A possible mechanism to explain wave-particle duality L D HOWE No current affiliation PACS Numbers: r, w, k

Transcription

1 A possible mechanism to explain wae-particle duality L D HOWE No current affiliation PACS Numbers: r, w, k Abstract The relationship between light speed energy and the kinetic energy of non-relatiistic particles is considered and an equation proposed to unify relatiistic and non-relatiistic energy. The proposed equation leads naturally to the postulation of the de Broglie ector to represent any relatiistic or non-relatiistic particle. By adopting the concept of the de Broglie ector, it is possible to account for the wae-particle duality associated with quantum mechanics. This is demonstrated for both diffraction and refraction. The angular momentum of the de Broglie ector can be considered as the spin of a particle, pointing along the axis of rotation. The isualisation of a particle as simply a de Broglie ector leads to the idea that all matter is an embodiment of energy trapped within a potential well. If energy, in the form of a photon, escapes the potential well and reacts in a negatie manner to graitation, it would accelerate away from the potential well to the speed of light within an immeasurably short distance. 1 Introduction There is no reason to suppose that there is any essential difference between space and matter. Nor is there any reason to suppose that matter and energy are not simply two embodiments of the same thing. There exist in space primary, directly detectable behaiours, which lead us to model them as the results of force fields. These are the electric field and the graitational field. We presume the apparent source of these fields to be respectiely charge and mass. When charges moe in space, a third force field is obsered, the magnetic field. When mass moes in space its moement is associated with momentum. Our modelling of physical matter leads us to assume that other forces exist between subatomic particles, namely the strong and weak forces. In this paper we try to rationalise the relationship between matter and energy and to explain the quantum mechanical mystery of wae particle duality together with its implications. 1

2 Energy and mass equialence If we consider a particular photon as a quantum of pure energy, we denote its energy e by the well known equation e hf [.1] where h is Planck s constant and f is the characteristic frequency associated with that particular photon. Now from basic wae theory we can define the frequency f by so or c f [.] hc e [.3] hc e [.4] Now there is an equialence between energy and mass ia Einstein s well known equation e mc [.5] where m is the rest mass of a physical entity. In this analysis, we use the term me to indicate that the photon has no mass but m is its mass equialence. We now hae rearranging we get E hc [.6] mec h m c [.7] E In Equation.7, is the Compton waelength of a physical particle with rest mass m E. This reminds us of the expression for the de Broglie waelength of a physical particle with rest mass m E and elocity h db m [.8] E Now we can define the de Broglie frequency ia Which leads to In other words db [.9] fdb m f h m [.10] db E E hf db [.11]

3 But remembering that, for a physical particle e ½mE [.1] We arrie at e ½hf db [.13] So there is a question as to how we can reconcile [.1] with [.13]. We surmise that both could be written as e ½hf 1 db [.14] c c, this equates to [.1], whereas when It can be seen that when c, it equates to [.13]. Combining [.14] with [.11] we hae e ½m E 1 [.15] 3 The de Broglie ector Here we hypothesise that eery particle with a de Broglie waelength has an associated de Broglie ector. We can rewrite [.15] as e ½mE db [3.1] where db is the elocity of the de Broglie ector db [3.] c corresponding to the rotation of the de Broglie ector in de Broglie space. If we consider db = r db, where is the rotational elocity in radians per second and r db is the amplitude of the de Broglie ector, we can calculate the amplitude of the de Broglie ector using = f db, where f db is the de Broglie frequency. From [.11] me fdb [3.3] h Therefore, the amplitude of the de Broglie ector, r db, is gien by: which reduces to: r c h db me [3.4] c r m c db [3.5] E Note that the expression for r db is identical to that for the reduced Compton waelength for a physical particle with rest mass m E and is independent of the frame of reference of any obserer. It can therefore be regarded as the de Broglie signature of a particle, whether a physical particle or a photon. 3

4 4 How a particle can behae as a wae By describing a particle as a de Broglie ector rotating in de Broglie space it becomes clear that both the classical concept of a particle and the quantum mechanical description of a particle as a wae packet can be unified within a single model. It is no longer necessary to describe a particle as two different manifestations, depending on the situation. The conundrum of wae-particle duality has long been reported in een the most elementary of texts. Waelike properties are required to account for diffraction. The difficulty has always been associated with the fact that indiidual particles behae like particles, but the aerage behaiour of a large number of particles resembles that of a wae.. Diffraction The simplest case used to demonstrate this behaiour is the double slit experiment, shown in Figure 4.1. Each particle can be detected at the detector as a discrete eent. Double slit apparatus Detector Path of incident particle Path of particle to detector Figure 4.1: The double slit experiment. Incident particles pass through the double slit apparatus and are detected as eents at the detector. The question arises, how can independent particles, each of which is recorded as a discrete eent at the detector, somehow collaborate to form a waelike pattern? Additionally, if an attempt is made to discoer through which slit each particle passes, the waelike effect disappears. This is usually accounted for by citing the Heisenberg uncertainty principle, but without any attempt to explain how the slit measurement interferes with the experiment. 4

5 Relatie Amplitude Before we consider the double slit experiment, we should also consider the single slit situation and a theoretical no slit experiment. In this thought experiment, we define the source of particles as a point source, the slit apparatus as a plane normal to a line drawn from the source to the apparatus and the detector as a linear array arranged in a line normal to the plane of the slit projected through the source. It is usual to think of the particles as photons, with the slit apparatus constructed of a material opaque to isible light. Also, electrons are often considered, with the slit apparatus constructed of a material imperious to electrons. Howeer, other particles, such as neutrons, can also be considered if the slit apparatus is imperious to them. In the case of neutrons, suitable materials would be cadmium or BF 4. The expected results from a beam of particles can be isualised by means of Figure Single Slit Double Slit Theoretical No-Slit Linear distance from Centre Line Figure 4.: Diffraction results expected at a linear detector in a plane normal to the plane of the slit through the point source. The Theoretical No-Slit distribution is that which would be expected from a theoretical apparatus if there were no phase effects caused by the apparatus. We assume that a quantum particle can be represented by nothing more than the de Broglie ector and a probability function in space. For the purpose of our thought experiment, it is sufficient to reduce the probability function to a single dimension, as shown in Figure 4.. The larger deflections in Figure 4. appear distended owing to the fact that the detector is linear but the outputs from the slit apparatus are proportional to the angle. Therefore the linear displacements recorded at the detector will be proportional to Tan( ). 5

6 The Theoretical No-Slit distribution represents the relatie probability of an indiidual particle arriing at any point along the detector if there were no phase effects. Note that, in this case, the oerall probability of transmission by the apparatus is 1. The theoretical No-Slit distribution therefore determines the exit angle of the particle from the apparatus. A suitable probability function would be a a P( ) e The alue of a in Equation [4.1] can be chosen to adjust the half-height half-width of the probability function. Although there can neer be any direct eidence of the results from a purely theoretical experiment, its existence can be inferred from the diffraction patterns from the single and double slit experiments because the obsered results depend on the conolution of a sine wae probability and the probability defined in Equation [4.1]. For the purposes of this thought experiment, the alue of a has been chosen to be 5. [4.1] In the case of the single slit apparatus, it is normal to use a wae theory approach, where is the waelength of the wae function. The geometry is shown in Figure 4.3, with W the slit width and the waelength. For this thought experiment we hae used W =.5. W Figure 4.3: The wae theory explanation of the single slit experiment. When (n + 1) = W Sin the wae front from the centre of the slit is anti-phase to those from the edges. 6

7 It is deduced that for any non-negatie integer n, when (n + 1) = W Sin, the phase of the wae function from the centreline of the slit will be anti-phase to the phase at the edges, so that the signal will cancel. Howeer, for a single particle, the wae theory approach is inappropriate. It would be more acceptable to define a new probability function for the transmission of the particle by the apparatus, based on the de Broglie waelength of the particle. Equation 4. defines a probability that fits with the wae theory approach, but can be applied to a single particle. 1 W P T ( S) 1 Cos Sin θ [4.] P T (S) is the probability of transmission by the single slit apparatus and describes a sine wae with a maximum probability of 1 and a minimum probability of 0. In order to produce the obsered single slit diffraction pattern, it is necessary to conolute this with P( ) to produce the oerall probability that a particle will be detected at any point along the detector, as shown for the single slit pattern in Figure 4.: P ( ) P( ). P ( S) [4.3] S Figure 4.4 illustrates the double slit experiment, with d the separation between the slits, θ the angle of deflection, n a non-negatie integer and the de Broglie waelength of the quantum particle. For this thought experiment, we hae used d = 10. T Figure 4.4: The geometry of the double slit experiment d In order to explain the double slit experiment in purely quantum mechanical terms it is useful to abandon the idea that a particle passes through only one slit. The double slit apparatus only detects the particle on the scale of the double slit and hence this is the scale to which the probability function can be said to collapse. So the de Broglie ector of the particle may be regarded as haing two components, one from each slit. Now the particle may reach the detector at any point defined by a new probability function associated with each slit (Equation 4.3). Howeer, at an angle, such that (n 1) λ d Sin θ [4.4] 7

8 where n is a non-negatie integer, is the de Broglie waelength and d the inter-slit distance, the de Broglie ector from the two slits will be anti-phase. In that case the particle will be transmitted with a probability of 0. On the other hand if nλ d Sin θ [4.5] the de Broglie ector from the two slits will be in phase and the particle will be transmitted with a probability of 1. More formally, we can express the probability, P T (D), as 1 d P T ( D) 1 Cos Sin θ [4.6] By conoluting P T (D) with the probability defined by Equation 4.3 we arrie at P ( ) P( ). P ( S). P ( D) [4.7] P D ( ) is shown as the double slit result in Figure 4.. D Detecting the slit through which the particle passes reduces the scale of detecting to that of a single slit. Hence a detected particle will hae its directional probability collapsed to a single slit and therefore the de Broglie ector will hae only a single component. Thus the distribution at the detector will be reduced to that of a single slit. Refraction First we consider refraction. If a particle can be regarded as a de Broglie ector rotating in de Broglie space, then as it approaches the boundary between two media there will be two extremes of phase, one closest to the boundary and one furthest from it. Here we assume, without proof, that the speed of the particle depends on the nature of the medium in which it is traelling, which is the wae theory assumption associated with Snell s law. If we consider the two extremities of the ector phase we can approximate a quantum particle as two particles with separation of twice the de Broglie amplitude r db, represented by Pa and Pb in Figure 4.5. T T 8

9 Interface 1 t x 1 t z f P b y O Pa Figure 4.5: A two particle isualisation of refraction. If two particles trael with a constant separation, x, they will describe a cured path at the refraction interface, which satisfies Snell's law. We use the following argument to show that the quantum particle will satisfy Snell's law. Assume that the separation between the particles is some arbitrary distance x and the speeds of the particles in the incident and emergent media are 1 and respectiely. When Pa crosses the interface, its speed will be reduced to 1, whereas Pb will continue to trael at until it reaches the interface. If the particles continue with a separation of x, they will describe the cured path around the point O, as shown in Figure 4.5. After Pb crosses the interface they will trael in a straight line once again. If Pb reaches the interface t seconds after Pa, it can be seen that and hence so Now and 1 t f x z [4.8] t fz [4.9] f 1 t ( x z) z 1 ( x z) z Sin [4.10] [4.11] y z 1 [4.1] 9

10 so From 4.4 and 4.7 Sin y ( x z) [4.13] Sin 1 ( x z) Sin z Sin [4.1] 1 1 QED Sin The analysis is completely independent of x, the inter-particle separation, which is effectiely twice the amplitude of the de Broglie ector. Thus it can be demonstrated that, if a quantum particle is regarded as a de Broglie ector rotating in de Broglie space, it will obey Snell's law without recourse to wae-particle duality. 5 Spin and the de Broglie ector The angular momentum, L, for the de Broglie ector is gien by L medb rdb [5.1] So L m E [5.] c mec This simplifies to L [5.3] c In the normal manner, L points along the axis of rotation of the de Broglie ector. Now for a photon, where = c L [5.4] Where < c, the requirements for conseration of linear momentum and conseration of energy mean that for particles that coalesce or particles subject to decay, the magnitude of will be the same for all particles taking part in the reaction. So, if we regard L as the spin of a particle, then in all cases spin will be consered. 6 Energy and matter It has long been the custom to express the rest mass of a particle in terms of its energy equialence ia e = mc. The inariance of r db demonstrated by Equation 3.5 suggests that it might be more appropriate to express the mass of a particle in terms of r db : me [6.1] r c db 10

11 Indeed it might be helpful to consider matter as simply captured energy trapped within a potential well. This would fit ery well with a model where photons react to graitation field in a negatie manner. That is to say they are repelled by matter rather than attracted to it. This would mean that, if a photon were to escape from, or be ejected by, a potential well, its small negatie mass would accelerate away from the relatiely large positie mass of the potential well. Now the force, F, between two masses in a graitational field is characterised by 1 [6.] F Gm m r where G is the graitational constant, m 1 and m the two masses and r the separation between them. Thus, if a particle with negatie mass had a mass seeral orders of magnitude smaller than that of a positie mass, m, the acceleration, a, of the negatie particle is gien by Gm a [6.3] r We can find the radial elocity,, of such a particle at a distance, r, from the positie mass, starting with an initial zero radial elocity at a distance r 0 from the mass, by taking the integral of both sides of Equation 6.3 to gie This ealuates to 0 dr 0. dv Gm. dr dt r 0 If the maximum alue of is c, the speed of light, this gies [6.4] Gm [6.5] r Gm r0 [6.6] c It is interesting to note that this is exactly the Schwarzschild radius of the mass. For a mass of the order of a proton, r 0 = m, which is an extremely small distance. When r is less than r 0, becomes imaginary, so r 0 might be considered as the limit of a potential well. It is not clear what mechanism could cause the ejection of a photon from the potential well, but on ejection, its radial elocity would accelerate to within 1ms -1 of c by the time it reaches m from the centre of the potential well which, in practical terms, would essentially be instantaneous. In the opposite case, a photon approaching a physical particle directly would be decelerated as it approached within, say m of the particle with its speed reduced to zero at the boundary of the potential well. From there it could either be absorbed or reflected. If absorbed, it would increase the captured energy within the potential well. 11

12 7 Conclusions A mechanism has been proposed to describe the motion and energy of a quantum particle, as obsered from any frame of reference, using a single quantum property, the de Broglie ector. The ector and its rotation completely describe the mass and energy of the particle. The mass of the particle is represented by the amplitude of the de Broglie ector and is independent of the frame of reference of the obserer. Furthermore, the de Broglie ector proides a possible explanation of the wae particle duality conundrum resulting from quantum mechanics. Its application has been demonstrated for both refraction and reflection. The de Broglie ector also offers a mechanism to account for spin. A logical extension is to assume that matter is nothing more than energy captured within a potential well and that photons could be emitted and absorbed by such a potential well. 1