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140-662: The Aharonov–Bohm effect , sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum-mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential ( φ {\displaystyle \varphi } , A {\displaystyle \mathbf {A} } ), despite being confined to a region in which both the magnetic field B {\displaystyle \mathbf {B} } and electric field E {\displaystyle \mathbf {E} } are zero. The underlying mechanism

280-480: A {\displaystyle a} larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle. As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that

420-947: A is the unit vector in direction a , the divergence is The use of local coordinates is vital for the validity of the expression. If we consider x the position vector and the functions r ( x ) , θ ( x ) , and z ( x ) , which assign the corresponding global cylindrical coordinate to a vector, in general r ( F ( x ) ) ≠ F r ( x ) {\displaystyle r(\mathbf {F} (\mathbf {x} ))\neq F_{r}(\mathbf {x} )} , θ ( F ( x ) ) ≠ F θ ( x ) {\displaystyle \theta (\mathbf {F} (\mathbf {x} ))\neq F_{\theta }(\mathbf {x} )} , and z ( F ( x ) ) ≠ F z ( x ) {\displaystyle z(\mathbf {F} (\mathbf {x} ))\neq F_{z}(\mathbf {x} )} . In particular, if we consider

560-413: A source-free part B ( r ) . Moreover, these parts are explicitly determined by the respective source densities (see above) and circulation densities (see the article Curl ): For the irrotational part one has with The source-free part, B , can be similarly written: one only has to replace the scalar potential Φ( r ) by a vector potential A ( r ) and the terms −∇Φ by +∇ × A , and

700-417: A Euclidean coordinate system with coordinates x 1 , x 2 , ..., x n , define In the 1D case, F reduces to a regular function, and the divergence reduces to the derivative. For any n , the divergence is a linear operator, and it satisfies the "product rule" for any scalar-valued function φ . One can express the divergence as a particular case of the exterior derivative , which takes

840-443: A closed loop γ {\displaystyle \gamma } is e i ∫ γ A {\displaystyle e^{i\int _{\gamma }A}} (one can show this does not depend on the trivialization but only on the connection). For a flat connection one can find a gauge transformation in any simply connected field free region(acting on wave functions and connections) that gauges away

980-442: A closed loop must be an integer multiple of 2 π {\displaystyle 2\pi } (with the charge q = 2 e {\displaystyle q=2e} for the electron Cooper pairs ), and thus the flux must be a multiple of h / 2 e {\displaystyle h/2e} . The superconducting flux quantum was actually predicted prior to Aharonov and Bohm, by F. London in 1948 using

1120-466: A definite prediction of what the quantum state ψ ( t ) {\displaystyle \psi (t)} will be at any later time. Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian . Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example,

1260-464: A discussion, see Schwarzschild (1986) and Imry & Webb (1989). Bachtold et al. (1999) detected the effect in carbon nanotubes; for a discussion, see Kong et al. (2004). The magnetic Aharonov–Bohm effect is also closely related to Dirac's argument that the existence of a magnetic monopole can be accommodated by the existing magnetic source-free Maxwell's equations if both electric and magnetic charges are quantized. A magnetic monopole implies

1400-510: A family of unitary operators parameterized by a variable t {\displaystyle t} . Under the evolution generated by A {\displaystyle A} , any observable B {\displaystyle B} that commutes with A {\displaystyle A} will be conserved. Moreover, if B {\displaystyle B} is conserved by evolution under A {\displaystyle A} , then A {\displaystyle A}

1540-404: A flux of gas particles passing out of it, so there is positive divergence at that point. However any closed surface not enclosing the point will have a constant density of gas inside, so just as many fluid particles are entering as leaving the volume, thus the net flux out of the volume is zero. Therefore, the divergence at any other point is zero. The divergence of a vector field F ( x ) at

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1680-576: A gauge field acting in the space of control parameters. Quantum mechanics Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms . It is the foundation of all quantum physics , which includes quantum chemistry , quantum field theory , quantum technology , and quantum information science . Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary ( macroscopic and (optical) microscopic ) scale, but

1820-454: A hermitian metric and a U(1)- connection ∇ {\displaystyle \nabla } . The curvature form of the connection, i F = ∇ ∧ ∇ {\displaystyle iF=\nabla \wedge \nabla } , is, up to the factor i, the Faraday tensor of the electromagnetic field strength . The Aharonov–Bohm effect is then a manifestation of

1960-471: A loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system. Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory. As described above, entanglement

2100-426: A mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ψ {\displaystyle \psi } belonging to a ( separable ) complex Hilbert space H {\displaystyle {\mathcal {H}}} . This vector

2240-749: A mathematical singularity in the vector potential, which can be expressed as a Dirac string of infinitesimal diameter that contains the equivalent of all of the 4π g flux from a monopole "charge" g . The Dirac string starts from, and terminates on, a magnetic monopole. Thus, assuming the absence of an infinite-range scattering effect by this arbitrary choice of singularity, the requirement of single-valued wave functions (as above) necessitates charge-quantization. That is, 2 q e q m ℏ c {\displaystyle 2{\frac {q_{\text{e}}q_{\text{m}}}{\hbar c}}} must be an integer (in cgs units) for any electric charge q e and magnetic charge q m . Like

2380-417: A measurement of its position and also at the same time for a measurement of its momentum . Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference , which is often illustrated with the double-slit experiment . In the basic version of this experiment, a coherent light source , such as a laser beam, illuminates a plate pierced by two parallel slits, and

2520-494: A particle with electric charge q {\displaystyle q} traveling along some path P {\displaystyle P} in a region with zero magnetic field B {\displaystyle \mathbf {B} } , but non-zero A {\displaystyle \mathbf {A} } (by B = 0 = ∇ × A {\displaystyle \mathbf {B} =\mathbf {0} =\nabla \times \mathbf {A} } ), acquires

2660-582: A particle, through regions of zero electric field, an observable Aharonov–Bohm interference phenomenon from the phase shift has been predicted; again, the absence of an electric field means that, classically, there would be no effect. From the Schrödinger equation , the phase of an eigenfunction with energy E {\displaystyle E} goes as e − i E t / ℏ {\displaystyle e^{-iEt/\hbar }} . The energy, however, will depend upon

2800-431: A perfectly uniform current distribution) encloses a magnetic field B {\displaystyle \mathbf {B} } , but does not produce any magnetic field outside of its cylinder, and thus the charged particle (e.g. an electron ) passing outside experiences no magnetic field B {\displaystyle \mathbf {B} } . (This idealization simplifies the analysis but it's important to realize that

2940-476: A phase shift φ {\displaystyle \varphi } , given in SI units by Therefore, particles, with the same start and end points, but traveling along two different routes will acquire a phase difference Δ φ {\displaystyle \Delta \varphi } determined by the magnetic flux Φ B {\displaystyle \Phi _{B}} through

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3080-425: A phase shift: where t is the time spent in the potential. For example, we may have a pair of large flat conductors, connected to a battery of voltage Δ V {\displaystyle \Delta V} . Then, we can run a single electron double-slit experiment, with the two slits on the two sides of the pair of conductors. If the electron takes time t {\displaystyle t} to hit

3220-462: A phenomenological model. The first claimed experimental confirmation was by Robert G. Chambers in 1960, in an electron interferometer with a magnetic field produced by a thin iron whisker, and other early work is summarized in Olariu and Popèscu (1984). However, subsequent authors questioned the validity of several of these early results because the electrons may not have been completely shielded from

3360-457: A point x 0 is defined as the limit of the ratio of the surface integral of F out of the closed surface of a volume V enclosing x 0 to the volume of V , as V shrinks to zero where | V | is the volume of V , S ( V ) is the boundary of V , and n ^ {\displaystyle \mathbf {\hat {n}} } is the outward unit normal to that surface. It can be shown that

3500-454: A positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value. In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source or a sink at a given point. It is a local measure of its "outgoingness" – the extent to which there are more of the field vectors exiting from an infinitesimal region of space than entering it. A point at which

3640-471: A probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to

3780-418: A pure gauge vector potential A = d ϕ {\displaystyle A=d\phi } . There is no real asymmetry because representing the former in terms of the latter is just as messy as representing the latter in terms of the former. This means that it is physically more natural to describe wave "functions", in the language of differential geometry , as sections in a complex line bundle with

3920-450: A region of non trivial field: The monodromy of the flat connection only depends on the topological type of the loop in the field free region (in fact on the loops homology class). The holonomy description is general, however, and works inside as well as outside the superconductor. Outside of the conducting tube containing the magnetic field, the field strength F = 0 {\displaystyle F=0} . In other words, outside

4060-438: A result of the enclosed magnetic field, despite the magnetic field being negligible in the region through which the particle passes and the particle's wavefunction being negligible inside the solenoid. This phase shift has been observed experimentally. There are also magnetic Aharonov–Bohm effects on bound energies and scattering cross sections, but these cases have not been experimentally tested. An electric Aharonov–Bohm phenomenon

4200-405: A single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus , whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an s orbital ( Fig. 1 ). Analytic solutions of

4340-551: A single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: The general solution of the Schrödinger equation is given by which is a superposition of all possible plane waves e i ( k x − ℏ k 2 2 m t ) {\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}} , which are eigenstates of

Aharonov–Bohm effect - Misplaced Pages Continue

4480-628: A stationary magnetic field as the curl of a vector potential (then a new concept – the idea of a scalar potential was already well accepted by analogy with gravitational potential). The language of potentials generalised seamlessly to the fully dynamic case but, since all physical effects were describable in terms of the fields which were the derivatives of the potentials, potentials (unlike fields) were not uniquely determined by physical effects: potentials were only defined up to an arbitrary additive constant electrostatic potential and an irrotational stationary magnetic vector potential. The Aharonov–Bohm effect

4620-557: A trivialization of the line-bundle, a non-vanishing section, the U(1)-connection is given by the 1- form corresponding to the electromagnetic four-potential A as ∇ = d + i A {\displaystyle \nabla =d+iA\,} where d means exterior derivation on the Minkowski space . The monodromy is the holonomy of the flat connection. The holonomy of a connection, flat or non flat, around

4760-406: A vector field is often illustrated using the simple example of the velocity field of a fluid, a liquid or gas. A moving gas has a velocity , a speed and direction at each point, which can be represented by a vector , so the velocity of the gas forms a vector field . If a gas is heated, it will expand. This will cause a net motion of gas particles outward in all directions. Any closed surface in

4900-473: Is and this provides the lower bound on the product of standard deviations: Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of

5040-421: Is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product : take the components of the ∇ operator (see del ), apply them to the corresponding components of F , and sum the results. Because applying an operator is different from multiplying the components, this is considered an abuse of notation . For a vector expressed in local unit cylindrical coordinates as where e

5180-430: Is a function of position which depends on the coordinate system. In Cartesian, cylindrical and spherical coordinates, using the same conventions as before, we have ρ = 1 , ρ = r and ρ = r sin θ , respectively. The volume can also be expressed as ρ = | det g a b | {\textstyle \rho ={\sqrt {\left|\det g_{ab}\right|}}} , where g ab

5320-478: Is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence . This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic. There are many mathematically equivalent formulations of quantum mechanics. One of

5460-424: Is a valid joint state that is not separable. States that are not separable are called entangled . If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes

5600-413: Is assumed). Expressions of ∇ ⋅ A {\displaystyle \nabla \cdot \mathbf {A} } in cylindrical and spherical coordinates are given in the article del in cylindrical and spherical coordinates . Using Einstein notation we can consider the divergence in general coordinates , which we write as x , …, x , …, x , where n is the number of dimensions of

5740-404: Is called solenoidal . If the gas is heated only at one point or small region, or a small tube is introduced which supplies a source of additional gas at one point, the gas there will expand, pushing fluid particles around it outward in all directions. This will cause an outward velocity field throughout the gas, centered on the heated point. Any closed surface enclosing the heated point will have

Aharonov–Bohm effect - Misplaced Pages Continue

5880-405: Is conserved under the evolution generated by B {\displaystyle B} . This implies a quantum version of the result proven by Emmy Noether in classical ( Lagrangian ) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law . The simplest example of a quantum system with a position degree of freedom is a free particle in

6020-1066: Is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics. The Hamiltonian H {\displaystyle H} is known as the generator of time evolution, since it defines a unitary time-evolution operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} for each value of t {\displaystyle t} . From this relation between U ( t ) {\displaystyle U(t)} and H {\displaystyle H} , it follows that any observable A {\displaystyle A} that commutes with H {\displaystyle H} will be conserved : its expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator A {\displaystyle A} can generate

6160-509: Is crucially quantum mechanical; quantum mechanics is well known to feature non-local effects (albeit still disallowing superluminal communication), and Vaidman has argued that this is just a non-local quantum effect in a different form. In classical electromagnetism the two descriptions were equivalent. With the addition of quantum theory, though, the electromagnetic potentials Φ and A are seen as being more fundamental. Despite this, all observable effects end up being expressible in terms of

6300-454: Is defined as the scalar -valued function: Although expressed in terms of coordinates, the result is invariant under rotations , as the physical interpretation suggests. This is because the trace of the Jacobian matrix of an N -dimensional vector field F in N -dimensional space is invariant under any invertible linear transformation . The common notation for the divergence ∇ · F

6440-463: Is equally valid to declare the zero momentum eigenfunction to be e − i ϕ ( x ) {\displaystyle e^{-i\phi (x)}} at the cost of representing the i-momentum operator (up to a factor) as ∇ i = ∂ i + i ( ∂ i ϕ ) {\displaystyle \nabla _{i}=\partial _{i}+i(\partial _{i}\phi )} i.e. with

6580-448: Is given by The operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} is known as the time-evolution operator, and has the crucial property that it is unitary . This time evolution is deterministic in the sense that – given an initial quantum state ψ ( 0 ) {\displaystyle \psi (0)} – it makes

6720-468: Is important conceptually because it bears on three issues apparent in the recasting of ( Maxwell 's) classical electromagnetic theory as a gauge theory , which before the advent of quantum mechanics could be argued to be a mathematical reformulation with no physical consequences. The Aharonov–Bohm thought experiments and their experimental realization imply that the issues were not just philosophical. The three issues are: Because of reasons like these,

6860-406: Is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ⟨ ψ , P λ ψ ⟩ {\displaystyle \langle \psi ,P_{\lambda }\psi \rangle } , where P λ {\displaystyle P_{\lambda }} is the projector onto its associated eigenspace. In

7000-726: Is known as wave–particle duality . In addition to light, electrons , atoms , and molecules are all found to exhibit the same dual behavior when fired towards a double slit. Another non-classical phenomenon predicted by quantum mechanics is quantum tunnelling : a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential. In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling radioactive decay , nuclear fusion in stars, and applications such as scanning tunnelling microscopy , tunnel diode and tunnel field-effect transistor . When quantum systems interact,

7140-444: Is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states. Instead, we can consider a Gaussian wave packet : which has Fourier transform, and therefore momentum distribution We see that as we make a {\displaystyle a} smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making

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7280-628: Is not sufficient for describing them at very small submicroscopic (atomic and subatomic ) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation, valid at large (macroscopic/microscopic) scale. Quantum systems have bound states that are quantized to discrete values of energy , momentum , angular momentum , and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both particles and waves ( wave–particle duality ), and there are limits to how accurately

7420-815: Is part of quantum communication protocols, such as quantum key distribution and superdense coding . Contrary to popular misconception, entanglement does not allow sending signals faster than light , as demonstrated by the no-communication theorem . Another possibility opened by entanglement is testing for " hidden variables ", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly Bell's theorem , have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of local hidden variables, then

7560-540: Is postulated to be normalized under the Hilbert space inner product, that is, it obeys ⟨ ψ , ψ ⟩ = 1 {\displaystyle \langle \psi ,\psi \rangle =1} , and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ψ {\displaystyle \psi } and e i α ψ {\displaystyle e^{i\alpha }\psi } represent

7700-466: Is replaced by − i ℏ ∂ ∂ x {\displaystyle -i\hbar {\frac {\partial }{\partial x}}} , and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times − ℏ 2 {\displaystyle -\hbar ^{2}} . When two different quantum systems are considered together,

7840-452: Is similar to that of the superconducting quantum interference devices (see SQUID ). The Aharonov–Bohm effect can be understood from the fact that one can only measure absolute values of the wave function. While this allows for measurement of phase differences through quantum interference experiments, there is no way to specify a wavefunction with constant absolute phase. In the absence of an electromagnetic field one can come close by declaring

7980-467: Is symmetric A ij = A ji then div ⁡ ( A ) = ∇ ⋅ A {\displaystyle \operatorname {div} (\mathbf {A} )=\nabla \cdot \mathbf {A} } . Because of this, often in the literature the two definitions (and symbols div and ∇ ⋅ {\displaystyle \nabla \cdot } ) are used interchangeably (especially in mechanics equations where tensor symmetry

8120-813: Is taken in order to handle the general case where the determinant might be negative, such as in pseudo-Riemannian spaces. The reason for the square-root is a bit subtle: it effectively avoids double-counting as one goes from curved to Cartesian coordinates, and back. The volume (the determinant) can also be understood as the Jacobian of the transformation from Cartesian to curvilinear coordinates, which for n = 3 gives ρ = | ∂ ( x , y , z ) ∂ ( x 1 , x 2 , x 3 ) | {\textstyle \rho =\left|{\frac {\partial (x,y,z)}{\partial (x^{1},x^{2},x^{3})}}\right|} . Some conventions expect all local basis elements to be normalized to unit length, as

8260-527: Is the coupling of the electromagnetic potential with the complex phase of a charged particle's wave function , and the Aharonov–Bohm effect is accordingly illustrated by interference experiments . The most commonly described case, sometimes called the Aharonov–Bohm solenoid effect , takes place when the wave function of a charged particle passing around a long solenoid experiences a phase shift as

8400-406: Is the metric tensor . The determinant appears because it provides the appropriate invariant definition of the volume, given a set of vectors. Since the determinant is a scalar quantity which doesn't depend on the indices, these can be suppressed, writing ρ = | det g | {\textstyle \rho ={\sqrt {\left|\det g\right|}}} . The absolute value

8540-415: Is the reduced Planck constant . The constant i ℏ {\displaystyle i\hbar } is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the correspondence principle . The solution of this differential equation

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8680-469: Is then If the state for the first system is the vector ψ A {\displaystyle \psi _{A}} and the state for the second system is ψ B {\displaystyle \psi _{B}} , then the state of the composite system is Not all states in the joint Hilbert space H A B {\displaystyle {\mathcal {H}}_{AB}} can be written in this form, however, because

8820-505: The Born rule : in the simplest case the eigenvalue λ {\displaystyle \lambda } is non-degenerate and the probability is given by | ⟨ λ → , ψ ⟩ | 2 {\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}} , where λ → {\displaystyle {\vec {\lambda }}}

8960-597: The Lagrangian approach to dynamics , based on energies , is not just a computational aid to the Newtonian approach , based on forces . Thus the Aharonov–Bohm effect validates the view that forces are an incomplete way to formulate physics, and potential energies must be used instead. In fact Richard Feynman complained that he had been taught electromagnetism from the perspective of electromagnetic fields, and he wished later in life he had been taught to think in terms of

9100-481: The Laplacian of the connection for the (free) Hamiltonian Equivalently, one can work in two simply connected regions with cuts that pass from the tube towards or away from the detection screen. In each of these regions the ordinary free Schrödinger equations would have to be solved, but in passing from one region to the other, in only one of the two connected components of the intersection (effectively in only one of

9240-494: The Lorentz force law . In this framework, because one of the observed properties of the electric field was that it was irrotational , and one of the observed properties of the magnetic field was that it was divergenceless , it was possible to express an electrostatic field as the gradient of a scalar potential (e.g. Coulomb 's electrostatic potential, which is mathematically analogous to the classical gravitational potential) and

9380-713: The canonical commutation relation : Given a quantum state, the Born rule lets us compute expectation values for both X {\displaystyle X} and P {\displaystyle P} , and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation , we have and likewise for the momentum: The uncertainty principle states that Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. This inequality generalizes to arbitrary pairs of self-adjoint operators A {\displaystyle A} and B {\displaystyle B} . The commutator of these two operators

9520-432: The curl and reads as follows: or The Laplacian of a scalar field is the divergence of the field's gradient : The divergence of the curl of any vector field (in three dimensions) is equal to zero: If a vector field F with zero divergence is defined on a ball in R , then there exists some vector field G on the ball with F = curl G . For regions in R more topologically complicated than this,

9660-400: The electromagnetic four-potential , ( Φ , A ), must be used instead. By Stokes' theorem , the magnitude of the Aharonov–Bohm effect can be calculated using the electromagnetic fields alone, or using the four-potential alone. But when using just the electromagnetic fields, the effect depends on the field values in a region from which the test particle is excluded. In contrast, when using just

9800-461: The electromagnetic potential A the Dirac string is not gauge invariant (it moves around with fixed endpoints under a gauge transformation) and so is also not directly measurable. Just as the phase of the wave function depends upon the magnetic vector potential, it also depends upon the scalar electric potential. By constructing a situation in which the electrostatic potential varies for two paths of

9940-423: The photoelectric effect . These early attempts to understand microscopic phenomena, now known as the " old quantum theory ", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr , Erwin Schrödinger , Werner Heisenberg , Max Born , Paul Dirac and others. The modern theory is formulated in various specially developed mathematical formalisms . In one of them, a mathematical entity called

10080-577: The volume element and F are the components of F = F i e i {\displaystyle \mathbf {F} =F^{i}\mathbf {e} _{i}} with respect to the local unnormalized covariant basis (sometimes written as e i = ∂ x / ∂ x i {\displaystyle \mathbf {e} _{i}=\partial \mathbf {x} /\partial x^{i}} ) . The Einstein notation implies summation over i , since it appears as both an upper and lower index. The volume coefficient ρ

10220-562: The wave function provides information, in the form of probability amplitudes , about what measurements of a particle's energy, momentum, and other physical properties may yield. Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms, but its application to human beings raises philosophical problems, such as Wigner's friend , and its application to

10360-496: The Aharonov-Bohm effect does not rely on it, provided the magnetic flux returns outside the electron paths, for example if one path goes through a toroidal solenoid and the other around it, and the solenoid is shielded so that it produces no external magnetic field.) However, there is a ( curl -free) vector potential A {\displaystyle \mathbf {A} } outside the solenoid with an enclosed flux, and so

10500-481: The Aharonov–Bohm effect was chosen by the New Scientist magazine as one of the "seven wonders of the quantum world". Chen-Ning Yang considered the Aharonov–Bohm effect to be the only direct experimental proof of the gauge principle . The philosophical importance is that the magnetic four-potential ( ϕ , A ) {\displaystyle (\phi ,\mathbf {A} )} over describes

10640-561: The Aharonov–Bohm effect. Application of these rings used as light capacitors or buffers includes photonic computing and communications technology. Analysis and measurement of geometric phases in mesoscopic rings is ongoing. It is even suggested they could be used to make a form of slow glass . Several experiments, including some reported in 2012, show Aharonov–Bohm oscillations in charge density wave (CDW) current versus magnetic flux, of dominant period h /2 e through CDW rings up to 85  μm in circumference above 77 K. This behavior

10780-431: The Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors C 2 {\displaystyle \mathbb {C} ^{2}} with the usual inner product. Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are Hermitian (more precisely, self-adjoint ) linear operators acting on

10920-411: The Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces H A {\displaystyle {\mathcal {H}}_{A}} and H B {\displaystyle {\mathcal {H}}_{B}} , respectively. The Hilbert space of the composite system

11060-432: The Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate , and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition . When an observable is measured, the result will be one of its eigenvalues with probability given by

11200-489: The Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator , the particle in a box , the dihydrogen cation , and the hydrogen atom . Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form . However, there are techniques for finding approximate solutions. One method, called perturbation theory , uses

11340-417: The Schrödinger equation for the particle in a box are or, from Euler's formula , Divergence As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have

11480-417: The above limit always converges to the same value for any sequence of volumes that contain x 0 and approach zero volume. The result, div F , is a scalar function of x . Since this definition is coordinate-free, it shows that the divergence is the same in any coordinate system . However it is not often used practically to calculate divergence; when the vector field is given in a coordinate system

11620-403: The analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy . Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. One consequence of

11760-483: The area between the paths (via Stokes' theorem and ∇ × A = B {\displaystyle \nabla \times \mathbf {A} =\mathbf {B} } ), and given by: In quantum mechanics the same particle can travel between two points by a variety of paths . Therefore, this phase difference can be observed by placing a solenoid between the slits of a double-slit experiment (or equivalent). An ideal solenoid (i.e. infinitely long and with

11900-606: The basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. Both position and momentum are observables, meaning that they are represented by Hermitian operators . The position operator X ^ {\displaystyle {\hat {X}}} and momentum operator P ^ {\displaystyle {\hat {P}}} do not commute, but rather satisfy

12040-404: The collection of probability amplitudes that pertain to another. One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for

12180-626: The continuous case, these formulas give instead the probability density . After the measurement, if result λ {\displaystyle \lambda } was obtained, the quantum state is postulated to collapse to λ → {\displaystyle {\vec {\lambda }}} , in the non-degenerate case, or to P λ ψ / ⟨ ψ , P λ ψ ⟩ {\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}} , in

12320-507: The coordinate definitions below are much simpler to use. A vector field with zero divergence everywhere is called solenoidal – in which case any closed surface has no net flux across it. In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field F = F x i + F y j + F z k {\displaystyle \mathbf {F} =F_{x}\mathbf {i} +F_{y}\mathbf {j} +F_{z}\mathbf {k} }

12460-431: The dependence in position means that the momentum operator is equivalent (up to an i / ℏ {\displaystyle i/\hbar } factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space . This is why in quantum equations in position space, the momentum p i {\displaystyle p_{i}}

12600-460: The domain. Here, the upper index refers to the number of the coordinate or component, so x refers to the second component, and not the quantity x squared. The index variable i is used to refer to an arbitrary component, such as x . The divergence can then be written via the Voss - Weyl formula, as: where ρ {\displaystyle \rho } is the local coefficient of

12740-562: The eigenfunction of the momentum operator with zero momentum to be the function "1" (ignoring normalization problems) and specifying wave functions relative to this eigenfunction "1". In this representation the i-momentum operator is (up to a factor ℏ / i {\displaystyle \hbar /i} ) the differential operator ∂ i = ∂ ∂ x i {\displaystyle \partial _{i}={\frac {\partial }{\partial x^{i}}}} . However, by gauge invariance, it

12880-535: The electromagnetic fields, E and B . This is interesting because, while you can calculate the electromagnetic field from the four-potential, due to gauge freedom the reverse is not true. The magnetic Aharonov–Bohm effect can be seen as a result of the requirement that quantum physics must be invariant with respect to the gauge choice for the electromagnetic potential , of which the magnetic vector potential A {\displaystyle \mathbf {A} } forms part. Electromagnetic theory implies that

13020-402: The electromagnetic potential instead, as this would be more fundamental. In Feynman's path-integral view of dynamics , the potential field directly changes the phase of an electron wave function, and it is these changes in phase that lead to measurable quantities. The Aharonov–Bohm effect shows that the local E and B fields do not contain full information about the electromagnetic field, and

13160-420: The electrostatic potential V {\displaystyle V} for a particle with charge q {\displaystyle q} . In particular, for a region with constant potential V {\displaystyle V} (zero field), the electric potential energy q V {\displaystyle qV} is simply added to E {\displaystyle E} , resulting in

13300-423: The experiment, ultra-cold rubidium atoms in superposition were launched vertically inside a vacuum tube and split with a laser so that one part would go higher than the other and then recombined back. Outside of the chamber at the top sits an axially symmetric mass that changes the gravitational potential. Thus, the part that goes higher should experience a greater change which manifests as an interference pattern when

13440-435: The fact that a connection with zero curvature (i.e. flat ), need not be trivial since it can have monodromy along a topologically nontrivial path fully contained in the zero curvature (i.e. field-free) region. By definition this means that sections that are parallelly translated along a topologically non trivial path pick up a phase, so that covariant constant sections cannot be defined over the whole field-free region. Given

13580-442: The flux is outgoing has positive divergence, and is often called a "source" of the field. A point at which the flux is directed inward has negative divergence, and is often called a "sink" of the field. The greater the flux of field through a small surface enclosing a given point, the greater the value of divergence at that point. A point at which there is zero flux through an enclosing surface has zero divergence. The divergence of

13720-410: The four-potential, the effect only depends on the potential in the region where the test particle is allowed. Therefore, one must either abandon the principle of locality , which most physicists are reluctant to do, or accept that the electromagnetic four-potential offers a more complete description of electromagnetism than the electric and magnetic fields can. On the other hand, the Aharonov–Bohm effect

13860-401: The gas will enclose gas which is expanding, so there will be an outward flux of gas through the surface. So the velocity field will have positive divergence everywhere. Similarly, if the gas is cooled, it will contract. There will be more room for gas particles in any volume, so the external pressure of the fluid will cause a net flow of gas volume inward through any closed surface. Therefore,

14000-415: The general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr–Einstein debates , in which the two scientists attempted to clarify these fundamental principles by way of thought experiments . In the decades after the formulation of quantum mechanics,

14140-467: The identity function F ( x ) = x , we find that: In spherical coordinates , with θ the angle with the z axis and φ the rotation around the z axis, and F again written in local unit coordinates, the divergence is Let A be continuously differentiable second-order tensor field defined as follows: the divergence in cartesian coordinate system is a first-order tensor field and can be defined in two ways: and We have If tensor

14280-462: The interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave). However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior

14420-423: The journal Physical Review A in 2017 have demonstrated a quantum mechanical solution for the system. Their analysis shows that the phase shift can be viewed as generated by a solenoid's vector potential acting on the electron or the electron's vector potential acting on the solenoid or the electron and solenoid currents acting on the quantized vector potential. Similarly, the Aharonov–Bohm effect illustrates that

14560-523: The last equality with the contravariant element e ^ i {\displaystyle {\hat {\mathbf {e} }}^{i}} , we can conclude that F i = F ^ i / g i i {\textstyle F^{i}={\hat {F}}^{i}/{\sqrt {g_{ii}}}} . After substituting, the formula becomes: See § In curvilinear coordinates for further discussion. The following properties can all be derived from

14700-559: The latter statement might be false (see Poincaré lemma ). The degree of failure of the truth of the statement, measured by the homology of the chain complex serves as a nice quantification of the complicatedness of the underlying region U . These are the beginnings and main motivations of de Rham cohomology . It can be shown that any stationary flux v ( r ) that is twice continuously differentiable in R and vanishes sufficiently fast for | r | → ∞ can be decomposed uniquely into an irrotational part E ( r ) and

14840-430: The light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere , producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves;

14980-424: The magnetic fields. An early experiment in which an unambiguous Aharonov–Bohm effect was observed by completely excluding the magnetic field from the electron path (with the help of a superconducting film) was performed by Tonomura et al. in 1986. The effect's scope and application continues to expand. Webb et al. (1985) demonstrated Aharonov–Bohm oscillations in ordinary, non-superconducting metallic rings; for

15120-432: The momentum operator with momentum p = ℏ k {\displaystyle p=\hbar k} . The coefficients of the superposition are ψ ^ ( k , 0 ) {\displaystyle {\hat {\psi }}(k,0)} , which is the Fourier transform of the initial quantum state ψ ( x , 0 ) {\displaystyle \psi (x,0)} . It

15260-478: The nuclear path. Werner Ehrenberg (1901–1975) and Raymond E. Siday first predicted the effect in 1949. Yakir Aharonov and David Bohm published their analysis in 1959. After publication of the 1959 paper, Bohm was informed of Ehrenberg and Siday's work, which was acknowledged and credited in Bohm and Aharonov's subsequent 1961 paper. The effect was confirmed experimentally, with a very large error, while Bohm

15400-413: The oldest and most common is the " transformation theory " proposed by Paul Dirac , which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg ) and wave mechanics (invented by Erwin Schrödinger ). An alternative formulation of quantum mechanics is Feynman 's path integral formulation , in which a quantum-mechanical amplitude

15540-412: The one-dimensional case in the x {\displaystyle x} direction, the time-independent Schrödinger equation may be written With the differential operator defined by with state ψ {\displaystyle \psi } in this case having energy E {\displaystyle E} coincident with the kinetic energy of the particle. The general solutions of

15680-458: The ordinary differentiation rules of calculus . Most importantly, the divergence is a linear operator , i.e., for all vector fields F and G and all real numbers a and b . There is a product rule of the following type: if φ is a scalar-valued function and F is a vector field, then or in more suggestive notation Another product rule for the cross product of two vector fields F and G in three dimensions involves

15820-455: The original quantum system ceases to exist as an independent entity (see Measurement in quantum mechanics ). The time evolution of a quantum state is described by the Schrödinger equation: Here H {\displaystyle H} denotes the Hamiltonian , the observable corresponding to the total energy of the system, and ℏ {\displaystyle \hbar }

15960-464: The particles from external electric fields in the regions where they travel, but still allow a time dependent potential to be applied by charging the cylinders. This proved difficult to realize, however. Instead, a different experiment was proposed involving a ring geometry interrupted by tunnel barriers, with a constant bias voltage V relating the potentials of the two halves of the ring. This situation results in an Aharonov–Bohm phase shift as above, and

16100-448: The physicality of electromagnetic potentials, Φ and A , in quantum mechanics. Classically it was possible to argue that only the electromagnetic fields are physical, while the electromagnetic potentials are purely mathematical constructs, that due to gauge freedom are not even unique for a given electromagnetic field. However, Vaidman has challenged this interpretation by showing that the Aharonov–Bohm effect can be explained without

16240-475: The physics, as all observable phenomena remain unchanged after a gauge transformation. Conversely, the Maxwell fields under describe the physics, as they do not predict the Aharonov-Bohm effect. Moreover, as predicted by the gauge principle, the quantities that remain invariant under gauge transforms are precisely the physically observable phenomena. It is generally argued that the Aharonov–Bohm effect illustrates

16380-428: The position becomes more and more uncertain. The uncertainty in momentum, however, stays constant. The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region. For

16520-400: The question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of " wave function collapse " (see, for example, the many-worlds interpretation ). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that

16660-530: The relative phase of particles passing through one slit or the other is altered by whether the solenoid current is turned on or off. This corresponds to an observable shift of the interference fringes on the observation plane. The same phase effect is responsible for the quantized-flux requirement in superconducting loops. This quantization occurs because the superconducting wave function must be single valued: its phase difference Δ φ {\displaystyle \Delta \varphi } around

16800-413: The result can be the creation of quantum entanglement : their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "... the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought". Quantum entanglement enables quantum computing and

16940-566: The results of a Bell test will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables. It is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also linear algebra , differential equations , group theory , and other more advanced subjects. Accordingly, this article will present

17080-463: The same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space . The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions L 2 ( C ) {\displaystyle L^{2}(\mathbb {C} )} , while

17220-409: The screen, then we should observe a phase shift e Δ V t / ℏ {\displaystyle e\Delta Vt/\hbar } . By adjusting the battery voltage, we can horizontally shift the interference pattern on the screen. The initial theoretical proposal for this effect suggested an experiment where charges pass through conducting cylinders along two paths, which shield

17360-460: The slits) a monodromy factor e i α {\displaystyle e^{i\alpha }} is picked up, which results in the shift in the interference pattern as one changes the flux. Effects with similar mathematical interpretation can be found in other fields. For example, in classical statistical physics, quantization of a molecular motor motion in a stochastic environment can be interpreted as an Aharonov–Bohm effect induced by

17500-421: The source density div v by the circulation density ∇ × v . This "decomposition theorem" is a by-product of the stationary case of electrodynamics . It is a special case of the more general Helmholtz decomposition , which works in dimensions greater than three as well. The divergence of a vector field can be defined in any finite number n {\displaystyle n} of dimensions. If in

17640-455: The superconductor and only describe the physics in the outside region, it becomes natural and mathematically convenient to describe the quantum electron by a section in a complex line bundle with an "external" flat connection ∇ {\displaystyle \nabla } with monodromy rather than an external EM field F {\displaystyle F} . The Schrödinger equation readily generalizes to this situation by using

17780-625: The superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if ψ A {\displaystyle \psi _{A}} and ϕ A {\displaystyle \phi _{A}} are both possible states for system A {\displaystyle A} , and likewise ψ B {\displaystyle \psi _{B}} and ϕ B {\displaystyle \phi _{B}} are both possible states for system B {\displaystyle B} , then

17920-441: The theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a complex number , known as a probability amplitude. This is known as the Born rule , named after physicist Max Born . For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space

18060-412: The tube the connection is flat, and the monodromy of the loop contained in the field-free region depends only on the winding number around the tube. The monodromy of the connection for a loop going round once (winding number 1) is the phase difference of a particle interfering by propagating left and right of the superconducting tube containing the magnetic field. If one wants to ignore the physics inside

18200-437: The universe as a whole remains speculative. Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy . For example, the refinement of quantum mechanics for the interaction of light and matter, known as quantum electrodynamics (QED), has been shown to agree with experiment to within 1 part in 10 when predicting the magnetic properties of an electron. A fundamental feature of

18340-425: The use of potentials so long as one gives a full quantum mechanical treatment to the source charges that produce the electromagnetic field. According to this view, the potential in quantum mechanics is just as physical (or non-physical) as it was classically. Aharonov, Cohen, and Rohrlich responded that the effect may be due to a local gauge potential or due to non-local gauge-invariant fields. Two papers published in

18480-526: The value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle ). Quantum mechanics arose gradually from theories to explain observations that could not be reconciled with classical physics, such as Max Planck 's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein 's 1905 paper , which explained

18620-494: The vector potential. However, if the monodromy is nontrivial, there is no such gauge transformation for the whole outside region. In fact as a consequence of Stokes' theorem , the holonomy is determined by the magnetic flux through a surface σ {\displaystyle \sigma } bounding the loop γ {\displaystyle \gamma } , but such a surface may exist only if σ {\displaystyle \sigma } passes through

18760-422: The velocity field has negative divergence everywhere. In contrast, in a gas at a constant temperature and pressure, the net flux of gas out of any closed surface is zero. The gas may be moving, but the volume rate of gas flowing into any closed surface must equal the volume rate flowing out, so the net flux is zero. Thus the gas velocity has zero divergence everywhere. A field which has zero divergence everywhere

18900-413: The wave packets recombine resulting in a measurable phase shift. Evidence of a match between the measurements and the predictions was found by the team. Multiple other tests have been proposed. In 1975 Tai-Tsun Wu and Chen-Ning Yang formulated the non-abelian Aharonov–Bohm effect, and in 2019 this was experimentally reported in a system with light waves rather than the electron wave function. The effect

19040-443: Was also predicted, in which a charged particle is affected by regions with different electrical potentials but zero electric field, but this has no experimental confirmation yet. A separate "molecular" Aharonov–Bohm effect was proposed for nuclear motion in multiply connected regions, but this has been argued to be a different kind of geometric phase as it is "neither nonlocal nor topological", depending only on local quantities along

19180-424: Was done in the previous sections. If we write e ^ i {\displaystyle {\hat {\mathbf {e} }}_{i}} for the normalized basis, and F ^ i {\displaystyle {\hat {F}}^{i}} for the components of F with respect to it, we have that using one of the properties of the metric tensor. By dotting both sides of

19320-458: Was observed experimentally in 1998, albeit in a setup where the charges do traverse the electric field generated by the bias voltage. The original time dependent electric Aharonov–Bohm effect has not yet found experimental verification. The Aharonov–Bohm phase shift due to the gravitational potential should also be possible to observe in theory, and in early 2022 an experiment was carried out to observe it based on an experimental design from 2012. In

19460-467: Was produced in two different ways. In one light went through a crystal in strong magnetic field and in another light was modulated using time-varying electrical signals. In both cases the phase shift was observed via an interference pattern which was also different depending if going forwards and backwards in time. Nano rings were created by accident while intending to make quantum dots . They have interesting optical properties associated with excitons and

19600-564: Was still alive. By the time the error was down to a respectable value, Bohm had died. In the 18th and 19th centuries, physics was dominated by Newtonian dynamics, with its emphasis on forces . Electromagnetic phenomena were elucidated by a series of experiments involving the measurement of forces between charges, currents and magnets in various configurations. Eventually, a description arose according to which charges, currents and magnets acted as local sources of propagating force fields, which then acted on other charges and currents locally through

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