Particles which pass through slits can be classified as three types: Type (I): particles which only pass through slit A and arrive in the screen position x meaning the probability of passing through slit A is 1; Type (II): particles which only pass through slit B and arrive in the screen position x meaning the probability of passing through slit B is 1; Type (III): the probability of passing through slit A is c1(0 < c1 < 1), and the probability of passing through slit B is c2 (0 < c2 < 1). When only slit A is open, the particles in the neighborhood of x(x is interference position) consist of Type (I) particles. In this case, the particle’s density in position x is |ψ1|, where ψ1 is the state function of particles which pass through slit A. When only slit B is open, the particles in the neighborhood of x consist of Type (II) particles. In this case, the particle’s density in position x is |ψ2|, where ψ2 is the state function of particles which pass through slit B. When A and B open simultaneously, Type (III) particles should be taken into account: if they pass through slit A, their particle density dedication to position x is c1|ψ1|; if they pass through slit B, their particle density dedication to position x is c2|ψ2|. The average dedication in the neighborhood of x should be the geometric average of c1|ψ1| and c2|ψ2| , that is: √ c1|ψ1| c2|ψ2| = c1c2|ψ1||ψ2|.
Particles which pass through slits can be classified as three types: Type (I): particles which only pass through slit A and arrive in the screen position x meaning the probability of passing through slit A is 1; Type (II): particles which only pass through slit B and arrive in the screen position x meaning the probability of passing through slit B is 1; Type (III): the probability of passing through slit A is c1(0 < c1 < 1), and the probability of passing through slit B is c2 (0 < c2 < 1). When only slit A is open, the particles in the neighborhood of x(x is interference position) consist of Type (I) particles. In this case, the particle’s density in position x is |ψ1|, where ψ1 is the state function of particles which pass through slit A. When only slit B is open, the particles in the neighborhood of x consist of Type (II) particles. In this case, the particle’s density in position x is |ψ2|, where ψ2 is the state function of particles which pass through slit B. When A and B open simultaneously, Type (III) particles should be taken into account: if they pass through slit A, their particle density dedication to position x is c1|ψ1|; if they pass through slit B, their particle density dedication to position x is c2|ψ2|. The average dedication in the neighborhood of x should be the geometric average of c1|ψ1| and c2|ψ2| , that is: √ c1|ψ1| c2|ψ2| = c1c2|ψ1||ψ2|.
In a double slit experiment, a screen position x could be endowed with two probabilities: the probability of finding particles which passed through slit A and arrived in x, and the probability of finding particles which passed through slit B and arrived in x.Analysis showed that, the probability of finding particles in x when two slits opened simultaneously was greater than when only one slit opened; as long as the time that the two slits opened simultaneously was long enough, the particle interference phenomenon in x would occur with certainty.Sur la base du point de vue de la classification des particules, une analyse supplmentaire est fournie: dans une exprience double fente, la probabilit de trouver une particule dans l'cran lorsque deux fentes s'ouvrent simultanment est deux fois suprieure celle d'une seule fente.Ds que le temps d'ouverture des deux fentes est suffisamment long, le phnomne d'interfrence de particules se produira certainement.
A quantum experiment consists of a preparation procedure followed by the measurement of a specified observable. In the experiments considered here, the y -component of momentum is measured after a particle passes through a system of slits. The source–slit system is the preparation apparatus that determines the initial state |ψ⟩. Recognizing that a system of slits is a position-measuring device allows us to determine |ψ⟩. Then, a straightforward calculation gives P ( p y ) = |⟨ p y |ψ⟩| 2 the probability that a particle will pass through a system of slits with momentum p y = p sin θ. Interference effects are inherent in this probability function. We perform the calculation for four different slit systems. The method is suitable for an upper level course in quantum mechanics.
In classical physics, interactions appear to be of the form of cause and effect, as in Newton’s second law is: F = dp/dt. The force F governs the interaction and dp/dt is the effect. In quantum mechanics, there is a probability for an impulse hit interaction by a particle given by exp(ipx) within a wavelength region of size hbar/p. As a result, instead of an interaction occurring within dx->0, the particle itself dictates an interaction region of the size hbar/p. Consequently, in a 2-slit interaction, the slits should be about a wavelength hbar/p apart. In classical physics, an interaction occurs within the interaction region (dx->0), but in the 2-slit case the interaction region includes the space from the first slit to the second, so to ask: What is happening as the particle interacts within this region? means that a particle can pass through one slit and inteact with the other, or it can interact with the slit through which it passes. The interaction region is the smallest region in which the interaction occurs (as if it were a black box). The particle physically passes through one slit or the other, but interacts either way with both slits probabilisitcally because otherwise the interaction region would only be one slit or the other contradicting the probabilistic range hbar/p of interaction. As a result, a two-slit calculation is really a calculation of an interaction within an hbar/p region including both slits. This interaction is probabilistic due to the exp(ipx) interaction hit nature of the particle and so it is the particle which dictates the interaction to a large degree. The fact that the particle must pass through one slit or the other, together with a probabilistic interaction over the hbar/p region means that two key notions are at work, one being the physical location of the particle (passing through one slit or the other) which governs how the probabilistic interaction (due to the particle) operates. In particular, one draws two parallel rays, one from the center of each slit towards a fixed point on a screen far away. The interaction is based on the exp(ipx) probabilistic interaction of the particle and the separation of the two slits because the particle interacts with both as it passes through only one. In other words, reality is not simply based on what one can measure during an interaction. The fact that one does measure through which slit the particle passes does not mean that the positions of the slits cannot stil helpl lead to determinate statements such as zero particles arrive at certain x points on the screen. The slit separation is part of the physical reality of the problem. (Note: there is also diffraction from the two edges of each slit, i.e. diffraction, which follows the same ideas and so one has a combination of diffraction from each slit together with interference.) As a result, we suggest that one consider the entire region from one slit to the other as a quantum interaction region similar to the dx->0 used in Newtonian mechanics. Quantum mechanics then provides a probabilistic, not deterministic, descripition of possible interactions and their probability weights, unlike the cause (force) and effect (deterministic change in p of the particle) of Newtonian mechanics. The exp(ipx), hbar/p gradation form of probability, allows for probabilities to add or subtract at different x points. The walls of the 2-slit apparatus are linked with probabilistic momentum angle changing, with each angle having the same probability (uniform distribution), but otherwise much of the interaction is dictated by the particle, unlike in Netwonian mechanics. For a V(x) = Sum over i Vk exp(ikx), both exp(ipx) and exp(ikx) are one a similar footing.
The double-slit experiment has been called “the only mystery” of quantum mechanics. For one hundred years the question “which slit does the electron pass through?” has lacked a physical answer: the Copenhagen interpretation declares the question meaningless until measurement; the many-worlds interpretation posits branching universes; the pilot-wave interpretation introduces an undetected guiding field. We present a complete physical answer within the stochastic electrodynamic (SED) framework established in Papers 1–6 of this series. The answer rests on three propositions, all following from first principles: 1. The electron follows a definite trajectory—one that passes through exactly one slit—described by Nelson’s stochastic differential equation (SDE) with diffusion coefficient D = ℏ/2m. 2. The electron’s zero-point field (ZPF), which fills all space as the electromagnetic quantum field of the Coulomb interaction, carries information about both slits through the osmotic velocity u = D∇ln |ψA + ψB|^2. This non-local ZPF correlation guides the electron toward the interference maxima. 3. “Measurement” is a physical disturbance of the ZPF. It replaces the joint probability density ρ = |ψA + ψB|^2 with the conditioned density ρ → |ψA|^2 (Bayes update), which eliminates the interference term 2 Re[ψ∗A ψB] and destroys the fringes. The result is confirmed by Monte Carlo simulation of both the analytical |ψ|^2 distribution (Approach A) and the Nelson SDE (Approach B). Both reproduce the interference pattern; both show its disappearance when which-slit information is available. “The observer creates reality” is replaced by “physical disturbance of the ZPF updates the probability density.” The hundred-year paradox is resolved by a concrete physical mechanism requiring no new postulates beyond those established in Paper 1.
Introduction In this chapter we show how the quantized detector network (QDN) formalism describes the double-slit (DS) experiment. This is arguably the simplest experiment that demonstrates quantum affects such as wave–particle duality and quantum interference. It continues to be the focus of much debate and experiment (Mardari, 2005), because theoretical modeling of what is going on reflects current understanding of quantum physics and hence physical reality.We will apply QDN to two variants: the original DS experiment and the monitored DS experiment, where an attempt is made to determine the imagined path of the particle. The DS experiment is widely acknowledged by physicists to be of importance to the understanding of quantum mechanics (QM). So much so that in 2002, the single electron version, first performed by Merli, Missiroli, and Pozzi (Merli et al., 1976), was voted by readers of Physics World to be “the most beautiful experiment in physics” (Rosa, 2012). The DS experiment can be discussed in terms of three stages, shown in Figure 10.1. By the end of the preparation stage, Σ 0 , a monochromatic beam of light or particles has been prepared by a source P , such as a laser. The beam emerges from point O and then passes through an information void V 1 to the first stage, Σ 1 , which consists of a wall or barrier W . This wall has two openings denoted A and B that allow parts of the beam to pass through into another information void V 2 and onto the second and final stage Σ 2 , which consists of a detecting screen S . The screen S is in general some material that can absorb and record particle impacts. In reality, any screen will consist of a finite number of signal detectors, such as photosensitive molecules, but the typical QM modeling is done as if there were a continuum of sites on the screen, such as C , that could register particles.
In the experiments considered here, we measure the y-component of momentum for a particle passing through a system of slits. The source-slit system is the preparation apparatus that determines the state vector. Recognizing that a system of slits is a position-measuring device allows us to ascertain that the state vector is a position state. Then, writing the state vector in momentum space provides a straightforward calculation for the probability amplitude and its corresponding probability function. Interference effects, if any, are inherent in the probability function We determine the statistical distribution of scattered particles for four different slit systems. The results are in agreement with the well-known interference patterns obtained in classical wave optics.