You are watching: What does the square of the wave function represent?
Print version ISSN 0124-6127
discus.filos vol.14 no.22 Manizales Jan./June 2013
See more: Why Did The Transition To Collectivization Result In Widespread Starvation? ?
Interpreting the wave function what are electrons? And exactly how perform they move? Interpretación de la función de onda ¿qué child los electrones? Y ¿cómo se mueven?
Shan Gao Institute for the History of Natural Sciences, Chinese Academy of Sciences, China. gaoshan
Recibido el 18 de marzo de 2013 y aprobado el 13 de abril de 2013Abstract
In quantum mechanics, the physical state of an electron is explained by a wave attribute. According to the standard probability interpretation, the wave attribute of an electron is probability amplitude, and its modulus square offers the probcapability density of finding the electron in a details position in space. In this short article, we show that this main assumption of quantum mechanics may have actually an ontological extension. It is suggested that microscopic pposts such as electrons are indeed pposts, yet their activity is not consistent however discontinuous and also random. On this see, the modulus square of the wave feature not only offers the probability thickness of the particles being discovered in certain areas, but likewise gives the probability density of the pshort articles being there. In other words, the wave function in quantum mechanics can be regarded as a depiction of the state of random disconsistent activity of particles, and also at a deeper level, it might recurrent the dispositional residential or commercial property of the pshort articles that determines their random disconsistent activity.Key words
electrons, dispositional residential or commercial property, probcapability density, random disconsistent activity, wave attribute.Resumen
En la mecánica cuántica, el estacarry out físico de un electrón es descrito por una función de onda. Según la interpretación de probabilidad estándar, la función de onda de un electrón es amplitud de probabilidad, y su modulo cuadracarry out da la densidad de probabilidad de encontrar el electrón en una cierta posición en el espacio. En este artículo, se muestra que esta suposición main de la mecánica cuántica puede tener una extensión ontológica. Se argumenta que las partículas microscópicas como los electrones boy realmente partículas, pero su movimiento no es continuo, sino discontinuo y aleatorio. Desde esta perspectiva, el modulo cuadracarry out de la función de onda no sólo da la densidad de probabilidad de que las partículas se encuentren en ciertos lugares, sino que también da la densidad de probabilidad de que las partículas estén allí. En otras palabras, la función de onda en la mecánica cuántica se puede considerar como una representación del estaperform de movimiento discontinuo aleatorio de las partículas, y en un nivel más profuncarry out, puede representar la propiedad disposicional de las partículas que determina su movimiento discontinuo aleatorio.Palabras clave
electrones, propiedad disposicional, densidad de probabilidad, movimiento discontinuo aleatorio, función de onda.
The wave function offers not the density of stuff, yet provides fairly (on squaring its modulus) the thickness of probability. Probcapability of what, exactly? Not of the electron being tbelow, yet of the electron being uncovered tright here, if its place is 'measured'. Why this aversion to 'being' and insistence on 'finding'? The founding fathers were unable to form a clear photo of points on the remote atomic scale. (Bell)
The physical meaning of the wave function is an important interpretative trouble of quantum mechanics. The standard assumption is that the wave attribute of an electron is a probcapability amplitude, and also its modulus square offers the probcapacity thickness of finding the electron in a particular location at a provided immediate. This is generally dubbed the probcapacity interpretation of the wave function. Notwithstanding its good success, the probcapability interpretation is not wholly satismanufacturing facility because of resorting to the vague idea of measurement (Cf. Bell).
Recently a new penetrating evaluation mirrors that the wave feature not only provides the probcapability of getting different outcomes, yet also might sell a faithful depiction of fact (Pusey, Barrett and also Rudolph). This evaluation confirms the earlier outcome obtained based on protective measurements ((Aharonov and Vaidman) (Aharonov, Anandan and also Vaidman, "Meaning of")), and shows that the standard probcapability interpretation of the wave function is ripe for rereasoning. In reality, the realistic check out of the wave feature is currently a common presumption in the primary alternatives to quantum mechanics such as the de Broglie-Bohm theory and the many-worlds interpretation ((de Broglie) (Bohm) (Everett) (DeWitt and Graham)). Unfortunately, however, the specific interpretation of the wave attribute is still an unrefixed problem in these theories.
What, then, does the wave attribute truly represent? In this short article, we will try to answer this standard question with a brand-new evaluation of protective dimensions and also the mass and charge distributions of a quantum system. The answer might help to understand also the deep nature of quantum reality.
Measuring the state of a quantum system
The definition of the wave feature is often analyzed in the context of typical (impulsive) dimensions, for which the coupling interaction between the measured device and the measuring device is of short duration and solid. As a result, also though the wave function of a quantum device is in general extfinished over area, a suitable place measurement deserve to just detect the system in a random position in space1. Then it is unsurprising that the wave attribute is assumed to be pertained to the probability of the random measurement outcome by the conventional probability interpretation. This likewise indicates that traditional measurements cannot attain sufficient information about a solitary quantum device to identify what physical state its wave feature represents.
Fortunately, it has actually been recognized that the physical state of a solitary quantum device can be protectively measured ((Aharonov and also Vaidman) (Aharonov, Anandan and also Vaidguy "Meaning of") (Aharonov, Anandan and Vaidguy, "The meaning of") (Vaidman))2. A basic method is to let the measured system be in a nondegeneprice eigenstate of the totality Hamiltonian using an ideal protective interaction (in some situations the defense is offered by the measured mechanism itself), and then make the measurement adiabatically so that the state of the device neither collapses nor becomes entangled with the measuring device appreciably. In basic, the measured state needs to be recognized beforehand in order to arvariety a correct defense. In this way, such protective dimensions can meacertain the expectation values of observables on a single quantum mechanism, and also in certain, the mass and charge distributions of a quantum device as one component of its physical state, and also its wave function, can be measured as expectation values of particular observables. Because the principle of protective measurement is independent of the controversial collapse postulate and only based on the direct Schrödinger evolution (for microscopic units such as electrons) and also the Born rule3, which are two establiburned components of quantum mechanics, its result as predicted by quantum mechanics can be offered to investigate the definition of the wave function4.
According to protective measurement, the charge of a charged quantum mechanism such as an electron is dispersed throughout area, and also the charge density in each position is proportional to the modulus square of the wave function of the mechanism there. Historically, the charge thickness interpretation for electrons was initially said by Schrödinger once he introduced the wave attribute and also started wave mechanics (Schrödinger). Schrödinger plainly realized that the charge density cannot be timeless because his equation does not encompass the usual timeless interactivity between the densities. Presumably given that civilization believed that the charge thickness might not be measured and also lacked a continuous physical photo, this initial interpretation of the wave attribute was quickly rejected and reput by Born's probability interpretation (Born). Now protective measurement re-endows the charge circulation of an electron via truth by a more convincing argument. The question then is how to discover a continual physical explacountry for it5. Our following evaluation have the right to be regarded as a additionally advancement of Schrödinger's original concept to some degree. The twist is: that the charge distribution is not classic does not indicate its non-existence; quite, its existence may allude to a brand-new, non-timeless image of quantum reality that hides behind the mathematical wave function.
Electrons are particles
The vital to unveil the meaning of the wave function is to discover the physical origin of the charge distribution. The charge circulation of a quantum device such as an electron has actually 2 feasible existent forms: It is either actual or effective. The distribution is genuine indicates that it exists throughout space at the same time, e.g. tright here are different charges in different positions at any type of prompt. The distribution is efficient implies that tbelow is only a localized pwrite-up via the total charge of the device in one place at eexceptionally prompt, and the time average of its activity (in the time of an infinitesimal time interval) forms the reliable circulation in the whole room. Moreover, given that the integral of the created charge thickness in any kind of area is required to be equal to the average value of the full charge in the area, the motion of the pshort article is ergodic.
These two existent creates of the charge circulation of a quantum system have actually different physical impacts, and therefore they have the right to be distinguimelted. Experiments show that different charges in various positions at a given instant have electrostatic interactivity, while a charge at one immediate has actually no electrostatic interactivity with the charge at an additional immediate. Therefore, if the charge circulation is efficient, then tright here will certainly exist no electrostatic self-interaction of the circulation, while if the charge distribution is actual, then tbelow will certainly exist electrostatic self-interactivity of the distribution. In short, the initially form entails the existence of electrostatic self-interaction of the charge circulation of a quantum mechanism, while the second develop does not.
Due to the fact that the visibility of electrostatic self-interactivity is inregular via the superposition principle of quantum mechanics, and also especially, the existence of such electrostatic self-interactivity for individual electrons already contradicts speculative observations (e.g. the results of the double-slit experiments through single electrons)6, the charge distribution of a quantum mechanism such as an electron have to be efficient. This means that at eextremely instant there is just a localized pshort article via the full mass and charge of the system, and also during an infinitesimal time interval the time average of the ergodic activity of the particle develops the efficient mass and also charge distributions of the system. In short, electrons are pshort articles, and also their charge distributions in room, which are measureable by protective dimensions, are developed by the ergodic motion of these particles.
Particles move in a disconstant and also random way
The following question is which sort of ergodic motion the pwrite-ups undergo. If the ergodic activity of a pshort article is constant, then it deserve to only develop the mass and charge distributions throughout a finite time interval. But the mass and charge distributions of a quantum device at each instant, which is proportional to the modulus square of the wave attribute of the mechanism at the prompt, is required to be formed throughout an infinitesimal time interval near the immediate. Therefore it appears that the ergodic activity of the ppost cannot be continuous.
We have the right to also reach this conclusion by analyzing a concrete instance. Consider an electron in a superposition of 2 power eigenclaims in two separated boxes Y1(x) + Y2(x). In this example, even if one assumes that the electron as a localized pshort article have the right to relocate through boundless velocity, it cannot consistently move from one box to another because of the restriction of box wall surfaces. Thus, any type of type of constant movement cannot geneprice the effective charge thickness e|Y1(x) + Y2(x)|2. One may object that this is simply an artifact of the idealization of infinite potential. However before, also in this ideal situation, the version should also be able to generate the reliable charge circulation by indicates of some sort of ergodic movement of the electron; otherwise it will certainly be inconsistent via quantum mechanics7.
On the various other hand, if the activity of a particle is disconsistent, then the ppost deserve to conveniently move throughout all regions where the wave feature is nonzero in the time of an arbitrarily brief time interval at a offered prompt. Additionally, if the probcapability thickness of the pwrite-up showing up in each position is proportional to the modulus square of its wave feature tright here at every immediate, the discontinuous movement have the right to also geneprice the right effective mass and charge distributions. This might deal with the troubles plagued by the classic ergodic models. The discontinuous ergodic motion needs no visibility of a finite ergodic time. A pshort article undergoing disconsistent activity have the right to additionally relocate from one area to an additional spatially separated region, no matter whether tbelow is an unlimited potential wall between them, and also such disconstant activity is not influenced by the setting and also setup in between these regions either.
In conclusion, we have actually suggested that the mass and also charge distributions of a quantum system such as an electron are created by the disconstant activity of a localized pwrite-up via the complete mass and also charge of the device, and also the probability density of the particle appearing in each position is proportional to the modulus square of its wave function tbelow.
Meaning of the wave function
According to the over evaluation, microscopic particles such as electrons are indeed pshort articles. Here the concept of particle is offered in its usual sense. A particle is a small localized object through mass and also charge, and it is only in one position in area at an immediate. Moreover, the movement of these pwrite-ups is not continuous but disconsistent in nature. We may say that an electron is a quantum particle in the feeling that its activity is not continuous movement defined by timeless mechanics, yet disconstant activity explained by quantum mechanics.
From a logical suggest of see, for the discontinuous motion of a quantum particle, tright here must exist a probabilistic instantaneous condition that determines the probcapacity thickness of the pwrite-up showing up in eincredibly place in area, otherwise it would certainly not "know" exactly how commonly they have to appear in each place in area. In other words, the pshort article must have an instantaneous property that determines its motion in a probabilistic way. This residential or commercial property is commonly referred to as indeterministic disposition or propensity in the literature8. As an outcome, the place of the pshort article at every prompt is random, and also its trajectory formed by the random place series is also disconstant. In short, the movement of the particle is fundamentally disconsistent and also random.
Unchoose the deterministic continuous motion, the trajectory attribute x(t) have the right to no longer administer a useful description for random disconstant movement. For a quantum pwrite-up, tright here is no consistent trajectory at all. Rather, the random disconstant activity of the particle develops a ppost "cloud" extfinishing throughout room (in an infinitesimal time interval), and also the state of activity of the particle is stood for by the thickness and flux thickness of the cloud, denoted by p(x, t) and also j(x, t), respectively. This is equivalent to the description of a classical liquid in hydrodynamics. But their physical definitions are different. The pwrite-up cloud is formed by the random discontinuous motion of a single pshort article, and also the thickness of the cloud, p(x, t), represents the objective probcapability thickness of the particle appearing in position x at immediate t. By assuming that the nonrelativistic equation of activity is the Schrödinger equation in quantum mechanics9, the facility wave function Y(x, t) have the right to be uniquely expressed by p(x, t) and also j(x, t) (except for a consistent phase factor):
In this means, the wave feature Y(x, t) likewise offers a finish description of the state of random disconsistent activity of a ppost.
The summary of the activity of a single ppost deserve to be extended to the motion of many type of pshort articles. At each immediate the quantum device of N pshort articles can be represented by a point in a 3N-dimensional configuration room, and also the motion of these particles creates a cloud in the configuration area. Then, equivalent to the single pwrite-up case, the state of the device is represented by the thickness and flux density of the cloud in the configuration room, p(x1, x2…, xN) and j(x1, x2…, xN), wright here the density p(x1, x2…, xN) represents the probcapability thickness of pshort article 1 showing up in place x1 and particle 2 showing up in place x2, …, and also pshort article N appearing in position xN. Because these two amounts are defined not in the actual three-dimensional space, but in the 3N-dimensional configuration space, the many-particle wave function, which is created of these two quantities, is additionally characterized in the 3N-dimensional configuration space.
One essential allude requirements to be stressed below. Due to the fact that the wave attribute in quantum mechanics is characterized at a provided instant, not during an infinitesimal time interval, it have to be concerned not sindicate as a description of the state of motion of pwrite-ups, yet more suitably as a description of the dispositional building of the particles that determines their random discontinuous activity at a deeper level10. In specific, the modulus square of the wave feature determines the probcapacity thickness of the pposts showing up in certain positions in area. By comparison, the density and flux thickness of the pshort article cloud, which are defined in the time of an infinitesimal time interval at a provided immediate, are just a description of the state of the resulting random disconsistent motion of pwrite-ups, and they are determined by the wave function. In this feeling, we may say that the motion of pshort articles is "guided" by their wave function in a probabilistic way.
In this short article, we have actually suggested that quantum mechanics may have actually already spelled out the definition of the wave function. Tright here are three primary actions to reach this conclusion.
First of all, protective measurement, whose principle is based on the establimelted parts of quantum mechanics, shows that the charge of a charged quantum mechanism such as an electron is distributed throughout space, and also the charge thickness in each position is proportional to the modulus square of its wave function tbelow. Next, the superposition principle of quantum mechanics needs that the charge distribution is efficient, that is, it is formed by the ergodic motion of a localized particle through the full charge of the mechanism. Lastly, the consistency of the developed circulation with that predicted by quantum mechanics needs that the ergodic movement of the pwrite-up is discontinuous, and the probability thickness of the pshort article showing up in eincredibly place is equal to the modulus square of its wave feature there.
As such, quantum mechanics appears to indicate that the wave attribute describes the state of random disconsistent activity of pposts, and at a deeper level, it represents the dispositional residential property of the pwrite-ups that determines their random disconsistent motion. In certain, the modulus square of the wave feature not just gives the probcapability thickness of the pshort articles being found in particular places as the traditional probcapability interpretation assumes, but additionally provides the probcapability thickness of the particles being tbelow. It will be interesting to view exactly how this new interpretation of the wave feature deserve to be extfinished to quantum area concept and also what it indicates for the solutions to the measurement difficulty.
1 In this article we only take into consideration the spatial wave functions of quantum devices. 2 Note that the earlier objections to the validity and also definition of protective dimensions have been answered ((Aharonov, Anandan and Vaidmale, "The meaning of") (Dass and also Qureshi) (Vaidman) (Gao, "Comment on")). 3 It is worth noting that the possible presence of extremely slow collapse of the wave feature for microscopic units does not affect the principle of protective measurement. 4 It have the right to be supposed that protective measurements will certainly be realized in the near future through the rapid advancement of quantum innovations (Cf. Lundeen et al.). 5 The proponents of protective measurement did not analyze the origin of the charge distribution. According to them, this form of measurement means that the wave attribute of a solitary quantum system is a actual physical wave (Aharonov, Anandan and also Vaidguy, "Meaning of"). 6 As one more example, take into consideration the electron in the hydrogen atom. If tbelow exists such electrostatic self-interaction for individual electrons, then because the potential of the electrostatic self-interaction is of the same order as the Coulomb potential produced by the nucleus, the energy levels of hydrogen atoms will be remarkably different from those predicted by quantum mechanics and also shown by experiments. For a in-depth analysis view Gao ("The wave function", "Meaning of the", "Interpreting quantum"). 7 It is exceptionally widespread in quantum optics experiments that a single-photon wave packet is split right into 2 branches moving along 2 well separated paths in room. In certain, the experimental results are not influenced by the setting and also setup between the 2 routes of the photon. Therefore it is incredibly difficult to imagine that the photon performs a continuous ergodic motion ago and also forth in the room between its two courses. 8 It is worth stressing that the propensities possessed by the pwrite-ups relate to their objective movement, not to the dimensions on them as in the existing propensity interpretations of quantum mechanics (Cf. Suarez). 9 For a derivation of the cost-free Schrödinger equation check out Gao ("Interpreting quantum"). 10 For a many-ppost mechanism in an entangled state, this dispositional residential property is possessed by the totality device.
Aharonov, Y., Anandan, J. and also L. Vaidmale. "Meaning of the wave function". Phys. Rev. 1993: 4616. Publish. < Links >
---. "The interpretation of protective measurements" Found. Phys. 1996: 117. Publish. < Links >
Aharonov, Y. and also L. Vaidman. "Measurement of the Schrödinger wave of a single particle". Phys. Lett. 1993: 178-238. Print. < Links >
Bell, J. S. "Against 'measurement'". Miller, A. I. (ed.). Sixty-Two years of uncertainty: Historical thoughtful and also physics enquiries right into the structures of quantum mechanics. Berlin: Springer, 1990. Publish. < Links >
Bohm, D. "A argued interpretation of quantum concept in terms of "hidden" variables, I and II". Phys. Rev. Jan. 1952: 166-193. Publish. < Links >
Born, M. "Quantenmechanik der Stovslrfc.orgänge". Z. Phys. 1926: 803-827. Print. < Links >
Dass, N. D. H. and also T. Qureshi. "Critique of protective measurements". Phys. Rev. 1999: 2590. Print. < Links >
DeWitt, B. S. and also N. Graham (eds.). The many-worlds interpretation of quantum mechanics. Princeton: Princeton University Press, 1973. Print. < Links >
Everett, H. "'Relative state' formulation of quantum mechanics". Rev. Mod. Phys. Jun. 1957: 454-462. Publish. < Links >
Gao, S. "Comment on "How to defend the interpretation of the wave function versus protective measurements" by Jos Uffink". Phil. Sci. Dec. 2011. Online. < Links >
---. "The wave function and also quantum reality". Khrennikov, A., Jaeger, G., Schlosshauer, M. and G. Weihs (eds.). Proceedings of the International Conference on Advances in Quantum Theory. Boston: Amerihave the right to Institute of Physics, 2011. Print. < Links >
---. "Meaning of the wave function". International Journal of Quantum Chemistry. Dec. 2011: 4124-4138. Print. < Links >
---. "Interpreting quantum mechanics in regards to random discontinuous movement of particles". Phil. Sci. Feb. 2013. Online. < Links >
Lundeen, J. S. et al. "Direct measurement of the quantum wave function". Physical Resee Letters. Feb. 2012: 188-191. Publish. < Links >
Pusey, M., Barrett, J. and also T. Rudolph. "On the fact of the quantum state". Nature Phys. 2012: 475-478. Online. < Links >
Schrödinger, E. "Quantizierung als Eigenwertdifficulty (Vierte Mitteilung)". Ann. d. Phys. 1926: 109-139. Online. < Links >
Suarez, M. "Quantum propensities". Studies in the History and Philosophy of Modern Physics. Jun. 2007: 418-438. Online. < Links >
Vaidguy, L. "Protective measurements". Greenberger, D., Hentschel, K. and also F. Weinert (eds.). Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy. Berlin: Springer-Verlag, 2009. Publish. < Links >