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# How to Understand the Motion of a Particle in Quantum Mechanics?

How to understand the of a in ?

These are some extracts of reflections in the scientific community on the problem of motion in that Zenone of Elea clearly envisioned with his paradoxes 2.600 ago!

In when we talk about the motion of a particle it is the same as talking about the idea of . The fact is that in Classical Mechanics, a particle has a definite given by a point while in Quantum Mechanics the best we can get is a amplitude of the particle presence around a particular point. Because of that in Quantum Mechancis the idea of trajectory is meaningless. But nonetheless a particle move around, otherwise it would stay always where it is. In that , how should we understand the motion of a particle in Quantum Mechanics?

For instance, I've seem some books talking about "a particle propagating from left to right along the OxOx axis" or "a particle comes from the left" when talking about potential barriers. This is of course connected to the idea of motion of the particle, but since we don't have , I don't know how to understand those statements. In that setting, how should one understand intuitively and mathematically the idea of motion of a particle in Quantum Mechanics?

Most people would say that |Ψ(a)|2|Ψ(a)|2 is the probability that the particle would be found in some small neighborhood of aa. Most people would not say it is the probability that it is in that location. This is because that same kind of thinking (that it has a probability of having a property and that a merely reveals the value of that property that already existed) will literally get you in trouble if used in other situations. For instance a measurement objectively does not reveal a preexisting value of a component of a vector. And in fact a spin measurement objectively changes the thing you "measure." But there is a tradition that does do so, and it found a way to make that for position and it is called the de Broglie-Bohm (dBB) interpretation (or the pilot theory). The price to be paid is that if you say the probability is accurate for an actual position then other so called can't be merely revealing preexisting values. This is because the measurements can be correlated with position so knowing the position the position to determine else (and the context and calibration of the exact up can be a factor too and again this is because those can correlate to position). If we used different words this wouldn't sound so strange. We could call them spin polarizations instead of spin measurements for instance. And in the trajectories are not meaningless. However they aren't what you observe. You observer through interactions. So all you know is the interactions, you don't know the trajectory. In that sense, how should we understand the motion of a particle in Quantum Mechanics? There is a notion of (or probability current density). It is, for instance, how we compute the fraction of a beam that reflects or transmits through a barrier. You can imagine streamlines that have that current as their tangents. And for the pilot wave theory a particle has a location and it follows one of those streamlines. And for everyone else it is just one streamline of many.

Almost no one uses pilot since the trajectories are much more detailed than is needed just to compute what fraction of your results come out a certain way. But they can still talk about the probability current and that is usually what they mean if they talk about a particular direction of motion. They just don't literally think there is a particle with a location. They are just talking about current.

So you can have locations and if you do then you need to update them. Most people don't imagine locations but still do talk about the current, they just call it a probability current instead of imagining a probability of located somewhere and then that updating in .

And you would have a problem if you thought there were x y and z components of a spin vector and that the interactions we call "measuring the z component of spin" merely passively revealed that value. So if thinking they don't exist for position reminds them to not think they exist for components of a spin vector then great. It is great to learn not to make mistakes. And you can still compute all the you need without imagining having actual locations and moving around.

In that setting, how should one understand intuitively and mathematically the idea of motion of a particle in Quantum Mechanics? You can not worry about it and stick to probability currents, they will track the flow of probability which is all you need. Or you can study pilot wave theory, which doesn't help you make any new and some of those trajectories are weird. For instance the wave is defined on configuration so it is the configuration that changes and so they can be highly nonlocal in many .

What????

A quantum is described by a suitable ∗∗- of . The quantum states are functionals of the observables, that when applied on observables yield the average value of it in the state. So, given an AA, and a state ωω, ω(A)ω(A) is the evaluation of AA, i.e. its average value on the state. Now the state (or equivalently the observables) evolve in time. This means that we have a map ω(⋅)ω(⋅) that describes how the state changes with time. In other words, ωt(A)ωt(A) would describe the average value of AA at time tt. If we choose the position xx as an observable, we get a function of time that we may call effective (averaged) trajectory: x¯(t)=ωt(x).x¯(t)=ωt(x). This averaged trajectory may be interpreted as a notion of motion of a QM particle. Obviously, x¯(t)=x0x¯(t)=x0 it is not a good notion to say that a particle is static (the particle may be moving but be on average on the same place); but a non-constant function x(t)x(t) is a good indication of the motion of a QM particle. In , the classical trajectory is actually x¯(t)x¯(t), in a regime where the quantum effects become negligible.

So???

Quantum mechanically, motion consists of a series of localizations due to repeated interactions that, taken close to the limit of the continuum, yields a -line. If a acts on the particle, its is accordingly modified. This must also be true for macroscopic objects, although now the description is far more complicated by the structure of and associated surface physics. The motion of macroscopic objects, as was illustrated in the context of electrostatic forces, is governed by the quantum mechanics of its constituent particles and their interactions with each other. The result may be characterized as: collective quantum mechanical motion is classical motion. Since electromagnetic forces may be represented as a gauge , electrostatic forces arise from the non-constant character of the affecting many-particle wave functions. The example used was that of the force on a charged or uncharged metallic, conducting sphere placed in an initially constant and uniform electric field. As was written in the introduction, there is little that is new in this essay. On the other hand, quantum mechanics is widely viewed as being imposed on the well- understood classical world of Newtonian mechanics and Maxwell’s . This dichotomy is part of the pedagogy of physics and leads to much cognitive dissonance. In the end, there is no classical world; only a many-particle quantum mechanical one that, because of localizations due to environmental interactions, allows the of the classical world of . Newtonian mechanics and Maxwell’s electromagnetism should be viewed as effective field theories for the “classical” world. blah, blah, blah!

ASSUMPTIONS i. Space is discontinuous. It breaks down to indivisible space quanta; the space elements. ii. Space elements are stationary. They accommodate and transfer . iii. Each space element can accommodate only a finite amount of information. iv. Information transfer is not instant. Processing is required.

Time Time is rate of change. It is the rate at which the properties of space elements are modified according to the information they receive or transfer. There are two components of time; conscious time and processing time. Conscious time (CT) is experienced through life and recorded through instrumentation. Processing time (ΔΤ) is undetectable. The flow of time is absolute and defined as the sum of CT + ΔΤ. Depending upon properties of events absolute time consists of different ratios between its conscious and processing components. Since ΔΤ is undetectable perception of time relates only to the conscious component of time. Perceived time is therefore relative. Motion cannot exist in quantum space; there can only be state of presence and intention of motion. Relative motion Any sequence of events occurring in the demands transition time between them or otherwise instantaneous (communication) is introduced. This would be in violation of everything and human stands for. Therefore, ΔΤ is required to protect what is already known or feels to be real. In a quantum space environment the mechanics of motion are different than in space-time continuum. Quantum mechanics demands that all motion occurs in pulses since time and space are quantized. The following diagram simplifies this type of motion in a synoptic graphical format. Information named “X” travels in slow through successive space elements from left to right:

Total (absolute) time = CT + ΔΤ = 4+1+4+1+4+1+4 = 19 pulses Conscious time (CT) = 4x4 = 16 pulses Undetected time (ΔΤ) = 1+1+1 = 3 pulses Therefore, traveler “X” has aged 16/19 pulses, meaning that only 16 pulses registered in the traveler’s and physical state.

A faster moving traveler “Y” experiences the same absolute time of 19 pulses as follows:

Total (absolute) time = CT + ΔΤ = 2+1+2+1+2+1+2+1+2+1+2 +1 = 19 pulses Conscious time (CT): 2x6 + 1 = 13 pulses Undetected time (ΔΤ): 1+1+1+1+1+1 = 6 pulses Therefore, traveler “Y” has aged 13/19 pulses, meaning that only 13 pulses registered in the traveler’s consciousness and physical state.

Even though the total time is constant at 19 pulses, travelers at different register different conscious time. If both travelers had synchronized clocks at the point of origin, then at t=19 the clock of the faster moving traveler would have registered slower time flow. This is a direct consequence of disallowing instant transition between any two events with the introduction of an additional time component.

The previous examples are obviously not aligned with human intuition. The main reason is the difference in order of magnitude. Conscious time is what we experience through life. It is largely macroscopic. Transition time (ΔΤ) is always microscopic. ΔΤ is constant in a homogeneous medium (for example in ). It does not fluctuate. Therefore, the effect of motion in the perception of time flow is only relevant when CT and ΔΤ are of comparable magnitude. This can only occur at such high speed where CT is forced to diminish into a magnitude comparable to ΔΤ. This means that ΔΤ, the introduced transition time between events (the new time component), is of such microscopic order of magnitude that only becomes apparent in the special condition of events occurring at a which is fast enough to be comparable to the universe’s frequency. This statement treats motion as a periodic event repeating itself between successive space elements as they accommodate and transfer the traveling information.

BS, reminds me the phantomatic existance of ether! Nevertheless, interesting the graphical description of discrete movements at the tick of Planck clock...