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Quantum Scattering With Pilot Waves

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

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Quantum Scattering With Pilot Waves

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  1. Quantum Scattering With Pilot Waves Going With the Flow

  2. 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

  3. 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)

  4. The Basics Schrodinger’s equation Probability current

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

  6. 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.)

  7. 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.

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

  9. 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.

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

  11. 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.

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

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