Quantum Hamilton dynamics in a quantum dot

The lack of equations of motion in quantum mechanics has prohibited the analysis of quantum dots from a viewpoint of nonlinear dynamics. For a complete description of the electronic motion in a quantum dot, we need a new method that can provide not only the particle behavior of an electron but also its wave behavior. In the present paper, quantum Hamilton mechanics is shown to possess the desired ability of manifesting the wave-particle duality of electrons moving in a quantum dot. Solutions of nonlinear Hamilton equations reveal the existence of electronic standing waves in a quantum dot. A commensurability condition is derived from the nonlinear Hamilton equations to characterize all the critical magnetic fields yielding standing waves, whose occurrence is found to be accompanied by a jump in the electronic resistance. The comparison with the experimental data shows that the predicted locations of the critical magnetic fields match closely with the peaks of the measured magneto-resistance.