An Alternative Model of the Double Slit Interference Experiment

[wmadden]

An Alternative Model of the
Double Slit Interference Experiment
W.H. Madden
November 20, 2007​

ABSTRACT

This post presents an alternative model of the double-slit interference (DSI) experiment. The model is based on a novel physical interpretation of the concept of the DeBroglie wavelength.

This alternative model is shown to fulfill the essential requirements for any plausible model of the DSI experiment: It can account for the observed wave-like interference pattern, is applicable to “single quanta” type variants of the DSI experiment, and is in complete agreement with the experimentally observed—and QM predicted—results. Further, the model is entirely particle based.

The proposed model has advantages: Being strictly particle based, the wave/particle duality paradox is avoided. Non-local “action at a distance” is not required to account for the individual quanta’s sensitivity to the global condition of both slits. Also, a concrete physical interpretation is given to the DeBroglie wavelength.

The main disadvantage of this model is that this physical interpretation calls into question the fundamental assumption of continuous motion.

INTRODUCTION

In the DSI experiment, the seemingly innocent question of “what goes through the slits?” remains unresolved, even after years of debate. Despite perfect success in predicting all experimental outcomes, QM formalism offers little help in answering this question (1,2).

Numerous variations of the DSI experiment have clearly shown the gradual build-up of the wave-like interference pattern by single quanta (for example, Tonomura, et al (3). Recently, Afshar (4) has demonstrated the simultaneous presence of both wave-like and particle-like behavior for single quanta, in violation of complementarity.

These experiments, along with many others, have served to reinvigorate—but not resolve—the debate on the underlying nature of wave/particle duality. A central theme of this debate is the physical meaning of the concept of particle wavelength, as first introduced (for non-relativistic cases) by DeBroglie:

$$\lambda = \frac{h}{mv} \ \ \ \ \ \ Eq. \ 1$$​

where $$\lambda$$ is the particle wavelength, h is Planck’s constant, and m and v are the particle mass and velocity, respectively.


A NEW INTERPRETATION OF PARTICLE WAVELENGTH

For this paper, the following concrete physical interpretation of particle wavelength is adopted:

The particle wavelength is the minimum, discrete spatial distance across which the particle can travel.

For unaccelerated motion, an immediate corollary to this discrete motion interpretation (DMI) is:

Only trajectories with lengths equal to integer multiples of wavelengths are allowed:

$$T_{DMI} = N\lambda \ \ \ \ \ Eq. \ 2$$​

where N is an integer.


Consequently, the difference between any two DMI trajectories $$T_{a}$$ and $$T_{b}$$ and must also be an integer multiple of the wavelength, $$\lambda$$:

$$T_{a} = N_{a}\lambda$$

$$T_{b} = N_{b}\lambda$$

$$T_{a}-T_{b} = (N_{a}-N_{b})\lambda$$

$$So,$$

$$T_{a}-T_{b} = M\lambda \ \ \ \ \ \ Eq. \ 3$$​

where M is an integer.


Applied to the DSI experiment, the DMI restriction of Equation 3 is immediately familiar as the wave model condition for an interference maxima at a given point, $$P_{Y}$$, on the screen:

$$P_{A}-P_{B} = M\lambda \ \ \ \ \ \ Eq. \ 4$$​

where $$P_{A}$$ and $$P_{B}$$ are, respectively, the path distances from slit A and slit B to the point, $$P_{Y}$$.

Unlike DMI, the wave model sets no restrictions on the allowed path lengths. In the wave model, path lengths can always be expressed in the form:

$$P_{A} = (N_{A}+K_{A})\lambda$$

$$P_{B} = (N_{B}+K_{B})\lambda \ \ \ \ \ \ Eq. \ 5$$​

where $$N_{A}$$ and $$N_{B}$$ are integers, and $$K_{A}$$ and $$K_{B}$$ are both non-integer values between 0 and 1. So, for example, $$P_{A}$$ could be 2,000,000.3$$\lambda$$, and $$P_{B}$$ could be 1,999,998.3$$\lambda$$. Although neither of the path lengths $$P_{A}$$ or $$P_{B}$$ is an integer multiple of the wavelength, $$\lambda$$, the difference between them, 2$$\lambda$$, is.

Because the wave model allows more possible solutions for interference maxima than does the DMI model, the DMI model—as considered so far —must be incomplete.

A more sophisticated version of the DMI model should include an important uncertainty inherent in the model itself—the uncertainty in the exact position of the trajectory’s starting point, $$\Delta$$. The term, $$\Delta$$, arises as a direct consequence of the Heisenberg Uncertainty Principle: if the trajectory start point is known to a precision smaller than the wavelength, the momentum of the particle involved—and therefore the very wavelength being considered—will be disturbed.

The starting point uncertainty, $$\Delta$$, can be physically interpreted as the distance the trajectory extends into the slit opening. $$\Delta$$ must always be between 0 and 1$$\lambda$$, and, for each event, $$\Delta$$ is definite but unknown. In a large sample of events, $$\Delta$$ will vary randomly for each event.



THE DMI MODEL OF THE SINGLE SLIT EXPERIMENT

Fig. 1 shows the DMI model of a single-slit interference experiment, including the start-point uncertainty, $$\Delta$$, and using ‘real-world’ experimental values.


In this example, the slit barrier to screen distance, R, is 1 meter; the slit width, W, is .1 millimeter, and the source wavelength, $$\lambda$$, is 500 nanometers.

In the DMI model, the basic unit of length is 1$$\lambda$$. Expressing the length, R, in terms of the wavelength:

$$R = (N_{R}+K_{R})\lambda \ \ \ \ \ \ Eq. \ 6$$​

with $$N_{R}$$ an integer, and $$K_{R}$$ a non-integer between 0 and 1.

For the example considered here, R is exactly 1 meter. Since $$N_{R}$$=$$\frac{R}{\lambda}$$, $$N_{R}$$ = 1 meter / 500 nanometers, and $$N_{R}$$ = 2,000,000; with the values chosen for this example, $$K_{R}$$ is 0.

Theoretically, R can be known to any precision, and can be measured from a point at the barrier not positioned on the slit, so the start point uncertainty, $$\Delta$$, does not apply to measurement of the distance, R.

On the other hand, DMI model trajectories must include the uncertainty, $$\Delta$$, and can be generally expressed as:

$$T_{DMI} = (N_{T}+K_{T}+\Delta)\lambda \ \ \ \ \ \ Eq. \ 7$$​

In keeping with the primary DMI condition, all DMI trajectories are restricted to:

$$T_{DMI} = N\lambda \ \ \ \ \ \ Eq. \ 8$$​

The only way equations 7 and 8 can be equal is if the term $$(K_{T}+\Delta)$$ is also an integer. As previously defined, both $$(K_{T}$$ and $$\Delta)$$ are less than 1$$\lambda$$, so their sum, $$(K_{T}+\Delta)$$, must be less than 2$$\lambda$$. To satisfy the DMI condition, $$(K_{T}+\Delta)$$, must equal 1.

Returning to the example of Fig. 1, consider a single event. For simplicity, assume the uncertainty, $$\Delta$$, for this particular event is equal to 0 (a possible, although statistically unlikely, assumption).

For this example, the DMI model predicts the following discrete set of allowed trajectories:

$$T_{M} = \{(N\lambda)...(N+1)\lambda...(N+2)\lambda...(N+M)\lambda\} \ \ \ \ \ \ Eq. \ 9$$​

with N = 2,000,000, and M = {0…1…2…3…etc.}.

Each of these possible trajectories will have an associated displacement at the screen, $$Y_{M}$$, given by basic trigonometry:

$$Y_{M^2} = (N+M)^2\lambda^2-N^2\lambda^2 \ \ \ \ \ \ Eq. \ 10$$​

Simplifying Eq.10, and discarding relatively small terms, gives a close approximation for the screen displacements:

$$Y_M=\sqrt{(2NM)}\ \lambda \ \ \ \ \ \ Eq. \ 11$$​

When M = 0, $$Y_M$$ is also 0, and there is no displacement at the screen.
Solving for the next few displacements gives:

$$Y_{1} \sim 1\ millimeter$$​

$$Y_{2} \sim 1.4142\ mm.$$​

$$Y_{3} \sim 1.7320\ mm.$$​

In general, the allowed displacements for this single event (where $$\Delta$$ is 0) are approximated by $$Y_M$$ = $$\sqrt{M}\ mm.$$

Obviously, this pattern is not what is observed at the screen, but these results are for a single event, with a single value—zero—of $$\Delta$$.

The actual pattern observed will be the composite of a large sample of events, with the variable, $$\Delta$$, different each time.

To include the uncertainty, $$\Delta$$, Eq.10 must be re-written:

$$Y_{M^2} = (N+M)^2\lambda^2-(N+\Delta)^2\lambda^2 \ \ \ \ \ \ Eq. \ 12$$​

After simplification, and again disregarding small terms, a close approximation for $$Y_M$$ is:

$$Y_{M} = \sqrt{2N(M-\Delta)} \ \lambda\ \ \ \ \ \ Eq. \ 13$$​

As $$\Delta$$, varies from 0 to approaching 1, Eq.13 varies from $$\sqrt{2NM}$$ to $$\sqrt{2N(M-1)}$$; for smaller values of M, this variation leads to a significant change in the pattern on the screen.

For example, consider the trajectory when M = 1. Its displacement at the screen is 1 mm when $$\Delta$$, is 0$$\lambda$$, but approaches 0 mm as $$\Delta$$ approaches 1$$\lambda$$.

As $$\Delta$$, varies across a large sample of events, the cumulative distribution of points on the screen becomes almost continuous, in agreement with what is actually observed.

THE DMI MODEL OF THE DOUBLE SLIT EXPERIMENT

Applying the DMI model to the double-slit interference experiment now becomes straightforward.

Figure (2a) shows the DSI experiment, with the same values for R, W, and $$\lambda$$, as in the single-slit example just discussed. The slit separation, d, is .2 mm. For clarity, the slit barrier side of the experiment has been greatly exaggerated.

Recall the wave-model condition for a bright interference fringe at a point, $$P_Y$$, on the screen:

$$P_{A}-P_{B}=M\lambda$$ and​

$$(N_{A}+K_{A})\lambda-(N_{B}+K_{B})\lambda=M\lambda \ \ \ \ \ \ Eq. \ 14$$​

where the left hand terms are the expressions for the wave-model path lengths.

In the DMI model, these expressions are simply replaced by ones which include the start point uncertainty, $$\Delta$$:

$$T_{a} = (N_{a}+K_{a}+\Delta)\lambda$$​

$$T_{b} = (N_{b}+K_{b}+\Delta)\lambda\ \ \ \ \ \ Eq. \ 15$$​

It is important to note that, for each event, the uncertainty, $$\Delta$$, is the same for both slits; this is because the uncertainty applies to the start point of a single particle trajectory—that is, a single event—which must originate at only 1 of the 2 possible slits.

Substitution of the DMI trajectories (Eq.15) into the wave-model interference condition (Eq.14) has no effect; since the variable $$\Delta$$ is the same for both trajectories $$T_{a}$$ and $$T_{b}$$, it simply cancels out.

The underlying geometrical framework of both the wave and DMI models is seen to be identical, and predictions from one model must carry over to the other. The observed outcomes, if not the underlying physical basis, of both models are the same.

Fig 2b shows the interference pattern resulting from the DSI experiment of Fig. 2a. The pattern is described by the conventional wave-model equation:

$$Y_{M}=\frac{(M \lambda R)}{d}\ \ \ \ \ \ Eq. \ 16$$​

where, again, M = {0…1…2…3…etc.}, R = 1 m, $$\lambda$$ = 500 nm, and
d = .2 mm.


Substituting the values {0,1,2,3…etc.} for M gives the displacements for the first few interference maxima:

$$Y_{0} \sim 0$$​

$$Y_{1} \sim 2.5\ millimeters$$​

$$Y_{2} \sim 5.0\ mm$$​

$$Y_{3} \sim 7.5\ mm$$...,etc.​

This result, as predicted by both the wave and DMI models, agrees exactly with that which is experimentally observed.

By incorporating both the discrete motion interpretation of particle wavelength and the uncertainty in the trajectory start point, the DMI model of the DSI experiment is shown to have the following features:


It can account for the observed wave-like interference pattern, although it is entirely particle-based, and it is applicable to individual events—that is, to single particles.

No “wave collapse” is required in the DMI model, and the apparent non-locality (that is, the instantaneous change in outcome dependent upon the state of both slits) is caused not by any interaction between the particle and the slits, but by the presence or absence of possible DMI trajectories. The non-local aspects of the DSI experiment are seen to be geometrical—and not physical—in nature.


VARIATIONS OF THE DSI EXPERIMENT

These include delayed-choice experiments and single quanta “self-interference” experiments. A very brief discussion of each follows:

1. Delayed Choice: in these types of experiments, the decision as to what type of detector to employ is made after the quanta (wave or particle) has already passed through the slit barrier (5). If two narrow field of view detectors are used, with each focused on one slit or the other, the detector outputs can be analyzed to determine which slit the quanta passed through. If, on the other hand, an extended detector (the phosphorescent screen) is used, an interference pattern is observed. The choice of detector appears to alter the nature (wave-like or particle-like) of the quanta involved, after it has already passed the slit barrier. In a sense, the choice of detector seems to influence the past.

2. Self Interference: in this variant, one photon at a time is directed through a beam splitter, with a 50% chance of passing through, and 50% chance of reflection. See Fig. 3. If the photon follows path A, it goes directly to the screen; if it follows path B, it is first reflected by a mirror before proceeding to the screen. The position of the mirror is adjustable along path B. As the mirror is moved, the intensity at the screen varies. Although only 1 photon at a time is considered, an interference-like effect occurs at the screen, as though the single photon were somehow interfering with itself (6).


The DMI model deals readily with both these variants. Delayed-choice experiments are essentially the DSI experiment posed in more dramatic terms, and, in DMI, the choice of detector is irrelevant; in all cases, a single particle-like quanta passes through one slit or the other; the interference pattern arises as a statistical consequence of the permitted DMI trajectories. Only those quanta following DMI trajectories are detected as point-like flashes on the screen; those which do not are not detected at all.

For the self-interference experiment, the same reasoning applies. The presence of the beam splitter provides for 2 separate paths to the screen; in effect, the beam splitter acts as a double slit, and the photon simply follows one of the available paths at random. When at least one of these paths is a DMI permitted trajectory, the photon can be detected at the screen. When both paths are DMI trajectories, the probability of detection is that much greater. If one of the path lengths is varied (by changing the position of the mirror very slightly), the intensity at the point on the screen will vary as well.


CONCLUSIONS

Returning to the original question—what goes through the slits? —the DMI model furnishes an entirely particle-based answer, which can also account for the observed wave-like interference pattern. DMI does so by adopting a concrete physical interpretation of the particle wavelength, and this interpretation then restricts the allowed trajectories the particle can follow to the screen. Furthermore, the DMI model can be applied to individual particles.

If the second slit is opened, a new set of possible DMI trajectories appears instantaneously, but again, for each event, the particle merely follows one of these at random. For a large sample of events, the probability of detecting the particle within a given region on the screen depends on the number of allowed DMI trajectories with endpoints in that region.

Non-locality is very much present in the DMI model, but it is of a geometrical—not physical—nature. It is the DMI trajectories which can interact non-locally, and the experimental setup can be manipulated to alter these instantaneously.

The central tenet of DMI—that the DeBroglie wavelength has a concrete physical interpretation—serves to clearly frame what Feynman (7) once called “the only mystery” of QM, but does not resolve it. In this paper, discrete motion is treated as a first principle, and no explanation is offered here for the mechanism of particle “jumps”, if such a mechanism exists at all.

Obviously, DMI has far-reaching implications, extending to the foundations of both quantum and classical mechanics, and these must be investigated more thoroughly. Some immediate tests of the validity of DMI could be found in applications to the “electron in a box” experiment, quantum tunneling, and development of a DMI model of the hydrogen atom.

References

1. Bell, J.S., “Six Possible Worlds of Quantum Mechanics”, Chapter 20, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press (1987)
2. Zeilinger, A., “On the Interpretation and Philosophical Foundation of Quantum Mechanics”, Vastakohtien Todellisins, Helsinki University Press (1996)
3. Tonomura, et al. “Demonstration of Single-Electron Buildup of an Interference Pattern,” Amer. Journal of Physics, vol.57, p.117 (1989)
4. Afshar, Shahriar S.; Flores, Eduardo; McDonald, Keith F., et al ,“Paradox in Wave Particle Duality”, Foundations of Physics, vol. 37, p. 295 (2007)
5. Helmuth, P., et al, “Delayed Choice Experiments in Quantum Interference”, Phys. Review A, vol. 35, p. 2532 (1987)
6. Granger, P., Adler, G., Aspect, A, “Experimental Evidence for a Photon Anti-Correlation Effect on a Beam Splitter”, Europhys. Letts., vol. 1, p. 173 (1986)
7. Feynman, R.P., “The Feynman Lectures on Physics”, vol. III, Addison-Wesley (1965)

 

[dtfroedge]

With:

An Alternative Model of the
Double Slit Interference Experiment

The underlying geometrical framework of both the wave and DMI models is seen to be identical, and predictions from one model must carry over to the other. The observed outcomes, if not the underlying physical basis, of both models are the same.
[/LIST]​

And:

Consequently, the difference between any two DMI trajectories and and must also be an integer multiple of the wavelength,---
If the second slit is opened, a new set of possible DMI trajectories appears instantaneously

Given these points which seem to presume the basis of QM I'm not sure I see an advantage for the theory :smile DTF​

 

[sweetser]

Quantum diffraction

Hello:

This proposal looks wrong to me based on the well understood observation of quantum diffraction that happens when one slit is used on a coherent light source. One sees an interference pattern after the peak. That is not what is shown in figure 1. Quantum diffraction doesn't get nearly the same amount of airplay as quantum interference, but shows Nature is consistent. Since figure 1 disagrees with observation, this alternative is not viable.

I am at peace with this issue. In my living room, I have a piece of art titled "Groups of coherent photons behave like waves and particles". Words #1 and #3 are critical to understand in this context.

It appears like your model does not deal with the issue of coherence, which indicates to me the logic is wrong. A coherent light source will show the interference pattern in the two slit experiment, and an incoherent light source will not. Therefore coherence is vital to understanding this issue.

 

[Mentz114]

Hi:

The theory is not about particles but waves, and so can only give the same answers as the usual wave treatments. I base this on the following argument.

If we define the dB wavelength in the way you do, then a trajectory becomes a series of points which lie close togther along the classical trajectory. Consider a second trajectory for a particle of equal momentum. Will the dots line up ? You have a hidden phase in your model which I suspect is your Delta (uncertainty). Sweetser above makes a similar point when he cites coherence, which depend on phase.

I don't think there's anything new in this theory.

You may be interested to look at Bohmian QM where trajectories are wavy and cross the classical trajectory at intervals equal to the dB wavelength.

 

[wmadden]

Reply to DTFROEDGE

Hello. Thank you for reading (and responding!) to this post.

I completely agree with you that this alternative model adds nothing new to conventional QM theory, but it was never intended to do so. QM formalism, as it stands, is a supremely successful theory, and has withstood every experimental test to which it has been put.
However, QM theory does leave the door open for interpretation of some of the underlying physical elements (specifically, for the DSI Experiment, the question of "what goes through the slits?")...and there are several such interpretations around (Everett's "Many Worlds", Cramer's "Transactional", the Bohm interpretation, etc.).
My proposal offers a new physical interpretation for DeBroglie's particle wavelength..."the discrete motion interpretation", and the consequence of this is the emulation of wave-like behavior by individual particles. The possible advantage of this approach is via Occam's Razor: the wave particle duality paradox is avoided (the quanta are always treated as particles), and physical non-locality is replaced by a hopefully more palatable form of geometrical non-locality. Of course, my DMI model comes with its own form of quantum "weirdness" - particle-like quanta "jump" across the DeBroglie wavelength - and this directly challenges the assumption of continuous motion.
Again, I thank you for your very thoughtful comment.

 

[wmadden]

Reply to Sweetser

Thank you for reading and responding to my post, and I apologize for the lengthy delay in my replying to you.

You raised many critical questions and I shall try to deal with these, though not necessarily in order. Your primary objection (and please correct me if I misunderstand) concerns Figure 1, and its failure to match the experimentally observed results. You are absolutely correct--and I did acknowledge this in the original posting (though I perhaps should have used a stronger disclaimer). Figure 1 was intended to show in schematic form the geometric relations between the possible DMI trajectories for a single slit. Fig. 1 shows the simplest case for a single "event" (that is, the passage of a single quanta--be it a photon, electron, atom, etc.) for a very specific set of idealized conditions (the slit to screen distance, R, of exactly 2,000,000 $$\lambda$$, and with the uncertainty factor $$\Delta$$ set to zero). As mentioned in the post, these conditions are highly unlikely -- $$\Delta$$ will vary randomly from 0 to 1$$\lambda$$ with each event, and the possible trajectories for a single quanta -- as shown in Fig. 1 -- will vary accordingly. The effect of a non-zero value of $$\Delta$$ on the outcome illustrated in Fig. 1 is given by Eq. 13.

I stress again the importance of single quanta variants of the DSI experiments (and these have actually been performed--see, for example, Refs. 3, 4 and 6 in the original post). When these experiments are performed, each event does not produce a very faint interference (or in the case of a single slit, diffraction) pattern. Instead, a single, point-like flash is observed at the screen. Only when numerous (hundreds--better yet, thousands) of single quanta events are observed successively, does the cumulative result appear as an interference pattern (for two slits) or a diffraction pattern (for a single slit).

To stay within the posting length restrictions, I did not discuss the issue of single slit diffraction--which I will attempt to do-- very briefly-- here. Quantum diffraction can be successfully accounted for by the conventional wave model (and again, please correct me if I am mistaken in this). Essentially, the single slit is subdivided into many sub-slits, each of a width approaching the wavelength. Each of the sub-slits then acts as a source of coherent Huygen wavelets and it is the interference of these wavelets which produces the observed diffraction pattern. The situation is analogous to replacing the single slit with an exceedingly fine-ruled diffraction grating. This, in turn, is equivalent to replacing the double slits with multiple slits, spaced very closely together.

Although more complicated than the double slit experiment, the same underlying geometrical framework applies, and, as discussed in the original post, the discrete motion interpretation of DeBroglie wavelength permits individual particles to emulate wave-like behavior, as a consequence of the "discrete" trajectories that must be followed to the screen. Coherence, while absolutely essential in a wave based model, is not applicable to the particle based model I propose here. I will prepare a more formal post addressing this issue, if the good moderators of this forum so permit.

Finally, you said:

I am at peace with this issue. In my living room, I have a piece of art titled "Groups of coherent photons behave like waves and particles". Words #1 and #3 are critical to understand in this context.

This puts you in the best of company (Bohr, Feynman, etc.!). But...we should still try, should we not? As mentioned in the original post, if a particle based model can be developed which recovers the experimentally observed facts (each quanta detected as a point-like flash; cumulative build-up of the wave-like interference pattern, etc.) such a model deserves further consideration. Even if shown to be wrong, perhaps we can learn something by the attempt.

Thanks again for your reply.

 

[dtfroedge]

wmadden:
The double slit undoubtably is the most intriguing of physics experiments, so simple yet so enigmatic. Everyone knows the mathematical model, it works so well, yet we humans have a problem wrapping our mind around the actual physical process. People who value sanity just leave it alone. I think your view is as good as the next, but I'm not sure it distinguishably testable, and doesn’t illuminate the core issue, most in need of explanation: the non-locality.
A while back I wrote an article, perhaps further complicating the issue of trajectory for the double slit, you may find interesting.
http://www.arxdtf.org/css/slit2.pdf
DTF

 

[wmadden]

Reply to Mentz114

Thank you for raising many interesting points regarding my post. The first of these-regarding phase and coherence- are crucial to any wave based model of the double slit experiment. But please bear in mind that, for single quanta experiments, the conventional wave model predicts a very faint interference pattern for each event – and this is not what is experimentally observed. Instead, each event is observed as a single point-like “flash” at the screen. It is the cumulative result of many such individual events which produces a wave-like interference pattern.
This is the crux of wave/particle duality: both the particle and wave models are necessary – but neither alone is sufficient – to fully account for the observed results.
This is where my proposal comes in: it is a modification of the conventional particle model, which, by “discretizing” the possible particle trajectories, results in emulation of wave-like behavior – each individual event is observed as a point-like flash, and these flashes are much more likely to occur at locations matching the wave model interference fringes. For single quanta experiments, no “wave” is required, and the question of coherence – a wave model attribute -- simply does not arise.
If a suitably modified particle model can account for the single quanta experiments, the same principles should extend to multi-quanta experiments.
The modification to the particle model I propose is to adopt a novel physical interpretation of the DeBroglie “wavelength”, and I suspect the terminology here is giving rise to some confusion. Mathematically, the DeBroglie “wavelength” is simply a measure of distance. Historically, it has been applied in a wave model context. However, as discussed above, a strictly wave-based model cannot fully account for the experimental results. The physical meaning of the DeBroglie wavelength (distance) remains open to interpretation.

You correctly point out the similarity between the $$\Delta$$ factor in my proposal and the idea of phase in the wave model. There are some functional similarities, but, again, phase is a wave-based concept, and my model deals strictly with particles.
The $$\Delta$$ in my model has a definite physical interpretation: it is the uncertainty of the trajectory starting point for each single particle, and I justify its inclusion by the following reasoning:

1. It is required by the Heisenberg Uncertainty Principle: if a particle’s momentum (hence, wavelength) is known to a certain precision, the particle’s position is uncertain by an amount at best equal to that wavelength.

2. For a particle, there is nothing special about the slit “face” (with the “face” being co-planar to the slit barrier). The slit is, after all, just empty space.

From the second point, the $$\Delta$$ uncertainty can be further interpreted as the distance the particle’s starting point extends into the slit (that is, behind the slit “face”).
This uncertainty is definite, but unknown, for each and every single event, and varies randomly with each event.

Also, as you point out, my “discrete motion” proposal is, in many ways, similar to the DeBroglie-Bohm model, but there is an important difference: in DeBroglie-Bohm, each particle-like quanta, at the moment of creation, is accompanied by a physically real “quantum potential” field, which instantaneously ( that is, non-locally) permeates the entire experimental apparatus. In my proposal, no physical non-locality is required – the quantum potential field is replaced by simple, geometrical relations among the possible discrete trajectories. These trajectories have no independent, physical reality – a trajectory only becomes “real” if the particle happens to follow it.
Non-locality is very much present in my model, but it is of a geometrical – not physical – nature.
(A simplistic example to illustrate the difference: consider a triangle with sides with 3, 4 and 5 ft. Now, “instantaneously” extend the sides to 3, 4, and 5 lightyears. Nothing physical has changed, and no violation of the speed of light limitation has occurred. However, were a particle to be launched along a trajectory given by any of these sides, it would travel a distance given by the side.
Geometry can be non-local; this non-locality, however, is not physically real.)

Finally, you ask the question:

Consider a second trajectory for a particle of equal momentum. Will the dots line up ?

Well...yes, though not at the same time, anymore than say, 2 golf balls hit at exactly the same angle, with exactly the same speed, from exactly the same starting point, can simultaneously follow the same trajectory. 2 particles cannot occupy the same location at the same time. And, in light of the definition of the $$\Delta$$ uncertainty given above, sharing an identical starting point is a statistically very unlikely occurrence.

I must disagree with your statement this model is "about waves". I hope I have demonstrated that my proposal is really about a modification of the conventional particle model...and how such a modification produces "wave-like" behavior, even for single quanta experiments.

I hope this reply has clarified some of the points you raised -- if not, please let me know. Again, I thank you very much for your reply.

 

[Mentz114]

Dr. Madden,
thanks for answering my questions in such detail. I have to agree now that my statement "this is really about waves" was rather hasty. In fact I've not found any other work which begins with the notion of discrete motion, governed by the de Broglie wavelength, so it does look like a new way of looking at motion. Of course there are theories where space and/or time are quantised, or in a lattice, but not quite the DM you take as an axiom.

Let me say that I have no objection to your uncertainty concept, it is physically justified and reminded me immediately of the unknowable quantum phase and HUP. Hence my remark.

Also, as you point out, my “discrete motion” proposal is, in many ways, similar to the DeBroglie-Bohm model, but there is an important difference...it would travel a distance given by the side.Geometry can be non-local; this non-locality, however, is not physically real.)

(I've filleted this to save space)

This is very interesting and I'm going to think about it. The idea of the quantum potential being replaced by non-local geometry is new to me.

Consider a second trajectory for a particle of equal momentum. Will the dots line up ?

I meant to say '..parallel trajectory ...' but my point is covered by the phase discussion.

I must disagree with your statement this model is "about waves". I hope I have demonstrated that my proposal is really about a modification of the conventional particle model...and how such a modification produces "wave-like" behavior, even for single quanta experiments.

I'm convinced.

Regards,
M

 

[Mentz114]

Thinking about collisions brings in relativity, and the de Broglie wavelength is not invariant under boosts. If two particles interact, then one observer may see an interaction between lambda_A and lambda_B, but another observer sees an interaction between different wavelengths. If the outcome of the interaction depends directly on the deB wavelength, something unphysical is predicted.

Maybe it will work if one uses a different definition of momentum, and hence wavelength, based on a relativistic invariant.

Obviously I haven't worked this through...

 

[wmadden]

Reply #2 to dtfroedge

Happy New Year. Thank you for your recent reply and for posting the link to your paper. I found it very interesting.

A conceptually similar experiment to yours was performed by Pfleegor and Mandel back in 1967 (Pfleegor and Mandel, “Interference of Independent Photon Beams”, Physical Review, Vol. 159, pp. 184-188, (1967)). Instead of radio frequency sources, their experiment involved two independent lasers with beams that converged at a very small angle.

In the conventional wave model, two such independent sources would normally be phase- incoherent to each other, but two independent lasers can achieve coherence for brief periods (in this case, for approx. 20 microseconds), the so-called “coherence time”. When measurements were restricted to a “window” no longer than this, their experiment showed interference effects.

The conventional wave model accounts for this nicely – wave trains from both lasers were interfering with each other. However, when the laser sources were severely attenuated – so that, with high probability, only one photon at a time was present in their apparatus – a very weak interference pattern persisted (again, it must be stressed, this pattern was cumulative – that is, composed of many individual point-like detections).

The dilemma here is similar to the one you raised in your paper: If we adhere to Dirac’s dictum (“…each photon interferes only with itself. Interference between different photons never occurs.”), it seems that each single quanta is being “co-produced” by two separate sources.

QM formalism, however, does not demand such extremely counter-intuitive interpretations…it simply permits them, as well as other, less radical models. In essence, for the Pfleegor-Mandel experiment, the two slits have been replaced by two lasers. In my proposal, each quanta – that is, each individual photon – regardless of which laser produced it, follows a distinct single trajectory to the screen. It is the possible trajectories themselves which are unconventional – they are restricted by the “discrete motion” interpretation of the DeBroglie wavelength, and this restriction limits the possible points on the screen where the photon can be detected. When a large sample of such individual events is considered, a statistical consequence is the emulation of wave-like interference.

You commented:

wmadden:
I think your view is as good as the next, but I'm not sure it distinguishably testable
DTF

We are in absolute agreement on this, but I must point out, that neither is my alternative proposal ruled out by current experiments. In fact, recent extremely elegant single quanta experiments by Afshar have narrowed the field somewhat – two “contenders” (Bohr’s Complementarity and Everett’s "Many-Worlds”) appear to have been eliminated.

You also commented:

wmadden:
and doesn’t illuminate the core issue, most in need of explanation: the non-locality.
DTF

Again we are in agreement, but only with respect to a much broader context (EPR, Bell’s Inequality, “hidden variable” conjectures, etc.) Within the strictly limited confines of the DSI Experiment, however, my proposal eliminates the need for any physical “spooky action at a distance”. Near instantaneous changes in the observed pattern at the screen result from geometrical changes in the apparatus – when both slits are opened, there are more allowed trajectories to the screen, and an individual particle-like quanta (photon, electron, atom), simply has more single trajectories to choose from. Only one of these is actually followed, and that at random.

Consider a conventional wave model for a single quanta -- it is detected at the screen at just one location, and – somehow – must non-locally “know” not to be detected at other interference fringes (this is, perhaps, the crudest example of the “collapse of the wave function” problem).

On the other hand, a conventional particle-like quanta must “know” – in some non-local fashion – the open or closed state of the slit it did not pass through, so as to contribute to an emergent double-slit interference pattern, or not. My DMI proposal, as discussed above, avoids both these forms of physical non-locality; although non-local effects are present, they arise from geometrical – not physical – causes. Both QM formalism (and experimental results) demand non-locality, but leave the door open as to the underlying nature of its cause.

I thank you again for your thoughtful comments.

 

[wmadden]

Reply #2 to Mentz114

Happy New Year. Thank you so much for your kind comments…it’s very gratifying to know my ideas have, perhaps, provided some food for thought, as have yours. Also (just to keep me honest), I must say, while your salutation was exremely complimentary, I hold no advanced physics degrees.

I look forward to any additional comments you may have. Thanks again.

 

[Patricio]

If you admit an uncertainty in the beginning of trajectory, shouldn’t you not also admit an uncertainty in the momentum of the particle?
If so what one get’s is a set of possible trajectories corresponding to a continuous interval of wave length’s

 

[wmadden]

Reply to Patricio

Thank you for your question regarding the role of the trajectory start point uncertainty, $$\Delta$$, in my proposal.

If this term applied to the quanta itself, it would indeed have the consequences you describe (introducing a spread in the momentum, energy and wavelength of each quanta).

However, the $$\Delta$$ term applies not to the quanta, but rather to the measurement of the overall trajectory length followed by it.

In the conventional wave model, the slit to screen path length(s) are treated as originating precisely at the face of each slit, with no regard given to any positional uncertainty at all.

The Heisenberg Uncertainty Principle, however, limits the position measurement of a quanta along its trajectory to a minimal uncertainty of one wavelength.

In my particle-based model, the $$\Delta$$ term acknowledges the effect of this Heisenberg positional uncertainty on measurement of the slit to screen trajectory length. For each quanta, $$\Delta$$ is a definite--though unknown—distance, and must be less than the quanta’s wavelength. Physically, $$\Delta$$ can be interpreted as a minute extension of the trajectory start point into the slit opening…which, after all, is just empty space.

Please see my response to Mentz114 (Post No. 8 in this thread) for additional information regarding this topic.

I hope this response was helpful in addressing the questions you raised. If not, please let me know!

Thanks again.

 

[ZapperZ]

Thank you for your question regarding the role of the trajectory start point uncertainty, $$\Delta$$, in my proposal.

If this term applied to the quanta itself, it would indeed have the consequences you describe (introducing a spread in the momentum, energy and wavelength of each quanta).

However, the $$\Delta$$ term applies not to the quanta, but rather to the measurement of the overall trajectory length followed by it.

In the conventional wave model, the slit to screen path length(s) are treated as originating precisely at the face of each slit, with no regard given to any positional uncertainty at all.

The Heisenberg Uncertainty Principle, however, limits the position measurement of a quanta along its trajectory to a minimal uncertainty of one wavelength.

Are you sure?

If I put a slit at a position x with a width of $\Delta x$ that is less than the wavelength, I can certainly get a photon to pass through it. Try it some time! And when it does, and I detect it after it passed through the slit, I can say that that photon was at position x with an uncertainty in its position measurement LESS than its wavelength.

The HUP has nothing to do with the uncertainty of a single measurement.

Zz.

 

[wmadden]

Reply to ZapperZ

Thank you for your comments on my reply to Patricio.

You said:

If I put a slit at a position x with a width of $\Delta x$ that is less than the wavelength, I can certainly get a photon to pass through it. Try it some time!

No argument there – I never disputed a photon can pass through a slit narrower than its wavelength – it absolutely can.

You also said,

And when it does, and I detect it after it passed through the slit, I can say that that photon was at position x with an uncertainty in its position measurement LESS than its wavelength.

An unassailable argument, but, one based on an underlying assumption: that the photon’s trajectory is continuous, which brings us to the central tenet of my original posting on An Alternative Model of the DSI Experiment.

My conjecture is that a quanta’s wavelength is the minimum, discrete spatial distance through which that quanta can travel…and the corollary is that, for unaccelerated motion, the total length of a quanta’s trajectory must be an integer multiple of its wavelength. The motivation for introducing this “discrete motion interpretation” (DMI) of wavelength is twofold:

1. To provide a concrete physical interpretation of the DeBroglie wavelength.
2. To entirely avoid wave particle duality, and provide an answer to a fundamental question regarding DMI: What goes through the slits?

Please refer to my original posting for a more detailed development of these ideas, and bear in mind it was within this context that I replied to Patricio. At no time do I dispute the mathematical formalism of quantum mechanics, nor question its results…I simply note (as have others) that the formalism alone does not – and, by its very nature cannot – address the above two points.

That being said, my response to Patricio was based on the following reasoning:

1. The photon wavelength is given by $$\lambda = h / p$$ with the momentum, p, equal to mc. Rewriting this: $$\lambda\cdot p = h$$
2. One form of the Heisenberg Uncertainty Principle is $$\Delta x \cdot \Delta p \geq h$$ with $$\Delta x$$ being the position uncertainty along the x axis, in units of distance and $$\Delta p$$ being the uncertainty in momentum also along the x axis, measured in units of mass times velocity. The minimum value for this is then $$\Delta x \cdot \Delta p = h$$ .

Side by side then we have:

$$\lambda \cdot p = h$$ and $$\Delta x \cdot \Delta p = h$$

Assuming the discrete motion conjecture is correct, I then make the identification

$$\lambda = \Delta x$$ , and $$mc = \Delta p$$

In other words, if -- and only if – the discrete motion conjecture is accepted, a photon’s position along its trajectory can never be measured to an accuracy smaller than its wavelength.

Lastly, you stated

The HUP has nothing to do with the uncertainty of a single measurement.

In “The Physical Principles of the Quantum Theory” Heisenberg himself discusses at length what we now call “Heisenberg’s Microscope” to illustrate his uncertainty principle, and repeatedly refers to single photon-particle measurement—though admittedly, this is regarding the uncertainty in the particle—not photon—position.

Again, I thank you for your thoughtful comments, and look forward to your reply.

 

[ZapperZ]

An unassailable argument, but, one based on an underlying assumption: that the photon’s trajectory is continuous, which brings us to the central tenet of my original posting on An Alternative Model of the DSI Experiment.

Why is it "unassailable"? I have an opening, it is at a particular location with some width. Therefore, if ANYTHING has passed through, then when it was passing through the opening, it has THAT position and THAT uncertainty. If you doubt this, then you might as well doubt ALL experiments. It also implies that your "alternative" model can't be verified, because I can use the same argument against any experimental evidence that you can come up with.

Lastly, you stated
In “The Physical Principles of the Quantum Theory” Heisenberg himself discusses at length what we now call “Heisenberg’s Microscope” to illustrate his uncertainty principle, and repeatedly refers to single photon-particle measurement—though admittedly, this is regarding the uncertainty in the particle—not photon—position.

Again, I thank you for your thoughtful comments, and look forward to your reply.

I still don't see anything contradictory here. After I make a determination of the position of one photon, my ability to predict what its momentum will be very much depends on my knowledge of the position that I just measured. This is exactly the HUP. However, WHEN I make a measurement of the momentum, I can do that with arbitrary accuracy, limited only by my measurement technology. This accuracy is NOT the HUP! Make any measurement that you like. Each one of this has its own measurement accuracy or uncertainty. It does not have anything to do with the HUP. This distinct difference is what most people tripped on.

So you still think that this statement of yours is valid?

The Heisenberg Uncertainty Principle, however, limits the position measurement of a quanta along its trajectory to a minimal uncertainty of one wavelength.

Zz.

 

[wmadden]

Looks like I'm going to have to rethink my use of the HUP regarding the trajectory start point uncertainty in my original posting, and response to Patricio.

Meanwhile, I'd be very interested in your thoughts on the DSI experiment, especially regarding the physical interpretation of the quanta's DeBroglie wavelength, and the question: What goes through the slits?

 

[skynelson]

Interesting. Thanks for adding new ideas to the field.

 

[Carid]

Can we breathe some life into this dead thread?
I have one point though I don't know what it means:
The de Broglie wavelength is infinitely long when a body is at absolute rest.
This curiously fits in with your notion that movement is discretised.
The wavelength got too long for the body to go anywhere!

 

[ZapperZ]

Can we breathe some life into this dead thread?
I have one point though I don't know what it means:
The de Broglie wavelength is infinitely long when a body is at absolute rest.
This curiously fits in with your notion that movement is discretised.
The wavelength got too long for the body to go anywhere!

If you wish to ask a question that isn't related to the OP, please do so in main forum. This forum is ONLY for the OP to work out his/her idea.

Zz.

 

[wmadden]

Thanks for your response to this thread; and please excuse the long delay in my replying.

I agree with both your observations: Yes, this thread has grown far too quiet, and yes, a stationary object (velocity = 0) would have an infinite wavelength.

Perhaps addressing your second observation will stimulate some new activity on this thread, thereby also addressing your first.

It may be instructive to consider the situation when a macroscopic object has a very small -- though not zero – velocity, and therefore a comparatively large wavelength.

Let’s indulge in the following--very speculative--thought experiment. Consider a macroscopic object – say a billiard ball, with mass on the order of .1kg – with a macroscopic wavelength – say on the order of .1 meter. Solving the DeBroglie wavelength expression for such a combination gives a velocity of

$$v = \frac{h}{m \lambda} \approx \frac{10^{-34}}{10^{-1}x10^{-1}} \approx \frac{10^{-34}}{10^{-2}} \approx 10^{-32} \mbox {meters/sec.}$$

The time, T, required for a billiard ball with such an infinitesimal velocity to travel across .1 meter is

$$T = \frac{\lambda}{v} \approx \frac{10^{-1}}{10^{-32}} \approx 10^{31} \mbox {seconds}$$

…a time much, much larger than the estimated age of the universe. (Remember, this is just a “thought” experiment!)

Disregarding the obvious implausibility of ever conducting such an experiment, the question is: What is the physical meaning of the distance, $$\lambda$$ ?

3 possibilities come to mind:

I. The billiard ball “smears” out in some way across the wavelength.

II. The billiard ball remains motionless for the time interval, T, then instantaneously appears at its new position one wavelength away.

III. The billiard ball remains motionless for some time $$T_{STATIC}$$ (less than time, T), then travels at some finite velocity $$V_{TRANSIT}$$ for an interval $$T_{TRANSIT}$$ across the wavelength distance. The sum of these time intervals $$(T_{STATIC}+ T_{TRANSIT})$$ is the total time, T, mentioned above.

Possibility I is perhaps closest to the more conventional interpretations of QM: in between direct observations, the billiard ball has no definite position. The physical interpretation of the wavelength, however, remains unclear, as does the mechanism by which the billiard ball travels across it.

Possibility II is more in keeping with the discrete motion interpretation proposed in my original posting, but invokes instantaneous motion -- that is, infinite velocity – and the accompanying issue of physical non-locality.

This leaves Possibility III, also in keeping with DMI in that the wavelength has a clear physical interpretation, and instantaneous motion is not required. However, a new – and as yet undefined – velocity $$(V_{TRANSIT})$$ is introduced – associated with the time $$T_{TRANSIT}$$


$$V_{TRANSIT}= \frac{\lambda}{T_{TRANSIT}}$$ .

Since the discrete motion interpretation should be applicable to all quanta – including photons – a conjecture regarding $$V _{TRANSIT}$$ is that it is equal to c.

The time interval $$T_{TRANSIT}$$ is then given by $$\frac{\lambda}{c}$$ and the time intervals $$T_{TRANSIT} \ and \ T_{STATIC}$$ are related to the total time,
$$T = \frac{h}{mv^2}$$ by:

$$T_{TRANSIT}=T ( \frac{v}{c} ) \ and \ T_{STATIC} = T- T_{TRANSIT} = T (1 - \frac{v}{c} )$$.

In Possibility III, zero rest mass quanta such as photons have $$T_{STATIC} = 0$$,
$$and \ T_{TRANSIT}=\frac{\lambda}{c}$$.


Again, I wish to stress that all the above is purely speculative, intended only to stir some activity on this thread. My formal position on the underlying mechanism of discrete motion remains as stated in the original post: that the discrete motion should be regarded as a “first principle”.


Much of the above, and indeed, DMI itself, seems somewhat contrived: dismissal of continuous motion, the idea of material particles somehow traveling at light speed, etc., etc. Why bother even entertaining such far-fetched ideas?

The answer is that the alternative interpretations of QM are even stranger.

To illustrate this, consider another, much more plausible thought experiment. This one involves just a slight modification of the DSI experiment. The slit barrier is replaced with a CCD detector (like the kind found in digital cameras) with 2 slits cut through it. With this modification, we can monitor the photons which are stopped by the slit barrier.

Again, a severely attenuated source of photons is used, so that, with high probability, only a single photon is present in the device at any given time.

What would we observe?

1. Most photons would be stopped by the slit barrier, and each would register on it as single, point-like detection.
2. Occasionally, a photon would not be detected at the slit barrier, but would instead pass through it, and register at the screen—again, as a single, point-like detection.

(So far, so good: a conventional, particle-like model of the photon adequately explains these first 2 results).

3. However, if the experiment is run for some time, so that the results of hundreds (better yet, thousands) of events are observed, the point-like detections at the screen begin to cluster in fringes on the screen—and these fringes form either a wave-like interference pattern (if both slits are open) or a diffraction pattern (if only a single slit is open).

The question now becomes: What goes through the slit(s)?

A conventional particle-like model of the photon will no longer do: it cannot account for the fringes seen when the cumulative results are considered.

A conventional wave-like model is also insufficient: it cannot account for the individual, point-like detections observed at either the slit barrier or the screen.

All these results are predicted perfectly by QM theory, but, because of its highly abstract mathematical nature, formal QM theory does not—and cannot—address the simple question of what goes through the slits. There are elements in the formal theory whose physical meaning remains unclear.

This confusing state of affairs has led to the development of numerous interpretations of QM theory, and all of these (at least, the serious ones) share 2 features:

All are in complete agreement with formal QM theory (and therefore, with any experiment performed).

All invoke some form of quantum “weirdness”.

A thorough review of these competing interpretations is way beyond the scope of this post, but a very brief discussion of some of the major contenders is given below.

Certainly, the most widely accepted is Bohr’s Copenhagen Interpretation, which in essence takes the position that formal QM theory, in and of itself, is enough. Any questions not answered by the theory (such as what goes through the slits) are meaningless, and any interpretations beyond this are unnecessary, and are simply “a matter of taste”.

The Copenhagen Interpretation, however, is, as the name implies, an interpretation itself…an extremely minimalist and pragmatic one, to be sure, but nonetheless also “a matter of taste”, and somehow unsatisfying (to some, at least-- including DeBroglie, Schrodinger, Einstein, Bohm, and Bell, to list a few).

There exist other serious alternatives to Copenhagen. Some quick examples:

The Everett-Wheeler “Many Worlds” interpretation, wherein each photon is detected at ALL possible locations—but each point-like detection occurs in a different universe. In short, each event (that is, each emission and subsequent detection of an individual photon) splits the universe into myriad copies of itself, and the one detection we observe occurs in the universe in which we happen to occupy.

Cramer’s “Transactional” interpretation, wherein a “handshake” occurs between the detection and emission of each photon—backwards in time.

The DeBroglie-Bohm interpretation, wherein the photons are treated at all times in a strictly particle-like fashion, but are steered by a quantum potential field, which instantaneously permeates the entire universe the moment the photon is created.

So why bother to add yet another interpretation, such as the discrete motion interpretation proposed in my original post?

First, it agrees with QM theory, thereby meeting the first—and most important—criteria any serious interpretation must fulfill.

And, in comparison to those interpretations listed above, it seems quite tame, and perhaps offers the best “economy of weirdness”, with the following advantages:

It answers the question of what goes through the slits, by eliminating wave/particle duality---the quanta are always treated as point-like particles. The wave-like behavior observed with a large number of events arises as a consequence of the allowed DMI trajectories, and has nothing to do with the individual quanta themselves.

Also, it provides a concrete physical interpretation for the DeBroglie wavelength.

Lastly, it deals with the necessary issue of non-locality by casting it in a different light: although non-locality is inherently present in the DMI model, it is of a geometrical—that is, non-physical—nature. The discrete motion trajectories can appear, interact, and disappear instantaneously because they are geometrical entities, and, as such, have no independent physical reality….they only “exist” when a physically real particle follows one .

Any comments, criticisms, or questions on any of the above remarks are most welcome.

 

[crk]

I believe you are absolutely correct in your conjecture. This duality issue is I believe, strictly an issue of semantics and always has been.

There is an ongoing confusion and always has been, about the difference between a "physical manifestation" and the "behaviour" of that manifestation. I believe as you do, that light (or more accurately, energy) is always physically manifest as discreet particles, "photons". Wave motion is merely the "behaviour" exhibited by photons interacting with each other and/or their environment. There is no conflict! There is no duality! One is apples, the other is oranges. Only one is the actual physical manifestation of light.

Even in ripple tank demonstrations of water wave behaviour and interference patterns, it is the energy in the water that is responsible for that behaviour. It is energy that makes the water mobile and energy that keeps the individual "particles"(water molecules) separated. It is therefore no mystery why this apparatus is so appropriate in demonstrating the phenomena of the "behaviour" of light "particles".

Keep on thinking outside the box.There is still a great deal to be learned.

Good luck and keep up the good work.

crk

 

[wmadden]

Thank you so much for your words of encouragement. Glad you enjoyed the thread.

 

[zbde0000]

Why you say" the uncertainty,[PLAIN]https://www.physicsforums.com/latex_images/17/1746150-0.png is the same for both slits " in double slit.
The Ta,Tb trajectory for a same particle,I think may not be the same..