Quantum Scattering With Pilot Waves Going With the Flow

Quantum Scattering With Pilot Waves Introduction – history Introduction – physics Scattering theory -- Time-independent formalism -- Time-dependent formalism -- An Exactly-Soluble Model Calculation procedure Results -- The R Function -- The S Function -- Trajectories -- Differential Cross Section Conclusions

History Louis deBroglie (1927) David Bohm (1953) David Bohm, The Undivided Universe (1993) John Bell, Speakable and Unspeakable in Quantum Mechanics (1985) Peter Holland, The Quantum Theory of Motion (1996) Sheldon Goldstein, Quantum Theory Without Observers, Physics Today (1998)

The Basics Schrodinger’s equation Probability current

Pilot wave interpretation This defines the real functions R and S. They are related to the current density as follows: is interpreted as the equation of motion for the particle whose coordinate is where

Recipe for Calculating Trajectories Find the time-dependent wave function by solving Schrodinger’s equation. Integrate the equations of motion. Distance and time are measured in units of and (Free particles have v=1.)

Alternative Formulation The potential V is the same potential that appears in Schrodinger’s equation. Q is the quantum potential. Note that because of the R in the denominator, Q can be large even when R is vanishingly small.

Time-independent Scattering Theory Asymptotic wave function Scattered probability current Differential cross section

Time-dependent Formalism is a gaussian wave packet centered at as before. Note: The approximations are difficult to quantify. The result seems to depend on the shape of the wave packet. It’s still an asymptotic theory.

Solve the time-independent Schrodinger equation. where The solutions will have the form is a function ofand Exact time-dependent model

Make it time-dependent where is the gaussian momentum-space wave function. The angular integrations can be done exactly leaving one numeric integration over q in the vicinity of k.

Choose Starting Coordinates Step forward in time using Runge-Cutta Integration Computational Procedure Start a new trajectory