# Testing Quantum Mechanics with an Ultra-Cold Particle Trap

## Abstract

**:**

## 1. Introduction

## 2. Background

_{x}, L

_{y}, L

_{z}with one corner taken as the origin of a Cartesian coordinate system has the simple form [6]:

_{nxnynz}= (8/L

_{x}L

_{y}L

_{z})

^{1/2}sin(n

_{x}πx/L

_{x}) sin(n

_{y}πy/L

_{y}) sin(n

_{z}πz/L

_{z}) exp (–iEt/ℏ)

_{nxnynz}= [(n

_{x}

^{2}/L

_{x}

^{2}) + (n

_{y}

^{2}/L

_{y}

^{2}) + (n

_{z}

^{2}/L

_{z}

^{2})] (π

^{2}ℏ

^{2}/2m) > 0

_{x}, n

_{y}, n

_{z}= 1, 2, 3, ...; ℏ is Planck’s constant divided by 2π; and t is time.

_{nxnynz}∣

^{2}> 0 for 0 < x < L

_{x}, 0 < y < L

_{y}, 0 < z < L

_{z}, OQT predicts that there is a nonzero probability of detecting a particle at any position inside the potential well, including near its walls. As the ground state (n

_{x}= n

_{y}= n

_{z}= 1) energy is greater than zero, OQT predicts that a particle in the well will have a nonzero momentum value [8,9]. Nonzero momentum is also required by the Uncertainty Principle as understood in OQT [10].

**p**of the particle is given by [11,12,13,14,15,16]:

**p**= m

**v**=

**∇**S

**v**is the particle’s velocity. Equation (3) is a key expression in this experimental proposal and the cited references confirm its validity in deBB Theory. (Note that the expression for the momentum of a nonzero spin particle will have a more complicated form.) The deBB Theory predicts that a spinless particle inside a three-dimensional “infinite” well is at rest (in the laboratory frame of reference). This outcome is found by substituting the S-function from Equation (1) into Equation (3), then:

**p**=

**∇**(–Et) = 0 = m

**v**

## 3. Original Testing Proposal

^{2}θ − n

^{2})

^{½}/cosθ

_{in}is the incident density distribution of the “cloud” of atoms, κ = (2π/λ) (n

^{2}sin

^{2}θ − 1)

^{1/2}, I

_{sat}= 2π

^{2}ℏΓc/3λ

^{3}, I

_{ev}= (4n cos

^{2}θ) I

_{L}/(n

^{2}− 1), with I

_{L}being the intensity of the incident laser beam and c is the speed of light in a vacuum.

- Effect of GravityGravity will affect an atom’s vertical motion resulting in the matter wave not forming a standing wave pattern. Unless the effects of gravity could be counteracted without affecting the stationary state, the experiment would need to be done in a free-fall environment. Performing the experiment in such an environment would introduce significant practical complications to its conduct and greatly increase its cost.
- Vibration EliminationVibrations of the walls of the particle trap have the tendency to disrupt the standing matter wave and possibly heat the atoms if they approach too near to the walls. This could be eliminated by cooling the trap’s walls close to absolute zero. External sources of vibration would also need to be avoided by acoustically, mechanically, and thermally isolating the apparatus.
- Homodyne DetectionEach of the incident laser beams used to generate the evanescent waves might be split into two parts with one part reflected from a wall of the trap and the other part used as a reference beam. This arrangement will greatly reduce fluctuations over what would result by having the reflected and reference beams produced by separate laser devices [25].

## 4. Discussion—A Feasible Test

## 5. Concluding Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

^{2}/2m) (

**∇**

^{2}R)/R = [(n

_{x}

^{2}/L

_{x}

^{2}) + (n

_{y}

^{2}/L

_{y}

^{2}) + (n

_{z}

^{2}/L

_{z}

^{2})] (π

^{2}ℏ

^{2}/2m)

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