Quantum particle in a V-shaped well of arbitrary asymmetry. Brownian motors

The dependence of the energy spectrum of a quantum particle in an infinite V-shaped potential well on its asymmetry parameter is analyzed. The relationship between the occurrence of this dependence and different depths of quantum particle penetration into classically forbidden regions under the well walls is shown. In particular, an estimate of the energy spectrum using the Bohr–Sommerfeld quantization rules, which does not take into account the particle penetration into subbarrier regions, shows that there is no dependence on the well asymmetry parameter. An exception is the transition to the case of a vertical wall, when the semiclassical approach is characterized by a special boundary condition. The results are illustrated by the dependence of the tunnel current of a Brownian motor (ratchet) with an applied fluctuating force on the asymmetry of the sawtooth potential. With a certain selection of ratchet parameters, considering zero-point oscillations describes the occurrence of a particle flow in the direction opposite to that obtained without such consideration.