Quantum Mechanics Term Paper, June 2002.
For my graduate Quantum Mechanics class at UCLA, I wrote the following paper on the connections between Feynman's Checkboard, Spin and the Dirac Equation, including some original thoughts on the subject. Any undergraduate with a solid background in quantum mechanics and physics should be able to digest the paper, but I have to warn you that it is pretty dense.
The abstract from the paper:
Spin is one of the fundamental observables in quantum mechanics lacking a satisfactory physical picture. Early attempts to explain the physical origins of spin are briefly discussed here. Feynman was able to demonstrate in one space dimension and one time dimension (1+1) the equivalence of a particular path integral with the one-dimensional Dirac equation, thereby presenting a simple process from which the analogue of spin in one dimension, helicity, naturally arose. This path integral became known as Feynman's Checkerboard or Chessboard, named for the appearance of possible paths on a spacetime diagram. Subsequent work by others attempted to generalize this to three space dimensions (3+1) to obtain the full Dirac equation. The remainder of this paper will concentrate on rederiving Feynman's results and describing the relation to the Dirac equation and, interestingly enough, the one-dimensional Ising model. This paper will then conclude with an original interpretation to the 1+1 dimensional checkerboard problem.
The paper is available for download as a Postscript (recommended) or PDF file. The latex source is also available (requires AASTEX macros available from AAS web site) and Figures 1 and 2.
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6/10/2002. Last updated 6/10/2002.