In the classical electromagnetic theory the wave-vector k = (2 π / λ ) σ underlies the Fourier space of propagating (or radiative) fields. The k -vector combines into a single entity the wavelength λ and the unit vector σ that signifies the beam's propagation direction. The Fourier transform relation between the three-dimensional space of everyday experience and the space of the wave-vectors (the so-called k -space) gives rise to relationships between the two domains analogous to Heisenberg's uncertainty relations. Considering that in quantum theory the electromagnetic k -vector is proportional to the photon's momentum ( p = ħ k , where ħ = h /2 π , h being the Planck constant), one should not be surprised to find relationships between dimensions of a beam in the X Y Z -space and its momentum spread in the k -space. Such relationships impose fundamental limits on the ability of measurement systems to determine the various properties of electromagnetic fields. In this chapter we address two problems that have widespread applications in optical metrology, spectroscopy, telecommunications, etc., and discuss the constraints imposed by the uncertainty principle on these problems. The first topic of discussion is the separation of two overlapping beams of identical wavelength having slightly different propagation directions. This will be followed by an analysis of the limits of separating co-propagating beams having slightly different wavelengths. Angular separation and the limit of resolvability Figure 19.1 shows an aperture of diameter D , which transmits two plane waves of the same wavelength λ propagating in slightly different directions.
In the classical electromagnetic theory the wave-vector k = (2 π / λ ) σ underlies the Fourier space of propagating (or radiative) fields. The k -vector combines into a single entity the wavelength λ and the unit vector σ that signifies the beam's propagation direction. The Fourier transform relation between the three-dimensional space of everyday experience and the space of the wave-vectors (the so-called k -space) gives rise to relationships between the two domains analogous to Heisenberg's uncertainty relations. Considering that in quantum theory the electromagnetic k -vector is proportional to the photon's momentum ( p = ħ k , where ħ = h /2 π , h being the Planck constant), one should not be surprised to find relationships between dimensions of a beam in the X Y Z -space and its momentum spread in the k -space. Such relationships impose fundamental limits on the ability of measurement systems to determine the various properties of electromagnetic fields. In this chapter we address two problems that have widespread applications in optical metrology, spectroscopy, telecommunications, etc., and discuss the constraints imposed by the uncertainty principle on these problems. The first topic of discussion is the separation of two overlapping beams of identical wavelength having slightly different propagation directions. This will be followed by an analysis of the limits of separating co-propagating beams having slightly different wavelengths. Angular separation and the limit of resolvability Figure 19.1 shows an aperture of diameter D , which transmits two plane waves of the same wavelength λ propagating in slightly different directions.
The uncertainty principle places fundamental limits on the accuracy with which we are able to measure the values of different physical quantities (Heisenberg 1949 The Physical Principles of the Quantum Theory (New York: Dover); Robertson 1929 Phys. Rev. 34 127). This has profound effects not only on the microscopic but also on the macroscopic level of physical systems. The most familiar form of the uncertainty principle relates the uncertainties in position and linear momentum. Other manifestations include those relating uncertainty in energy to uncertainty in time duration, phase of an electromagnetic field to photon number and angular position to angular momentum (Vaccaro and Pegg 1990 J. Mod. Opt. 37 17; Barnett and Pegg 1990 Phys. Rev. A 41 3427). In this paper, we report the first observation of the last of these uncertainty relations and derive the associated states that satisfy the equality in the uncertainty relation. We confirm the form of these states by detailed measurement of the angular momentum of a light beam after passage through an appropriate angular aperture. The angular uncertainty principle applies to all physical systems and is particularly important for systems with cylindrical symmetry.
This paper is a small discussion about the uncertainty principle. With the great discoveries and researches made by predecessors, I try to answer some questions within the uncertainty principle. Using the definition of momentum, the momentum in de Broglie wave, the property of cosine function, the Euler’s formula, the basic property of plane wave, the law of derivative, the energy of quantum discovered by Max Planck, the photon energy discovered by Albert Einstein, the energy equation of a system and the Hamiltonian operator, I figure out the Schrodinger equation successfully. With the help of Schrodinger equation, the possibility density of a wave function, the concept of normalization, the average value of position and momentum, the Fourier transform, the integration by parts and the limit of normalized wave functions, I find the position operator and the momentum operator. Then using some knowledge of the standard deviation and the variance, the concept of inner product of vectors, some property of Dirac symbol, the law of calculation of complex number and the law of operator calculation, I obtain the general uncertainty relationship of abstract mathematical quantities. According to the general relationship, I get the inequality of uncertainty principle, and discuss the specific relationship between the position and the momentum. After getting the inequality, I use Heisenberg’s experiment of measuring an electron by using the γ ray to hit the electron to explain why we cannot measure the position and the momentum simultaneously. Based on Heisenberg’s experiment, I give my own explanation by discussing the earth and a spacecraft. Transforming the perspective, we can easily see how the accuracies of position and momentum change. In addition, I attempt to figure out why the uncertainty exists. However, I can only explain it by discussing the time and frequency in mathematical method and have no idea to work it out in a pure physics approach.
This paper discusses how the two ways in which the wavefunction of a free-propagating photon can be introduced-starting from the relativistic energy-momentum relationship or based on the electromagnetic field, in particular on Riemann-Silberstein vectors-are not entirely equivalent since they can lead to different consequences regarding photon localization. In the first case, a phase space localization in regions of the order of Planck's constant, in agreement with the quantum uncertainty principle, could be unambiguously obtained. In the second case, the choice of canonically conjugate variables and Fourier transforms determines if the state is treated quantumly or classically. Both formalisms are, however, compatible.
Light has the dual characteristics of particles (photons) and electromagnetic waves. The photon has an energy of 𝐸 = ℎ𝜈 (𝜈: frequency, h: Planck constant) and the momentum of 𝑝⃗ = ℎ𝑘⃗⃗ (𝑘⃗⃗: wavenumber and 1|𝑘⃗⃗| is the wavelength). The photon density is proportional to the square of the amplitude of the electromagnetic waves. The fundamental aspect of quantum mechanics is that these characteristics apply to all matters. The properties of matters are described by wave functions. The probability of the existence of the matter is proportional to the square of the associated wave function. When a matter is localized in a limited region, it can only assume discrete values of energy because the wavelength of the matter wave must be an integral division of the region. The phase of the wavefunction has uncertainty on order 1/2 radians; therefore, position and momentum (time and energy) cannot be simultaneously determined. As the size of the localization area of the wavefunction becomes smaller, the minimum kinetic energy becomes larger because of the smaller wavelength (larger momentum uncertainty). The Schroedinger equation was derived based on the idea that the relationship between the frequency and the wavenumber corresponds to that between energy and momentum given by classical mechanics, which makes it possible to obtain the wave functions of matters in the energy eigenstates. Several examples of solutions to the Schroedinger equation are introduced. The mixture between different energy eigenstates and the shift in the energy eigenvalues are induced by electromagnetic fields. The temporal change of the wave function (transition between different energy states) is also obtained using the Schroedinger equation.
R. P. Feynman, by far, physics’ most successful communicator, said: The uncertainty principle protects quantum mechanics. W. Heisenberg recognized that if it were possible to measure the momentum and the position simultaneously with a greater accuracy, quantum mechanics would collapse. Quantum mechanics maintains its perilous but correct existence”. In his book on Quantum Mechanics and Path Integrals , Feynman makes use of the two-slit experiment to provide an approximate description of the physics behind the uncertainty principle. Feynman’s description is outlined using a slightly different notation. The uncertainty principle is widely applied in optics. It applies to interferometry, linewidth measurements, and beam divergence measurements. The chapter focuses on the uncertainty principle and beam divergence. An important application of the uncertainty principle manifests itself in calculations of the angular resolution limit of telescopes used in astronomical observations. The Heisenberg uncertainty principle can be used to derive some useful identities in optics and interferometry.
The idea to base the uncertainty relation for photons on the electromagnetic energy distribution in space enabled us to derive a sharp inequality that expresses the uncertainty relation [Phys. Rev. Lett. 108, 140401 (2012)]. An alternative version of the uncertainty relation derived in this paper is closer in spirit to the original Heisenberg relation because it employs the analog of the position operator for the photon---the center of the energy operator. The noncommutativity of the components of the center of the energy operator results in the increase of the bound $3\ensuremath{\hbar}/2$ in the standard Heisenberg uncertainty relation in three dimensions. This difference diminishes with the increase of the photon energy. In the infinite-momentum frame, the lower bound in the Heisenberg uncertainty relations for photons is the same as in nonrelativistic quantum mechanics. A similar uncertainty relation is also derived for coherent photon beams. This relation has direct experimental consequences since it gives a precise relationship between the spectral composition of the laser beam and the minimal focal volume.