By Xavier Oriols Pladevall, Jordi Mompart

Most textbooks clarify quantum mechanics as a narrative the place every one step follows certainly from the single previous it. despite the fact that, the advance of quantum mechanics used to be precisely the contrary. It was once a zigzagging path filled with own disputes the place scientists have been pressured to desert well-established classical options and to discover new and inventive routes. This booklet demonstrates the large sensible software of one other of those routes in explaining quantum phenomena in a number of examine fields. Bohmian mechanics―the formula of the quantum idea pioneered via Louis de Broglie and David Bohm―offers an alternate mathematical formula of quantum phenomena by way of quantum trajectories. It sheds mild at the limits and extensions of our current realizing of quantum mechanics towards different paradigms, comparable to relativity or cosmology.

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**Example text**

7) between two diﬀerent times, t0 and t f , provides a stationary value for the action function, S(x[t]; x0 , t0 ; x f , t f ), where x0 = x[t0 ] and x f = x[t f ]. 10) 2 with K (v) = mv /2 the kinetic energy of the particle. The Lagrangian equation can also be deﬁned as L (x, v) = pv − H (x, v), where H (x, p) = K (v) + V (x) is the Hamiltonian functionb and p = mv is the (linear) momentum. a The suﬃx “p” means physical in order to distinguish from nonphysical trajectories, “np,” but it will be omitted, when unnecessary.

And so inﬂuenced that the particle does not go where the waves cancel out, but is attracted to where they cooperate. This idea seems to me so natural and simple, to resolve the waveparticle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored. Now, with almost a century of perspective and the knowledge that both routes give exactly the same experimental predictions, it seems that such great scientists took the strangest route. In fact, Bell argued that their choice was unprofessional [20].

In order to simplify as much as possible our mathematical notation, we will ﬁrst study a single spinless particle living in a one-dimensional (1D) space. The spatial degree of freedom of the particle will be represented by x. The theoretical generalization of all the arguments mentioned in this section into a single particle in a 3D space is quite simple. However, the practical solution of a ¨ single-particle Schrodinger equation in a 3D space has considerable computational diﬃculties, as we will discuss in Sec.