We develop a modified Hamiltonian-Jacobi theory of classical mechanics following the early work of Van Vleck. This modified Hamiltonian-Jacobi theory, or quasi-classical theory, permits us to exhibit in classical mechanics many features that in the past have been exclusively associated with quantum mechanics. We deal with classical wave functions, classical operators, classical "eigenvalue" equations, a classical "sum over paths" formulation of classical mechanics, and with classical creation and destruction operators. Following Van Vleck, one can derive the W T KB approximate solutions to the Schrodinger equation from the solutions of the classical Hamilton-Jacobi equation. If we apply the methods of Keller to the nonrelativistic and relativistic Kepler problem, we derive eigenvalues from the requirement of singlevaluedness imposed on the WKB solutions. It turns out that the energy eigenvalues are those given by the Schrodinger equation and the Klein-Gordon equation, respectively. In the particular case of the harmonic oscillator there exists a canonical transformation which transforms the quasi-classical equation into an exact equation of quantum mechanics. We conjecture that if the WKB approximation and the Schrodinger equation predict the same eigenvalues, then there always exists a canonical transformation which transforms the quasi-classical equation into the corresponding Schrodinger equation. Finally we derive the quasiclassical equations in momentum space.
We present a mathematical model for the motion of a bacterial population in prescribed attractant or repellent gradients. The model is suggested by the observations of Mesibov et al. (1973, J. Gen. Physiol. 62:203) and Brown and Berg (1974, Proc. Natl. Acad. Sci. U.S.A. 71:1388) who found that the sensitivity of the chemotactic response depends on the concentration of attractant. Predictions of the theory are in general agreement with the experiments of Dahlquist et al. (1972, Nat. New Biol. 236:120) and of Mesibov et al. on populations of motile bacteria in fixed attractant gradients. Additional tests of the model are proposed.
From the WKB solutions to the squared Dirac equation we derive the classical trajectories and spin precessions originally postulated by Bargmann, Michel, and Telegdi for charged spin- § particles. Identical equations of motion are obtained from the WKB solutions to the Dirac equation.
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