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The Aharonov-Bohm effect is the phase $q\Phi /\hslash$ acquired when a charge $q$ encircles magnetic flux $\Phi$, even when the magnetic field vanishes along the charge's path.

In the magnetic Aharonov-Bohm effect, a coherent electron wave takes two routes around inaccessible magnetic flux. Recombination converts their relative wave phase into displaced interference fringes. In 1959, Yakir Aharonov and David Bohm proposed a shielded solenoid geometry and identified the obstruction to removing the magnetic vector potential throughout the accessible region.^[ref]

Use International System of Units, electric charge $q$, and magnetic minimal coupling $H=(-i\hslash \nabla -q\mathbf{A}{)}^2/(2m)$. Let ${\gamma }_1,{\gamma }_2$ run from the same source $S$ to the same detector point $D$. The oriented closed curve $C$ follows ${\gamma }_1$ and then ${\gamma }_2$ backwards. For a static magnetic vector potential, the magnetic contribution to the relative wave phase isHere ${\Phi }_C={\int }_{\Sigma }\mathbf{B}\cdot d\mathbf{S}$ is the oriented magnetic flux through a spanning surface. Equal or calibrated nonmagnetic propagation phases are assumed. For an electron, $q=-e$, where $e>0$ is the elementary charge; reversing $C$ reverses the sign. The observable is the gauge-invariant holonomy $e^{i\delta }$, rather than a unique local value of the magnetic vector potential.

The accessible region is a multiply connected region: $C$ cannot contract to a point without entering the excluded flux region. Stokes theorem does not set the line integral to zero, because a spanning surface must cross that excluded region, where the magnetic field need not vanish. There is real magnetic flux in the apparatus; only the accessible paths are idealized as free of the magnetic field. The exterior of a finite straight solenoid also permits routes around its ends, so its exact topology and leakage differ from those of an infinite solenoid or an excluded solid torus.

An explicit model makes the obstruction visible. Outside an infinite flux tube of radius $a$, choose cylindrical coordinates $(r,\theta ,z)$ andIts curl vanishes there, but a loop with winding number $n$ has ${\oint }_C\mathbf{A}\cdot d\mathbf{l}=n\Phi$. Locally $\mathbf{A}=\nabla (\Phi \theta /(2\pi ))$; the angular coordinate changes by $2\pi$ on a circuit. For generic $q\Phi /\hslash$, that candidate gauge removal changes the wave function's single-valued boundary condition. On a simply connected region, a smooth static magnetic vector potential with zero curl instead has a globally defined scalar primitive and is a pure gauge potential. Thus local zero magnetic field and globally removable magnetic vector potential are different conditions.

For an original interference example, take mutually coherent scalar amplitudes with constant individual intensities $I_1,I_2$ and an imposed carrier phase $ky$, as in off-axis holography. ThenThe interference fringe spacing is $d=2\pi /k$. An electron enclosing $\Phi =h/(4e)$ has $\delta =-\pi /2$, shifting maxima by $d/4$ for this orientation. Increasing the magnetic flux by $h/e$ changes $\delta$ by $-2\pi$ and restores the same pattern: the electron Aharonov-Bohm effect has magnetic flux quantum $h/e$. This uses $h=2\pi \hslash$, the Planck constant. It does not require the source's magnetic flux to vary continuously during a single exposure.

For $I_1=1$, $I_2=1/4$, the extrema are $I_{\max }=9/4$, $I_{\min }=1/4$, and the interferometric visibility is $4/5$. A fixed Aharonov-Bohm phase translates these fringes without changing their contrast. Averaging unresolved fluctuating phases can reduce the contrast; a darkened pattern alone cannot identify enclosed magnetic flux. These formulas assume coherence, indistinguishable alternatives and stable beam envelopes.

The appeal of the Aharonov-Bohm effect is that a local Lorentz force calculation along either ideal path misses information carried by their global arrangement. It gives a physical example of holonomy and supplies the encircling phase used in Chern--Simons flux attachment. This alone does not establish intrinsic topological order. The static comparison also matters: changing magnetic flux can induce an electric field, so a time-dependent experiment requires the scalar-potential contribution as well as the displayed spatial line integral.

First part of David Bohm's two-paper account of de Broglie-Bohm theory, published in Physical Review 85, starting at page 166.

Second part of David Bohm's account of de Broglie-Bohm theory, treating measurement and the relation to ordinary predictions. Published in Physical Review 85, starting at page 180.

David Bohm's 1952 formulation of de Broglie-Bohm theory supplements the wave function with definite particle positions guided by that wave function. For a spinless particle of mass $m$, writing $\psi =R{e}^{iS/\hslash }$ gives the guidance velocity $\nabla S/m$. The resulting trajectories and the Schrödinger equation together preserve the distribution $|\psi {|}^2$ when it holds initially, connecting the particle description with ordinary measurement statistics. His second paper examined the measurement process and its relation to the usual quantum mechanics predictions.^[ref][ref]

A physicist, Professor of theoretical physics at Birkbeck from 1961 to 1987.^[ref]

In de Broglie-Bohm theory, the system has both a wavefunction and an actual configuration of particle positions. The wavefunction obeys the Time-dependent Schrödinger equation; the configuration follows the guidance equation. For spinless particles without electromagnetic vector potentials, write $\Psi =R{e}^{iS/\hslash }$ using a dimensionful action $S$. ThenThe actual configuration selects a trajectory, while the guiding wave generally retains all branches. A measurement correlates system and apparatus configurations; the actual apparatus position lies in one recorded branch. Effective collapse follows when wave packets have separated sufficiently in configuration space, although the full wave continues its Schrödinger evolution.

Substituting the polar form into that equation yields the probability continuity equation and a modified Hamilton-Jacobi equation,The amplitude-dependent term is the quantum potential. Taking gradients gives a second-order force description, but initial velocities remain constrained by the guidance law. Nodes require care because the displayed quotient is singular there.

If the initial configuration distribution is $|\Psi {|}^2$, the same continuity equation preserves that distribution: quantum equilibrium equivariance. It then reproduces ordinary nonrelativistic quantum measurement statistics within the model's applicability. Equivariance preserves a distribution; it is not by itself a proof that every initial distribution equals $|\Psi {|}^2$.

For entangled particles the guiding wave lives on their joint configuration space, so one particle's velocity can depend on the distant actual configuration. This explicit nonlocality avoids the locality premise of a Bell inequality. In quantum equilibrium it does not permit controllable faster-than-light signalling. Spin extensions use a spinor wavefunction and its probability current; they need not assign a classical little spinning vector with simultaneous definite values for every spin component.

David Bohm developed this account in his two 1952 papers. The first sets out the dynamics and the second treats measurement and the relation to ordinary predictions.^[ref][ref] The theory supplies additional dynamical variables and a particular nonlocal account of outcomes, rather than deriving a unique interpretation from the standard wavefunction alone.

In Bohmian mechanics, the corpuscles supply an actual configuration of matter, not a further branch of the wave function. Their positions and the guiding wave together specify the physical state. In a common nonrelativistic spinless formulation, the wave obeys the Schrödinger equation while the actual configuration $Q=(Q_1,\dots ,Q_N)$ obeyswhere the expression is applied away from nodes of $\Psi$. These are coupled in the sense that the wave guides the configuration; in the standard theory the wave's Schrödinger evolution does not receive a reciprocal source term from the actual particle positions. This is not a model of classical little balls carrying every ordinary observable as a pre-existing intrinsic attribute.

The point of adding a configuration is especially clear in a measurement. A wave may evolve into several macroscopically separated pointer packets. The apparatus's actual particle configuration lies in one packet, so there is a definite pointer record even though the full wave has not undergone fundamental collapse. The corpuscles thereby connect the mathematical description with the location of actual apparatus, records and observers. David Bohm's original papers provide a theory of individual processes and analyse measurement in this spirit.^[ref][ref]

For a subsystem, inserting the actual environment configuration into the universal wave gives a conditional wave function, schematically $\psi (x,t)\propto \Psi (x,Y(t),t)$. If the branches are suitably disjoint in the environment's configuration coordinates and remain dynamically isolated, the occupied branch supplies an effective collapse in Bohmian mechanics. The unoccupied branches are not simply erased by definition: before permanent separation, wave packets can recombine and affect interference. The conditional and effective wave-function constructions make this distinction precise.^[ref]

The Born rule also needs an account. The guidance current is $j_k=(\hslash /m_k)\mathrm{Im}({\Psi }^{\ast }{\nabla }_k\Psi )$, and both $|\Psi {|}^2$ and an ensemble of guided configurations satisfy the associated continuity equation. Hence an initial quantum equilibrium distribution $\rho =|\Psi {|}^2$ is preserved. Equivariance supports the ordinary statistical predictions, but the guidance law alone does not prove that an arbitrarily prepared ensemble has that distribution.

The corpuscles are therefore neither detectors that a conscious observer adds to the theory nor dispensable labels attached after the statistics have been calculated. They constitute a primitive ontology that provides definite configurations, trajectories and records. Whether this is preferable to a wave-only ontology is a further explanatory judgement; its empirical equivalence in quantum equilibrium does not make the two accounts identical in their claims about what exists.

Yakir Aharonov and David Bohm showed how scalar potentials and vector potentials can affect quantum interference even when a charged particle travels through a region with vanishing magnetic field. In the ideal magnetic arrangement, two paths enclose a confined magnetic flux $\Phi$, and their relative quantum phase changes by $q\Phi /\hslash$ for charge $q$. This makes the observable interference shift sensitive to the enclosed flux and the geometry of the paths, rather than only the local magnetic force along either path.^[ref]

A physicist researching quantum theory and the Aharonov-Bohm effect.^[ref]