# Quantum Trajectories: Real or Surreal?

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## Abstract

**:**

## 1. Introduction

## 2. Re-Examination of the Analysis of ESSW

#### 2.1. General Results Using Wave Packets

#### 2.2. What Can Be Said about the Behaviour of Individual Particles?

## 3. The Bohm Approach When Spin Is Included

#### 3.1. Spin and the Use of the Pauli Equation

## 4. Detailed Calculation of the Trajectories

#### 4.1. One Stern-Gerlach Magnet

#### 4.2. Numerical Values for Single Stern-Gerlach Magnet

#### 4.3. Two Stern-Gerlach Magnets

#### 4.4. The Appearance of the Quantum Torque

#### 4.5. Detailed Calculation of the Quantum Potential

#### 4.6. Numerical Details: Quantum Potential Single Stern-Gerlach Magnet

#### 4.7. Numerical Details: Quantum Potential in Two SG Magnet Case

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Kocsis, S.; Braverman, B.; Ravets, S.; Stevens, M.J.; Mirin, R.P.; Shalm, L.K.; Steinberg, A.M. Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer. Science
**2011**, 332, 1170–1173. [Google Scholar] [CrossRef] [PubMed] - Mahler, D.; Rozema, L.; Fisher, K.; Vermeyden, L.; Resch, K.; Braverman, B.; Wiseman, H.; Steinberg, A.M. Measuring Bohm trajectories of entangled photons. In Proceedings of the CLEO: QELS-Fundamental Science, Optical Society of America, San Jose, CA, USA, 8–13 June 2014; p. FW1A-1. [Google Scholar]
- Coffey, T.M.; Wyatt, R.E. Comment on “Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer”. arXiv, 2011; arXiv:1109.4436. [Google Scholar]
- Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables I. Phys. Rev.
**1952**, 85, 166–179. [Google Scholar] [CrossRef] - Philippidis, C.; Dewdney, C.; Hiley, B.J. Quantum Interference and the Quantum Potential. Nuovo Cimento
**1979**, 52, 15–28. [Google Scholar] [CrossRef] - Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables II. Phys. Rev.
**1952**, 82, 180–193. [Google Scholar] [CrossRef] - Bohm, D.; Hiley, B.J.; Kaloyerou, P.N. An Ontological Basis for the Quantum Theory: II—A Causal Interpretation of Quantum Fields. Phys. Rep.
**1987**, 144, 349–375. [Google Scholar] [CrossRef] - Holland, P.R. The de Broglie-Bohm Theory of motion and Quantum Field Theory. Phys. Rep.
**1993**, 224, 95–150. [Google Scholar] [CrossRef] - Kaloyerou, P.N. The Causal Interpretation of the Electromagnetic Field. Phys. Rep.
**1994**, 244, 287–385. [Google Scholar] [CrossRef] - Flack, R.; Hiley, B.J. Weak Values of Momentum of the Electromagnetic Field: Average Momentum Flow Lines, Not Photon Trajectories. arXiv, 2016; arXiv:1611.06510. [Google Scholar]
- Bliokh, K.Y.; Bekshaev, A.Y.; Kofman, A.G.; Nori, F. Photon trajectories, anomalous velocities and weak measurements: A classical interpretation. New J. Phys.
**2013**, 15, 073022. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M. Quantum Mechanics: Non-Relativistic Theory; Pergamon Press: Oxford, UK, 1977; p. 2. [Google Scholar]
- Einstein, A. Albert Einstein: Philosopher-Scientist; Schilpp, P.A., Ed.; Library of the Living Philosophers: Evanston, IL, USA, 1949; pp. 665–676. [Google Scholar]
- Heisenberg, W. Physics and Philosophy: The Revolution in Modern Science; George Allen and Unwin: London, UK, 1958. [Google Scholar]
- Jammer, M. The Philosophy of Quantum Mechanics; Wiley: New York, NY, USA, 1974. [Google Scholar]
- Madelung, E. Quantentheorie in hydrodynamischer Form. Z. Phys.
**1926**, 40, 322–326. [Google Scholar] [CrossRef] - De Broglie, L. La mécanique ondulatoire et la structure atomique de la matière et du rayonnement. J. Phys. Radium
**1927**, 8, 225–241. [Google Scholar] [CrossRef] - De Gosson, M. The Principles of Newtonian and Quantum Mechanics: The Need for Planck’s Constant; Imperial College Press: London, UK, 2001. [Google Scholar]
- Englert, J.; Scully, M.O.; Süssman, G.; Walther, H. Surrealistic Bohm Trajectories. Z. Naturforsch.
**1992**, 47, 1175–1186. [Google Scholar] [CrossRef] - Scully, M. Do Bohm trajectories always provide a trustworthy physical picture of particle motion? Phys. Scr.
**1998**, 76, 41–46. [Google Scholar] [CrossRef] - Feynman, R.P.; Leighton, R.B.; Sands, M. The Feynman Lectures on Physics III; Addison-Wesley: Reading, MA, USA, 1965; Chapter 5. [Google Scholar]
- Hiley, B.J. Welcher Weg Experiments from the Bohm Perspective, Quantum Theory: Reconsiderations of Foundations-3, Växjö, Sweden 2005; Adenier, G., Krennikov, A.Y., Nieuwenhuizen, T.M., Eds.; AIP: College Park, MD, USA, 2006; pp. 154–160. [Google Scholar]
- Hiley, B.J.; Callaghan, R.E. Delayed Choice Experiments and the Bohm Approach. Phys. Scr.
**2006**, 74, 336–348. [Google Scholar] [CrossRef] - Bohm, D. Quantum Theory; Prentice-Hall: Englewood Cliffs, NJ, USA, 1951. [Google Scholar]
- Landau, L.D.; Lifshitz, E.M. Quantum Mechanics: Non-Relativistic Theory; Pergamon Press: Oxford, UK, 1977; pp. 56–57. [Google Scholar]
- Mott, N.F. The Wave Mechanics of α-Ray Tracks. Proc. R. Soc.
**1929**, 126, 79–84. [Google Scholar] [CrossRef] - Bohm, D.; Schiller, R.; Tiomno, J. A causal interpretation of the Pauli equation (A). Nuovo Cimento
**1955**, 1, 48–66. [Google Scholar] [CrossRef] - Dewdney, C. Particle Trajectories and Interference in a Time-dependent Model of Neutron Single Crystal Interferometry. Phys. Lett.
**1985**, 109, 377–384. [Google Scholar] [CrossRef] - Dewdney, C.; Holland, P.R.; Kyprianidis, A.; Vigier, J.-P. Spin and non-locality in quantum mechanics. Nature
**1988**, 336, 536–544. [Google Scholar] [CrossRef] - Dewdney, C.; Holland, P.R.; Kyprianidis, A. What happens in a spin measurement? Phys. Lett. A
**1986**, 119, 259–267. [Google Scholar] [CrossRef] - Dewdney, C.; Holland, P.R.; Kyprianidis, A. A Causal Account of Non-local Einstein-Podolsky-Rosen Spin Correlations. J. Phys. A Math. Gen.
**1987**, 20, 4717–4732. [Google Scholar] [CrossRef] - Holland, P.R. The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Penrose, R.; Rindler, W. Spinors and Space-Time; Cambridge University Press: Cambridge, UK, 1984; Volume 1. [Google Scholar]
- Bell, J.S. Speakable and Unspeakable in Quantum Mechanics; Cambridge University Press: Cambridge, UK, 1987. [Google Scholar]
- Hiley, B.J.; Callaghan, R.E. The Clifford Algebra approach to Quantum Mechanics A: The Schrödinger and Pauli Particles. arXiv, 2010; arXiv:1011.4031. [Google Scholar]
- Hiley, B.J. Weak Values: Approach through the Clifford and Moyal Algebras. J. Phys. Conf. Ser.
**2012**, 361, 012014. [Google Scholar] [CrossRef] - Bohm, D.; Hiley, B.J. Non-locality and Locality in the Stochastic Interpretation of Quantum Mechanics. Phys. Rep.
**1989**, 172, 93–122. [Google Scholar] [CrossRef] - Feynman, R.P.; Leighton, R.B.; Sands, M. The Feynman Lectures on Physics III; Addison-Wesley: Reading, MA, USA, 1965; Chapter 5.2. [Google Scholar]
- Schrödinger, E. Are There Quantum Jumps? Part I. Br. J. Philos. Sci.
**1952**, 3, 109–123. [Google Scholar] [CrossRef] - Bohm, D. The Implicate Order: A New Approach to the Nature of Reality; A Talk Given at Syracuse University; Syracuse University: Syracuse, NY, USA, 1982. [Google Scholar]
- Bohm, D. A proposed Explanation of Quantum Theory in Terms of Hidden Variables at a Sub-Quantum Mechanical Level. In Observation and Interpretation, Proceedings of the Ninth Symposium of the Colston Research Society, Bristol, UK, 1–4 April 1957; Korner, S., Ed.; Butterworth Scientific Publications: London, UK, 1957; pp. 33–40. [Google Scholar]
- Hiley, B.J.; Callaghan, R.E. The Clifford Algebra Approach to Quantum Mechanics B: The Dirac Particle and its relation to the Bohm Approach. arXiv, 2010; arXiv:1011.4033. [Google Scholar]
- Takabayasi, T. Remarks on the Formulation of Quantum Mechanics with Classical Pictures and on Relations between Linear Scalar Fields and Hydrodynamical Fields. Prog. Theor. Phys.
**1953**, 9, 187–222. [Google Scholar] [CrossRef] - Schwinger, J. The Theory of Quantised Fields I. Phys. Rev.
**1951**, 82, 914–927. [Google Scholar] [CrossRef] - Bohm, D.J.; Hiley, B.J. Measurement Understood Through the Quantum Potential Approach. Found. Phys.
**1984**, 14, 255–264. [Google Scholar] [CrossRef] - Bohm, D.; Hiley, B.J. The Undivided Universe: An Ontological Interpretation of Quantum Theory; Routledge: London, UK, 1993. [Google Scholar]
- Flack, R.; Hiley, B.J. Feynman Paths and Weak Values. Preprints
**2018**, 2018040241. [Google Scholar] [CrossRef] - Flack, R.; Hiley, B.J. Weak Measurement and its Experimental Realisation. J. Phys. Conf. Ser.
**2014**, 504, 012016. [Google Scholar] [CrossRef] - Monachello, V.; Flack, R.; Hiley, B.J.; Callaghan, R.E. A method for measuring the real part of the weak value of spin using non-zero mass particles. arXiv, 2017; arXiv:1701.04808. [Google Scholar]
- Morley, J.; Edmunds, P.D.; Barker, P.F. Measuring the weak value of the momentum in a double slit interferometer. J. Phys. Conf. Ser.
**2016**, 701, 012030. [Google Scholar] [CrossRef]

**Figure 7.**Trajectories with spin vectors overlaid on the spin quantum potential immediately on exiting a single SG magnet.

**Figure 8.**${Q}_{trans}$ (

**left**) and ${Q}_{spin}$ (

**right**) quantum potential for a two SG magnets system.

**Figure 9.**Trajectories with spin vectors overlaid on the spin quantum potential for a two SG magnet system.

Atom | Ag |
---|---|

Mass | $1.8\times {10}^{-25}$ Kg |

Width of magnets | 4 and $8\times {10}^{-4}$ m |

Length of magnets | 1 and $2\times {10}^{-2}$ m |

Velocity of atoms | ${v}_{y}=y/t=500$ m/s |

Time within magnets | $\Delta t=2$ and $4\times {10}^{-5}$ s |

Magnetic field strength at centre | ${B}_{0}=5$ Tesla |

Magnetic field gradient | ${B}_{0}^{\prime}=1000$ Tesla/m |

Wave packet width | $\sigma =1\times {10}^{-4}$ m |

Wave packet speed | $u={\mu}_{B}{B}_{0}^{\prime}\Delta t/m=1$ m/s |

${\Delta}^{\prime}={\mu}_{B}{B}_{0}^{\prime}\Delta t\hslash =mu/\hslash $ | ${\Delta}^{\prime}=1.717\times {10}^{9}$ m${}^{-1}$ |

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Hiley, B.J.; Van Reeth, P. Quantum Trajectories: Real or Surreal? *Entropy* **2018**, *20*, 353.
https://doi.org/10.3390/e20050353

**AMA Style**

Hiley BJ, Van Reeth P. Quantum Trajectories: Real or Surreal? *Entropy*. 2018; 20(5):353.
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**Chicago/Turabian Style**

Hiley, Basil J., and Peter Van Reeth. 2018. "Quantum Trajectories: Real or Surreal?" *Entropy* 20, no. 5: 353.
https://doi.org/10.3390/e20050353