根据Born对归一化波函数的概率解释，从量子力学的观点重新审视孪生双光子思想实验，着眼的对象不再是这对孪生光子本身，而是一群具有共同属性的光子集合。所有这些光子具有完全相同的能量和角动量特征，但是在动量的相反方向却具有相等的概率；或者说这是一群在相反动量方向具有相等的数量、而所有其它量子特征完全相同的光子集合。仿效先前对光学双孔效应进行量子力学分析的步骤，具体构造出描述孪生双光子思想实验总的归一化光子态矢函数，所描述的并不是两个光子之间的纠缠，而是单个光子的概率，从而得到光子的空间分布函数。当光子的总数达到形成完整图像的统计学要求时，宏观上将显示出明暗间隔的条纹。在追溯纠缠态历史渊源的基础上，返璞归真孪生双光子思想实验的量子力学观点，对所谓的EPR佯谬引发出来的双光子纠缠态作出了新的解释，提出了量子信息的新认识。 Based on Born’s statistical interpretation for the normalized wave function, the twin two-photon thought experiment is reexamined from point of view of quantum mechanics. The emphasis is no longer the pair of twin photons itself, but rather a group of photons with common attributes. All the photons have the characteristics of identical energy and angular momentum, but possess same probability in the opposite direction of momentum, that is to say, this is a group of photon collections, which have the same number in the opposite direction of momentum, while all other quantum characteristics are identical. Following the example of the previous procedure dealing with optical two-hole effect by use of quantum mechanics, the total normalized photonic state-vector function that described the twin two-photon thought experiment has been specifically constructed, which does not describe the entangled behavior of the two photons, but the probability of a single photon, and thus deduces the probability distribution of a photon in the space. When the total number of photons satisfies the statistical requirements to form a complete picture, some fringes of light and dark interval will be displayed. On the basis of retrospective historical origins of entanglement, the point of view of quantum mechanics about the twin two-photon thought experiment was reverted. The so-called EPR paradox, and the entangled two-photon state have been re-interpreted, and a new understanding of quantum information has been put forward.
根据Born对归一化波函数的概率解释，从量子力学的观点重新审视孪生双光子思想实验，着眼的对象不再是这对孪生光子本身，而是一群具有共同属性的光子集合。所有这些光子具有完全相同的能量和角动量特征，但是在动量的相反方向却具有相等的概率；或者说这是一群在相反动量方向具有相等的数量、而所有其它量子特征完全相同的光子集合。仿效先前对光学双孔效应进行量子力学分析的步骤，具体构造出描述孪生双光子思想实验总的归一化光子态矢函数，所描述的并不是两个光子之间的纠缠，而是单个光子的概率，从而得到光子的空间分布函数。当光子的总数达到形成完整图像的统计学要求时，宏观上将显示出明暗间隔的条纹。在追溯纠缠态历史渊源的基础上，返璞归真孪生双光子思想实验的量子力学观点，对所谓的EPR佯谬引发出来的双光子纠缠态作出了新的解释，提出了量子信息的新认识。 Based on Born’s statistical interpretation for the normalized wave function, the twin two-photon thought experiment is reexamined from point of view of quantum mechanics. The emphasis is no longer the pair of twin photons itself, but rather a group of photons with common attributes. All the photons have the characteristics of identical energy and angular momentum, but possess same probability in the opposite direction of momentum, that is to say, this is a group of photon collections, which have the same number in the opposite direction of momentum, while all other quantum characteristics are identical. Following the example of the previous procedure dealing with optical two-hole effect by use of quantum mechanics, the total normalized photonic state-vector function that described the twin two-photon thought experiment has been specifically constructed, which does not describe the entangled behavior of the two photons, but the probability of a single photon, and thus deduces the probability distribution of a photon in the space. When the total number of photons satisfies the statistical requirements to form a complete picture, some fringes of light and dark interval will be displayed. On the basis of retrospective historical origins of entanglement, the point of view of quantum mechanics about the twin two-photon thought experiment was reverted. The so-called EPR paradox, and the entangled two-photon state have been re-interpreted, and a new understanding of quantum information has been put forward.
Based on Born's statistical interpretation for the normalized wave function, the twin two-photon thought experiment is reexamined from point of view of quantum mechanics. The emphasis is no longer the pair of twin photons itself, but rather a group of photons with common attributes. All the photons have the characteristics of identical energy and angular momentum, but possess same probability in the opposite direction of momentum, that is to say, this is a group of photon collections, which have the same number in the opposite direction of momentum, while all other quantum characteristics are identical. Following the example of the previous procedure dealing with optical two-hole effect by use of quantum mechanics, the total normalized photonic state-vector function that described the twin two-photon thought experiment has been specifically constructed, which does not describe the entangled behavior of the two photons, but the probability of a single photon, and thus deduces the probability distribution of a photon in the space. When the total number of photons satisfies the statistical requirements to form a complete picture, some fringes of light and dark interval will be displayed. On the basis of retrospective historical origins of entanglement, the point of view of quantum mechanics about the twin two-photon thought experiment was reverted. The so-called EPR paradox, and the entangled two-photon state have been re-interpreted, and a new understanding of quantum information has been put forward.
Although most physics experiments are carried out with independent particles, it is the collective nature of entangled particles that reveals the most fascinating and unexpected aspects of the quantum world. It was Erwin Schrodinger who first said “entanglement is not one but rather the characteristic trait of quantum mechanics”. One curious aspect of the behavior of a pair of particles in an entangled state is that, though each individual particle exhibits an inherent uncertainty, the joint entity of an entangled pair can exhibit no such uncertainty. As an example, while the time of arrival of an individual particle may be totally random, an entangled pair must always arrive simultaneously. This property offers a unique tool for carrying out absolute measurements. Our goal here is to explore the myriad implications and significance of entanglement and to exploit it for the development of a new type of optical measurement—quantum optical metrology. The existence of unique non-classical correlations between twin photons generated in the nonlinear process of spontaneous parametric origins. The non-classical link between such twin quanta is not diminished by arbitrarily large separations between the twins, even when they lie outside the light cone. Twin states have been used with great effectiveness over the past two decades for carrying out definitive quantum experiments that lead to counterintuitive results; among these are those arising from the EinsteinPodolsky-Rosen (EPR) paradox such as various tests of Bell’s inequalities [1-12], as well as non-local dispersion cancellation, entangled-photon–induced transparency, and entangled-photon spectroscopy with monochromatic light. The availability of these twin beams has made it possible to conduct such experiments without having to resort to
One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev. A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all of the usual rules of quantum mechanics and provides a framework for understanding the relationships amongst different forms of the photon wave function in the literature. It also gives a quantum picture of the optical angular momentum of beams that applies to both one photon and coherent states. According to the rules of quantum mechanics, this wave function gives the probability to count a photon at any position in space.
Introduction: In the rapidly developing areas of photonics associated with quantum teleportation of photons, quantum cryptography, and quantum computing, ideas are becoming increasingly relevant about the certain localized states of single or several entangled photons moving in space and time. As is known, in quantum mechanics, localized states of particles can only be described using wave functions in coordinate representation – the corresponding wave packets. Aims: To explain Young's one- and two-photon experiments using coordinate 6-component quantum mechanical and 1-component quasi-classical photon wave functions. Methods: The article outlines the method developed in authors' previous work in constructing the 6-component coordinate wave function of free photons in the form of the wave packet (an integral over the entire momentum space and sum over two possible values +1 and -1 of the helicity). The basis functions are the circularly polarized monochromatic waves, corresponding to certain values of the momentum, helicity, and photon energy operators and forming a complete orthonormal set of generalized eigenfunctions of these operators. This photon wave function (PWF) is normalized to the unit probability of finding the photon anywhere in space. Both the basis functions and the coordinate PWF satisfy a Schrödinger-type equation derived from the Maxwell-Lorentz equations written in quantum mechanical form first introduced by Majorana. Then the results of modeling and consideration of the space-time evolution of the wave packet with Gaussian momentum distribution corresponding to the directed photon emission for 80 fs are briefly presented. Further, in view of the possibility of constructing the coordinate PWF, the basic formula for the waveparticle duality of photons and particles is formulated. It is emphasized that the photon is fundamentally different from “ordinary” particles, since, according to the authors, the photon is the result of the propagation of a certain complex spin wave in a physical vacuum, the details of which can only be considered simultaneously with the study of the structure of the physical vacuum at Planckian distances. Results: Young`s one- and two-photon experiments are explained using various options for modeling 6-component coordinate PWFs, including approximate functions corresponding to spherically symmetric and electric dipole photon radiation. Conclusion: This explanation is illustrated using specific examples of radiation and, at the same time, is compared with numerical simulations for spherically symmetric radiation and the use of 1-component quasi-classical PWFs.
This paper represents a further academic deepening and upgrading of the authors' 2019 publication A Hypothesis on the Spatial Motion Mode of Photons. It should be explicitly stated that this paper falls within the category of natural philosophical thought experiments—its core value lies in constructing a unified physical image of the nature of light through rigorous logical deduction, and proposing verifiable theoretical hypotheses and experimental schemes; the validity of all conclusions must ultimately be verified by rigorous and extensive scientific experiments before being incorporated into the theoretical system of physics. As a foundational concept of quantum mechanics, the wave-particle duality of light has been accompanied by profound philosophical perplexities and theoretical tensions since its proposal, becoming a core bottleneck in the integration of classical and quantum physics. This paper systematically sorts out the logical incompleteness in the current quantum interpretation system—including the self-negation of the complementarity concept, the problem of photon localization, the fundamental opposition between the statistical and non-statistical interpretations of the wave function, and the philosophical controversy over the Heisenberg Uncertainty Principle, revealing the inherent contradictions of the traditional wave-particle duality framework. On this basis, adopting classical physical images and the logic of reduction to absurdity, and based on six axioms and six preparatory propositions, this paper puts forward a natural philosophical hypothesis on the essence of photons: a photon is an energetic mass point with a diameter smaller than the Planck length, moving in a uniform spiral linear motion in space. The paper deduces the core characteristics such as velocity, frequency, and wavelength of the photon's uniform spiral linear motion, and designs three operable, repeatable, and quantifiable physical experimental schemes to provide specific paths for the empirical verification of the hypothesis. The research deduces that the angular momentum of photon spatial motion (excluding photon spin motion) is always the reduced Planck constant ℏ, the energy E=mc² is naturally unified with E=hν (the standard formula for wave energy), and the standard expression of the Heisenberg Uncertainty Principle ΔxΔpₓ≥ℏ/2 can be given a classical physical interpretation from the perspective of superposition of measurement deviations. This paper systematically responds to potential questions regarding the origin of photon particle nature, wave nature, and compatibility with relativity, arguing that the hypothesis provides a logically consistent and clearly visualized path for understanding the nature of light, builds a new natural philosophical framework for the integration of quantum and classical theories of light, and also offers a new thinking perspective for the paradigm shift in the study of the nature of light.
It appears from the description in Sec. 15.3 that a global state containing a single photon in each of the modes i = (q, s) and j = (q′, s′) has the form |ψ〉 = |ψij〉 ⊗ ∏ k 6=i,j |0k〉, (33.1) where |ψij〉 = |1i〉 ⊗ |1j〉 = aˆ†i (0)aˆ†j(0)|0i〉 ⊗ |0j〉. (33.2) A Fock state containing precisely two photons also can be constructed by putting the two quanta into the same mode (say i), that is |ψij〉 = |2i〉 ⊗ |0j〉 = 1√ 2 [ aˆ†i (0) ]2 |0i〉 ⊗ |0j〉. (33.3) By linear superposition of Fock modes of the type given in Eq. (33.1) a general polychromatic two-photon state, |Φ2〉, can be formed: |Φ2〉 = 1 2 L−3 ∑ i,j φij aˆ † i (0)aˆ † j(0)|0〉. (33.4) The factor 1/2 appearing in Eq. (33.4) originates in our wish to end up with a relation between the two-photon wave functions in direct and reciprocal space having the same structural form as in the one-photon case [Eq. (15.125)]. The quantities φij are amplitude weight factors characterizing the various |Φ2〉-states. The form of the superposition in Eq. (33.4) shows that only the sum φij + φji can play a physical role, and without loss of generality one therefore may set φji = φij . (33.5) The inner product of |Φ2〉 with itself is readily obtained utilizing that 〈0|aˆkaˆlaˆ†i aˆ†j|0〉 = δkiδlj + δkjδli. (33.6) of Thus, 〈Φ2|Φ2〉 =L −6 ∑ i,j,k,l = L−6 ∑ i,j,k,l φijφ ∗ kl (δkiδlj + δkjδli) = L−6 ∑ i,j φij ( φ∗ij + φ ) = L−6 ∑ i,j |φ∗ij |2. (33.7) The two-photon state is normalized, i.e., 〈Φ2|Φ2〉 = 1, (33.8) provided the weight factors satisfy the condition L−6 ∑ i,j |φij |2 → 1 |φs,s′(q,q′)|2 d 3q (2π)3 d3q′ (2π)3 = 1. (33.9) By acting on |Φ2〉 with the global number operator one obtains Nˆ |Φ2〉 = 1 2 L−3 ∑ i,k,l φklaˆ † i aˆiaˆ = 1 2 L−3 ∑ i,k,l ( aˆ†kaˆi + δik ) aˆ†l |0〉 = 1 2 L−3 ∑ i,k,l φkl [ aˆ†i aˆ † l δik + aˆ ( aˆ†l aˆi + δil )] |0〉 = 1 2 L−3 ∑ ∑ i,k = 1 2 L−3 ∑ i,j (φij + φji) aˆ † i aˆ † j |0〉. (33.10) In view of Eqs. (33.4) and (33.5), one finally has Nˆ |Φ2〉 = 2|Φ2〉. (33.11) The polychromatic state |Φ2〉 hence is an eigenstate for the global number operator with eigenvalue 2, a result which of course follows from the construction in Eq. (33.4).