A Foundational Principle for Quantum Mechanics
A simple foundational principle for quantum mechanics is proposed leading naturally to a complex "probability" density expressed as a sum of path integrals. The action is then derived in its relativistically-invariant form with the Lagrangian following from the principle instead of being imposed as an extra hypothesis. However, the sum is restricted to physical paths on which the particle has real vector momentum. This restriction evidently conflicts with the usual derivation of Schroedinger's equation and from it conservation of energy and probability as defined by Born.
To see if the restricted path integral sum leads to sensible predictions, the double-slit experiment is analyzed using a single-quantum transfer approximation. The result is found to be in general agreement with experiment provided we add a natural measurement hypothesis, which is motivated by the similarity of our complex probability density to Feynman's path integral. According to our hypothesis, the probability of recording a particle at some position is proportional to the square of the complex probability density, integrated over time. This calculation does predict, however, that the particle energy is not necessarily conserved. The maximum change (near the first side peak at the screen) is a loss that is proportional to the square of Planck's constant when expressed in terms of the slit geometry and the mass of the particle. The fractional change is largest for light particles at low energy and might be observable with electrons at around 10 eV.
According to the principle, the Fourier components of the potential are independently responsible for interactions. This means that the contribution from each path is already proportional to the Fourier transform of the potential at the wave vector corresponding to the momentum transferred. Therefore the first minimum in the diffraction pattern does not depend on cancellation of out of phase contributions from different paths. In the non-relativistic limit, the dominant contributions come from initial trajectories along which the particle can scatter before reaching the plane of the slits and then pass through one of the slits on its way to the target position. These contributions are only significant in a small range of arrival times, and as a result the momentum transfer is almost independent of the initial trajectory. Then the Fourier transform becomes a common factor in the sum over initial trajectories and, after squaring, reproduces the familiar interference pattern.
The image above illustrates the range of positions from which a particle initially moving parallel to the z-axis with velocity v0 can change direction and reach a target position after passing through an opening in the x-y plane. If only one of the slits is open, as shown here, we can say the bright and dark lines in the interference pattern disappear because the Fourier transform of the potential has changed accordingly. This should help to resolve the mystery of the persistence of the interference fringes when the incident particles are very widely spaced.
For more details, please see the preliminary write-up .Towards a Foundational Principle for Quantum Mechanics
(first posted January 12, 2009
last updated June 9, 2009)
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