One of the counterintuitive features of quantum mechanics is the phenomenon of nonlocality. In simple terms, this implies that, in some circumstances, particles that have interacted at some initial time and then become spatially separated remain "entangled", such that a measurement on one particle affects the other instantaneously, no matter how large the separation of the two particles has become. The standard interpretation of quantum mechanics and the de Broglie–Bohm interpretation are both consistent with experimental evidence. But it is useful to understand nonlocality if one visualizes particle trajectories rather than collapse of the wavefunction. In the trajectory approach, a measurement of one particle could lead to an incorrect prediction of the trajectory (or "surreal trajectories") of the entangled particles. The surreal trajectory is a consequence of nonlocality in which the particles are able to influence one another instantaneously. This Demonstration considers the motion of two orthogonal onedimensional entangled particles in a Calogero–Moser potential with constant phase shift. It shows that measurement of the initial starting position of one particle affects the trajectory of the other particle. The motion of the two particles in twodimensional configuration space might be described by a single trajectory, in which the motion is local. The projection of the trajectory onto twodimensional configuration space leads to a decomposition of two spatially divided motions in onedimensional real space, in which the motion becomes entangled and where quantum entanglement becomes equivalent to quantum nonlocality. Here, the configuration space represents a projection from real space, which can simulate quantum nonlocality. If the motion is entangled, chaotic, or ergodic, motion often results. Measurement of the particle position is determined by its initial choice. Mathematically, entanglement occurs if factorizability of the wavefunction is not possible; for example, when quantum superposition produces a product state, the motion becomes entangled. This means that the motion in one coordinate direction depends also on the other coordinate directions, whether the motion is periodic or not. The component motions are independent only if the wavefunction is factorizable in configuration space. The Demonstration shows the motion in configuration space, real space, and phase space, in which the phase space consists of all possible values of position and velocity variables. The degree of entanglement is represented by the parameter For , the wavefunction is factorizable in configuration space. The motion is periodic, and the particles behave independently of one another. The initial starting position of one particle does not affect the motion of the other particle. For , the motion of the two particles becomes entangled and chaotic, depending on the constant phase shift . The initial starting position of a particle affects the motion of the other particle in real space. The graphic shows the trajectory in configuration space (red) and the motion of the two particles in onedimensional real space (green and blue). If the phase space button is activated, these are shown: the and components of the velocity in configuration space (red), the position in the direction, the component of the velocity of particle 1 (blue) and the position in the direction, and the component of the velocity of particle 2 (green).
Consider the Schrödinger equation , with , , , , and so on. The solution involves associated Laguerre polynomials. An entangled, unnormalized wavefunction for two onedimensional particles, which cannot move along the entire and axes but are constrained to remain on the half and axes, can be expressed by a superposition state with a special parameter : , where , are eigenfunctions, and are permuted eigenenergies of the corresponding stationary onedimensional Schrödinger equation, with and . The eigenfunctions , are given by where , are associated Laguerre polynomials, and are the quantum numbers with . The wavefunction is taken from [1]. For , the velocity vector in one coordinate direction does not depend on the other coordinate direction: and , from which the trajectory is calculated, and where and are the initial starting positions, which can be freely chosen for numerical integration of the velocity vector. The initial starting positions (particle 1) and (particle 2) can be changed by using the controls. For , , and , both components of the velocity are equal, which produces a straight line in configuration space. For the special case , the trajectory becomes periodic, depending only on the constant phase shift in the term. In the program, if PlotPoints, AccuracyGoal, PrecisionGoal, and MaxSteps are increased (if enabled), the results will be more accurate. [1] M. Trott, The Mathematica GuideBook for Symbolics, New York: Springer, 2006. [2] B.G. Englert, M. O. Scully, G. Süssman, and H. Walther, "Surrealistic Bohm Trajectories," Zeitschrift für Naturforschung A, 47(12), 1992 pp. 1175–1186. [5] S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. Krister Shalm, and A. M. Steinberg, "Observing the Average Trajectories of Single Photons in a TwoSlit Interferometer," Science, 332(6034), 2011 pp. 1170–1173. doi:10.1126/science.1202218. [6] D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, and A. Steinberg,"Experimental Nonlocal and Surreal Bohmian Trajectories," Science Advances, 2(2), 2016 pp. 1–7. doi:10.1126/science.1501466.
