Quantum trajectories and the Bohm time constant

Abstract This work proposes a new logarithmic nonlinear Schrodinger equation to describe the dynamics of a wave packet under continuous measurement. Via the method of quantum trajectories formalism of the Bohmian model of quantum mechanics, it is shown that this continuous measurement alters the dynamical properties of the measured system. While the width of the wave packet may reach a stationary regime, its quantum trajectories converge asymptotically in time to classical trajectories. So, continuous measurements not only disturb the particle but compel it to eventually converge to a Newtonian regime. The rate of convergence depends on what is defined here as the Bohm time constant which characterizes the resolution time of the measurement. If the initial wave packet width is taken to be equal to 2.8×10 −15  m (the approximate size of an electron) then the Bohm time constant is found to be about 6.8×10 −26  s.

[1]  Bohmian trajectories and quantum phase space distributions , 2002, quant-ph/0208156.

[2]  Stability properties of |Ψ|2 in Bohmian dynamics , 2002, quant-ph/0206043.

[3]  L. Marchildon,et al.  Two-particle interference in standard and Bohmian quantum mechanics , 2003 .

[4]  J. Bell,et al.  Speakable and Unspeakable in Quatum Mechanics , 1988 .

[5]  D. Home,et al.  A critical re-examination of the quantum Zeno paradox , 1992 .

[6]  Peter Holland.  Computing the wavefunction from trajectories: particle and wave pictures in quantum mechanics and their relation , 2004 .

[7]  Transmission and reflection in a double potential well: doing it the Bohmian way , 2001, quant-ph/0111045.

[8]  A. Datta,et al.  Bohmian picture of Rydberg atoms , 2002, quant-ph/0201081.

[9]  I. Bialynicki-Birula.  On the linearity of the Schrödinger equation , 2005 .

[10]  E. Vrscay,et al.  Spin-dependent Bohm trajectories for hydrogen eigenstates , 2002, quant-ph/0308105.

[11]  M. D. Kostin.  On the Schrödinger‐Langevin Equation , 1972 .

[12]  D. Bohm.  A SUGGESTED INTERPRETATION OF THE QUANTUM THEORY IN TERMS OF "HIDDEN" VARIABLES. II , 1952 .

[13]  L. Shifren,et al.  Correspondence between quantum and classical motion: comparing Bohmian mechanics with a smoothed effective potential approach , 2000 .

[14]  G. Bowman.  Wave packets and Bohmian mechanics in the kicked rotator , 2002 .

[15]  Asher Peres,et al.  Zeno paradox in quantum theory , 1980 .

[16]  Wineland,et al.  Quantum Zeno effect. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[17]  A. K. Pan.  Understanding the spreading of a Gaussian wave packet using the Bohmian machinery , 2010 .

[18]  Ron,et al.  Incomplete "collapse" and partial quantum Zeno effect. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[19]  Roderich Tumulka,et al.  Understanding Bohmian mechanics: A dialogue , 2004 .

[20]  A. R. Plastino,et al.  Bohmian quantum theory of motion for particles with position-dependent effective mass , 2001 .

[21]  On the motion of a single particle near a nodal line in the de Broglie–Bohm interpretation of quantum mechanics , 2003 .

[22]  Roderich Tumulka,et al.  Bohmian mechanics and quantum field theory. , 2003, Physical review letters.

[23]  A. Nassar.  Time‐dependent invariant associated to nonlinear Schrödinger–Langevin equations , 1986 .

[24]  Chaotic causal trajectories: the role of the phase of stationary states , 2000 .

[25]  Finite resolution of time in continuous measurements: phenomenology and the model ∗ , 1997, quant-ph/0007096.

[26]  C Brooksby,et al.  Quantum backreaction through the Bohmian particle. , 2001, Physical review letters.

[27]  E. Sudarshan,et al.  Zeno's paradox in quantum theory , 1976 .

[28]  Bohmian description of a decaying quantum system , 2000, quant-ph/0005109.

[29]  Quantum Trajectories for Brownian Motion , 1999, quant-ph/9907100.

[30]  F. Borondo,et al.  Causal trajectories description of atom diffraction by surfaces , 2000 .

[31]  E. Sudarshan,et al.  Time evolution of unstable quantum states and a resolution of Zeno's paradox , 1977 .

[32]  Understanding quantum superarrivals using the Bohmian model , 2002, quant-ph/0206027.