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

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How to Understand the Motion of a Particle in Quantum Mechanics? How to understand the motion of a particle in Quantum Mechanics? These are some extracts of reflections in the scientific community on the problem of motion in quantum mechanics that Zenone of Elea clearly envisioned with his paradoxes 2.600 years ago! In Classical Mechanics when we talk about the motion of a particle it is the same as talking about the idea of trajectory. The fact is that in Classical Mechanics, a particle has a definite position given by a point while in Quantum Mechanics the best we can get is a probability 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 sense, 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 trajectories, 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? Answer 1 Most people would say that |Ψ(a)|2|Ψ(a)|2 is the probability density 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 measurement merely reveals the value of that property that already existed) will literally get you in trouble if used in other situations. For instance a spin 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 work for position and it is called the de Broglie-Bohm (dBB) interpretation (or the pilot wave theory). The price to be paid is that if you say the probability is accurate for an actual position then other so called measurements can't be merely revealing preexisting values. This is because the measurements can be correlated with position so knowing the position forces the position to determine everything else (and the context and calibration of the exact set 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 pilot wave theory 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 probability current (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 waves 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 being located somewhere and then that updating in time. 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 frequencies you need without imagining particles 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 predictions and some of those trajectories are weird. For instance the wave function is defined on configuration space so it is the configuration that changes and so they can be highly nonlocal in many senses. What???? Answer 2 A quantum system is described by a suitable ∗∗-algebra of observables. 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 observable 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 operator 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 addition, the classical trajectory is actually x¯(t)x¯(t), in a regime where the quantum effects become negligible. So??? Answer 3 Quantum mechanically, motion consists of a series of localizations due to repeated interactions that, taken close to the limit of the continuum, yields a world-line. If a force acts on the particle, its probability distribution is accordingly modified. This must also be true for macroscopic objects, although now the description is far more complicated by the structure of matter 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 field, electrostatic forces arise from the non-constant phase character of the electric field 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 electromagnetism. 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 emergence of the classical world of human perception. Newtonian mechanics and Maxwell’s electromagnetism should be viewed as effective field theories for the “classical” world. blah, blah, blah! Answer 4 ASSUMPTIONS i. Space is discontinuous. It breaks down to indivisible space quanta; the space elements. ii. Space elements are stationary. They accommodate and transfer information. 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 universe demands transition time between them or otherwise instantaneous action (communication) is introduced. This would be in violation of everything science and human intuition stands for. Therefore, ΔΤ is required to protect what is already known or feels to be real.
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