By Ángel S. Sanz
Trajectory-based formalisms are an intuitively attractive approach of describing quantum methods simply because they enable using "classical" strategies. starting as an introductory point appropriate for college students, this two-volume monograph offers (1) the basics and (2) the functions of the trajectory description of easy quantum tactics. This moment quantity is focussed on easy and uncomplicated purposes of quantum strategies resembling interference and diffraction of wave packets, tunneling, diffusion and bound-state and scattering difficulties. The corresponding research is performed in the Bohmian framework. through stressing its interpretational features, the ebook leads the reader to an alternate and complementary technique to greater comprehend the underlying quantum dynamics.
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Additional info for A Trajectory Description of Quantum Processes. II. Applications: A Bohmian Perspective
1) and, in particular, through the associated Bohmian trajectories . , x0 = 0). 10) can be expanded as an even-power series of . 12). For t/2mσ02 < 1, this expression becomes ∞ x(t) = x(0) + xt + (−1)(n−1) n=1 t (2n − 3)!! n 2 · n! 2mσ02 2n x(0). 40) As expected, the two first terms in the right-hand side of this expression are exactly the same that one would expect from a classical trajectory; note that the first term arises precisely when there is no spreading. However, the presence of the dependent (third) term makes that a divergence with respect to the classical motion is observable: it leads to a hyperbolic spreading of the trajectories.
60). Thus, the corresponding quantum mechanical process is not a Markovian process in the customary sense of the term [57, 58]. 66), since it is independent of the surface temperature. 68). 69), while the sum over n plays an important role only at very low, but non-vanishing surface temperatures. 68) becomes unity and F (γ t; 0) displays a different time-dependence. 577 is Euler’s constant, and Ei(−γ t) and Ei(γ t) are the exponential integrals . The environment no longer transfers energy to the adparticle due to the zero-point motion of the surface oscillators.
2 and 5, Vol. 1, Langevin equations can be recast in a Hamiltonian-like form within the Caldirola-Kanai model [50, 51]. This is a timedependent Hamiltonian that includes phenomenologically the friction produced by a viscid or viscous medium on the system—nonetheless, it is also possible to justify the model from more rigorous grounds [52, 53]. As shown below, within the context of single wave-packet dynamics, this model leads to results similar to those that we can find in classical physics, including localization of the quantum system .
A Trajectory Description of Quantum Processes. II. Applications: A Bohmian Perspective by Ángel S. Sanz