian which is the wave function of the harmonic oscillator ( and ). The middle panel gives the energy as a function of the number of iterations. The lower panel gives a sample of the paths (as a function of time in units of ) that contribute more to the path integral formula. All expressions are for and .
The button ``Potential'' opens a window with the potential and allows you to change it (clicking on the ``hand'' in the lower right corner). By changing it the simulation is restarted.
Yo can also ``pause'' and ``continue'' the simulation and print the panels.
All the simulations are performed for a lattice of sites and with . The plots are updated only every iterations and the initial conditions for the iterations are also restarted randomly every iterations.
For ( and ) you can see the nice Gaussian of the harmonic oscillator with .
For ( and , try for instance and , ) there are two degenerate minima. Classically the particle stays in one of the two minima. In quantum mechanics, however, we can see that in the typical contributing paths the particle stays most of the time in one of the minima but from time to time it jumps to the other minimum (these are the so-called instantonic solutions) where it stays also some time. This allows the particle to stay half of the time in each of the two minima generating a wave function completely symmetric.
You will need the Java plugin (version 1.4 or above) to run the Path Integral applet (Click on the "Launch Applet" button to open it)
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Copyright © 1997, 1998, 1999, Ross Moore, Mathematics Department, Macquarie University, Sydney.
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The translation was initiated by Arcadi Santamaria on 2004-12-09