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?
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.
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.
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!
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. 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 speed 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 consciousness 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 speeds 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 vacuum). 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 frequency 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...
As we know, what classical mechanics describes is continuous motion of particles. Then a natural question appears when we turn to quantum mechanics, i.e. which motion of particles does quantum mechanics describe? This is not an easy question. In fact, people have been arguing with each other about its solution since the founding of quantum mechanics. Present realistic interpretations of quantum mechanics all assume that the particles still undergo the continuous motion. But in order to hold this assumption, they must assume another kind of physical reality besides particles. Hidden variables theory assumes the physical existence of y-wave field, which provides the quantum potential to create the quantum behaviors of the classical particles undergoing continuous motion. Many-worlds interpretation also assumes the physical existence of y-wave field. In every world the particles undergo the continuous motion. Stochastic interpretation assumes the existence of the background fluctuation field, which is required to result in the Brown motion of classical particles for accounting for their quantum displays. Even though the present realistic interpretations of quantum mechanics are all dualistic , it doesn’t preclude the existence of a monistic realistic interpretation of quantum mechanics. In a recent paper , a theory of discontinuous motion of particles is presented. It only assumes one kind of physical reality---particles, and concludes that the bizarre quantum behavior of particles only results from the motion of particles itself. Concretely speaking, it is shown that the wave function in quantum mechanics is the very mathematical complex describing the discontinuous motion of particles in continuous space-time, and the simplest nonrelativistic evolution equation of such motion is the same as Schroedinger equation. Considering the fact that space-time may be essentially discrete when combining quantum theory and general relativity, the author further analyzes the discontinuous motion of particles in discrete space-time, and show that its evolution may naturally result in the dynamical collapse process of the wave function. These analyses strongly imply what quantum theory describes may be discontinuous motion of particles.
The continuity of spaced-time is just an assumption. In the nonrelativistic and relativistic domain this assumption can be applicable, and we find no essential inconsistency or paradox. But in the domain of general relativity, the motion of particle and the space-time background are no longer independent, and there exists one kind of subtle dynamical connection between them. Thus the combination of the above evolution law of discontinuous motion (or quantum mechanics) and general relativity may result in essential inconsistency, which requires that the assumption of continuous space-time should be rejected and may further result in the appearance of dynamical collapse. Now let’s have a close look at it.
According to general relativity, there exists one kind of dynamical connection between motion and space-time, i.e. on the one hand, space-time is determined by the motion of particles, on the other hand, the motion of particle must be defined in space-time. Then when we consider the superposition state of different positions, say position A and position B, one kind of basic logical inconsistency appears. On the one hand, according to the above evolution law of the discontinuous motion of particles (or quantum mechanics), the valid definition of this superposition requires the existence of a definite space-time structure, in which the position A and position B can be distinguished. On the other hand, according to general relativity, the space-time structure, including the distinguishability of the position A and position B, can’t be pre-determined, and it must be dynamically determined by the superposition state of particle. Since the different position states in the superposition state will generate different space-time structures, the space-time structure determined by the superposition state is indefinite. Thus an essential logical inconsistency does appear! Then what are the direct inferences of the logical inconsistency? First, its appearance indicates that the superposition of different positions of particle can’t exist when considering the influence of gravity, since it can’t be consistently defined in principle. It should be stressed that this conclusion only relies on the validity of general relativity in the classical domain, and is irrelevant to its validity in the quantum domain. Thus the existence of gravity described by general relativity will result in the invalidity of the linear superposition principle. This may be the physical origin of dynamical collapse of wave function. Secondly, according to the physical definition of the superposition state of different positions of particle, its existence closely relates to the continuity of space-time, concretely speaking, it requires that the particle in this state should move throughout these different positions during infinitesimal time interval. Thus the nonexistence of this superposition means that infinitesimal time interval based on continuous space-time will be replaced by finite time interval, and accordingly the space-time where the particles move will display some kind of discreteness. In this kind of discrete space-time, the particle can only move throughout the different positions during finite time interval, or we can say, the particle will stay for finite time interval in any position.
Besides, it can prove that when considering both quantum mechanics and general relativity, the minimum measurable time and space size will no longer infinitesimal, but finite Planck time and Planck length. Here we will give a simple operational demonstration. Consider a measurement of the length between points A and B. At point A place a clock with mass m and size a to register time, at point B place a reflection mirror. When t = 0 a photon signal is sent from A to B, at point B it is reflected by the mirror and returns to point A. The clock registers the return time. For the classical situation the measured length will be L ct 2 1 = , but when considering quantum mechanics and general relativity, the existence of the clock introduces two kinds of uncertainties to the measured length. The uncertainty resulting from quantum mechanics is: 1/ 2 ( ) mc L LQM h d ³ the uncertainty resulting from general relativity is: 2 c Gm dLGR ³ , then the total uncertainty is: dL = dLQM +dLGR 2 1 / 3 ( ) L Lp ³ × , where LP = 1 / 2 3 ( ) c Gh , is Planck length. Thus we conclude that the minimum measurable length is Planck length LP . In a similar way, we can also work out the minimum measurable time, it is just Planck time TP = 1 / 2 5 ( ) c Gh . Lastly, we want to denote that the existence of discreteness of space-time may also imply that the many worlds theory is not right, and the collapse of wave function does exist. Since there exists a minimal time interval in discrete space-time, and each parallel world must solely occupy one minimal time interval at least, there must exist a maximal number of the parallel worlds during any finite time interval. Then when the number of possible worlds exceeds the maximal number, they will be merged in some way, i.e. the whole wave function will collapse to a smaller state space.
A theory of dynamical collapse in discrete space-time In this section, we will briefly introduce the dynamical collapse theory based on the discontinuous motion of partic les in discrete space-time. In the paper , the discontinuous motion of particles in discrete space-time and its evolution are further analyzed, and a preliminary theory of dynamical collapse in such discrete space-time is presented. First, the motion state of a particle in discrete space-time is defined. According to the meaning of discrete space-time, the existence of a particle is no longer in one position at one instant as in the continuous space-time, but limited in a space interval LP during a finite time interval TP . This defines the instantaneous state of particle in discrete space-time. Similar to the situation in continuous space-time, the motion state of particle in discrete space-time is that during a finite time interval much larger than TP , the particle moves throughout the whole space, which proper description is still the measure density r(x,t) and measure flux density j(x,t) , but time-averaged. The visual physical picture of such motion will be that during a finite time interval TP the particle stays in a local region with size LP , then it will still stay there or appear in another local region, which may be very far from the original region, and during a time interval much larger than TP the particle will move throughout the whole space with a certain average position measure density r(x,t) . After defining the motion state of particle in discrete space-time, the evolution of the discontinuous motion of particles in discrete space-time is analyzed, and a preliminary nonrelativistic evolution equation is worked out. The evolution equation is a revised Schroedinger equation containing two evolution terms, in which the first term is the usual linear Hamiltonian in Schroedinger equation, the second term is a new stochastic nonlinear evolution term resulting from the stochastic change of the position measure density r(x,t) . It is stressed that the equation is essentially a discrete evolution equation in physics, and all of the quantities are defined relative to the Planck cells TP and LP . It is further demonstrated that the revised Schroedinger equation will naturally result in the dynamical collapse of wave function. As an example, the dynamical collapse time of a two-level system can be concretely calculated. It is 2 2 ( ) 2 E E k p c D » h t 1 , where DE is the difference of the energy between these two states. Thus the discontinuous motion of particles in discrete space-time turns out to be a possible basis of the theory of dynamical collapse, and may provide a preliminary framework for the complete quantum theory. In the following, we will further present some possible evidence for the conclusion that the evolution of the discontinuous motion of particles in discrete space-time will naturally result in the dynamical collapse of wave function. First, as to the discontinuous motion of particles in discrete space-time, since the particle does stay in a local region for a finite nonzero time interval, and appears stochastically in another local region during the next time interval, the position measure density r(x,t) of the particle, when changed due to the invalidity of the linear superposition principle, will be essentially changed in a stochastic way, which closely relates to the concrete stay time in different stochastic region 1 , and the corresponding wave function will be also stochastically changed. Thus the evolution of discontinuous motion in discrete space-time should be the combination of the deterministic linear evolution and stochastic nonlinear evolution. Secondly, we need to further find the concrete cause resulting in the stochastic change of the position measure density r(x,t) , and to see whether it will really result in the dynamical collapse of wave function. As we know, the evolution of wave function is determined by the Hamiltonian of the system, or the energy distribution of the system. Thus the stochastic change of the evolution may also relate to the energy distribution of the system. Now consider a simple two-level system, which state is a superposition of two static states with different energy levels E1 and E2 , and its position measure density r(x,t) will oscillate with the period of T = h / DE , where DE = E2 - E1 is the energy difference. Then if the energy difference DE is so large that it exceeds the Planck energy Ep , the position measure density r(x,t) will oscillate with a period shorter than the Planck time TP . But as we know, the Planck time TP is the minimum distinguishable time size in the discrete space-time, and there should be no changes during this minimal time interval. Thus the energy superposition state, in which the energy difference is larger than the Planck energy Ep , can’t hold all through, and must gradually collapse to one of the energy eigenstates. It can be further inferred that the dynamical collapse process must happen for any energy superposition state due to the general validity of the natural law including the collapse law. The above analysis has indicated that, when the energy difference between different branches of the wave function is large enough, say, for the macroscopic situation1 , the linear spreading of the wave function will be greatly suppressed, and the evolution of the wave function will be dominated by the localizing process. Thus a macroscopic object will be always in a local position, and it can only be still or continuously move in space in appearance. This is just the display of continuous motion in the macroscopic world. Furthermore, it is shown that the evolution law of continuous motion can also be derived from the evolution of the discontinuous motion in discrete space-time. In this section, we will demonstrate that the instant motion of particle should be essentially discontinuous and random. This will give the logical basis of discontinuous motion. Since what quantum mechanics describes should be the discontinuous motion of particles, this may also answer the question ‘why the quantum?’. As we know, the object can move or change its position when there is not any outer cause such as outer force. This is an experiential fact, for example, when you kick a ball, it can move freely afterwards. This fact is well summarized in Newton’s first law. Besides, there are also other similar phenomena in the microscopic world, for example, the emission of alpha particles by radioactive isotopes happens without outer cause, or we can say, the alpha particles can spontaneously move out from the radioactive isotopes. On the other hand, there may exist some deep reasons for this counterintuitive fact. One possible reason is that if the object can’t spontaneously move, then the whole world will hold still. As we have known from modern physics, the interaction or force between particles is transferred by the other particles. Now if the particle can’t move in a spontaneous way, or it can only move when there is an outer force, then on the one hand, the particle can’t move without outer force, on the other hand, the outer force can’t exist without the moving particles, which transfer the force. Thus all particles will be motionless, and all forces will not exist either. In one word, the whole world will be in a deadly still state. The direct inference of this conclusion may be that the world will not exist either. Since there is no motion and interaction, the properties of particle, which closely relates to motion and interaction, will disappear, and the particle devoid of any properties will not exist either. Then the world also disappears, and nothing exists1 . Thus it seems that the objects must move spontaneously in order to exist. We can define the ability of the spontaneously moving of object as the nature or activity of object. Then such nature may be taken as the inner cause for the spontaneous motion of object. Since the nature of spontaneous motion of object doesn’t change all the while, this kind of inner cause is independent of time and concrete motion processes. The object can move spontaneously, then how does it move spontaneously? This is a very interesting and important problem. In the following, we will find the instant moving way of the object. Since the activity of object or the only cause resulting in the spontaneous motion of object is irrelevant to time and concrete motion processes, there is no cause to determine how the object moves spontaneously in space and time. This means that there is no cause to determine the concrete instant motion of the object, i.e. the object is neither determined to move in one special way, nor determined to move in the other special way. Thus the object can only move in a completely random way, or we can say, the instant motion of object must be essentially discontinuous everywhere. As we can see, the reason why the object moves in a random way is just because there isn’t any cause to determine a special regular moving way. In short, the object must move, but it doesn’t know how to move, so it can only move in a random and discontinuous way. The above conclusion is also justifiable from a mathematical point of views . As we know, the motion state of an object in continuous space-time is the infinitesimal interval state, not the instantaneous state. Then the motion state of the object is a point set in space-time1 , but which type of point set is it? According to the mathematical analysis of point set2 , the natural assumption in logic is that it is a general dense point set in space-time, since we have no a priori reason to assume a special form, say a continuous point set. Thus during the infinitesimal interval near any instant the object will always move in a random and discontinuous way. One big obstacle to understand the above conclusion is that people usually think that there exist some laws, say Newton’s first law, to determine the existence of a special instantly moving way. Here we will further argue that there doesn’t exist such laws at all. Firstly, all laws referring to the time interval, including the infinitesimal time interval, can’t determine such instantly moving way. The reason is very simple, since these laws refers to the time-interval motion state of objects, and they are all based on the supposed instantly moving way of objects during the time interval, for example, Newton’s first law presupposes the existence of continuous moving way. But in the above discussions, what we consider is the instant motion, not the time-interval state and its evolution, and what we need to find is just the instantly moving way within the time interval. Secondly, physical laws only consider the time-interval motion state of objects. This can be easily seen from the mathematical quantities dt and dx which appear everywhere in physics. Furthermore, present physics doesn’t analyze the way of instant motion. It only supposes the way of instant motion, for example, classical physics presupposes that the instant motion is continuous. Now as an example, let’s see why the instant motion is not continuous. Since what we analyze is the instant motion, the velocity, which is defined on the time interval, doesn’t exist yet, and Newton’s first law can’t help either. Then the free object has no velocity to hold, and it really doesn’t know which direction to move along. Thus the object can’t move in a continuous way, since continuous motion requires a definite direction, for example, in one-dimensional situation, the object must select a preferred direction, right or left to move continuously.
This finally sounds as a serious attempt to explain the mechanics of discrete motion. One might suggest a collapse and a rebuild of the waveform at each transition from a point set to the next. Yet, it does not approach the fundamental question: "How does the particle know if and when to evolve from a given point set to the next? How is represented in the wave function the information of future point set? Otherwise, how do we distignuish two particles, one steady and the other waiting to move?"