Validity of a naïve approximation formula

for Bohmian velocity

Noam Erezand

Lev Vaidman

Gillie Naaman Marom

with

The Reality in Bohmian Quantum Mechanics or CanYou Kill with an Empty Wave Bullet

In 2005 Vaidman published a paper in the magazine `foundation of Physics` in which he dealt with the question of Bohm`s surrealistic trajectories. This paper was written on the background of the still ongoing debate relating to the existence or non-existence of a non local effect related to the Bohmian particle. Vaidman was motivated by the knowledge that the nature of the detector in the experiment is very crucial for that question.

Which way does the particle choose?- Illustrative approach.

Naive approximating formula.

Naive approximating formula - with spin.

• Accurate in cases involving spin.

Naive approximating formula - without spin.

• Approximation in cases without spin.

The approximated picture.Two single mode packets

with same amplitudes moving toward each other.

The Bohmian picture. Two single mode

packets with same amplitudes moving toward each other

Equal amplitudes, single mode packets - orbits comparison.

Plane waves with equal amplitudes

The approximated picture.Two single mode packets with different amplitudes moving toward each other.

The Bohmian picture.Two single mode packets with different amplitudes moving toward each other.

Different amplitudes, single mode packets - orbits comparison.

Plane waves with non-equal amplitudes

Test Case - Single mode packets part A

Test Case - Single mode packets part B

Two single modes packets moving in opposite directions .

General free packets.,

Two one-dimensional Gaussians moving towards each other.

- The characteristic expansion time.

-The central frequency of the packet.

- The group velocity of the packet.

ρBohm versus ρApprox - λ0=10

Bohmian orbits vs. approximated orbits - λ0=10.

ρBohm versus ρApprox - λ0=3

Bohmian orbits vs. approximated orbits - λ0=3.

Can an empty wave packet kill a cat?

Can an empty wave packet kill a super slow cat?

Bubble chamber! Actual or conceptual?

Delayed-choice-experiments and the Bohm approach.B J Hiley and R E Callaghan, 2006 Phys. Scr. 74 336

“Thus, when the particle enters the bubble chamber, the process that is central to the BI analysis is the ionization process that takes place in the molecules of the liquid. It is this ionization that leads to a loss of coherence not because of irreversibility, but because the wavefunctionsinvolved in the process no longer overlap and are spatially distinct.”

Conceptual Bubble chamber animation.

What is a “non Bohmian” detector?

A detector with a wave function that keeps its position in configuration space much after the particle’s split wave function already arrived to the overlapping zone.

Examples:A super slow cat.A very slow Gedanke bubble chamber.

A simple “non Bohmian” detector.

Schrodinger equation for a ring:

Two degenerate first excited states:

Surrealistic Bohm trajectories.

One spatial dimension is enough.

X0-100 100

MirrorMirror

Half reflecting half transmitting mirror.

X0-100 100

MirrorMirror

Half reflecting half transmitting mirror.

Step 1.The packet is arriving.

Step 2.Splitting the packet.

Surrealistic trajectories in one spatial dimension.

X0-100 100

MirrorMirror

Half reflecting half transmitting mirror.

Step 3.Placing the detector after the packet already passed.

Step 4.Entanglement is created.

The particle and the detector combined wave function.

After hitting the detector:

ρBohm Versus ρApprox at different times.

ρBohm Versus ρApprox at different times. T=80

ρBohm Versus ρApprox at different times. T=100

3 dimensional orbit of Bohmian particle vs. Lev`s particle

A Comparison of orbits with two different detectors, without a detector and an approximated orbit.

The effect of the quantum detector - Intuitive analysis.

The group velocity of the ground states:

A wave function of two overlapping single-mode packets, entangled with the detector:

The Wave function in configuration space

The effect of the quantum detector - Intuitive analysis.

- Direction of propagation.

- Direction of constant phase.

If mp=md , this transformation is a simple rotation.

Surrealistic trajectories Matlab animation

Conclusion.

1. The naive formula gives a good approximation for the Bohmian trajectories of overlapping packets.

2. The approximation is improved when using none Bohmian detector.

3. It is possible to kill a super slow cat with an empty wave bullet. Real cats should not be worry.