Tru Physics

Aharonov–Bohm Effect

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Introduction

The Aharonov–Bohm effect is a quantum mechanical phenomenon that demonstrates how a charged particle is affected by electromagnetic fields, despite being in regions of apparently zero electromagnetic fields. This effect illustrates the fundamental role of potentials in quantum mechanics, in contrast to classical physics where fields play the dominant role.

Aharonov–Bohm Effect. Artistic representation of an electron being influenced by electromagnetic fields.
Artistic representation of an electron being influenced by electromagnetic fields.

Aharonov–Bohm Electromagnetic Potential

The electromagnetic potential consists of the electric potential V and the magnetic vector potential \vec{A}. The electric and magnetic fields \vec{E} and \vec{B} can be obtained from the potentials:

\vec{E} = -\nabla V - \dfrac{\partial \vec{A}}{\partial t}

\vec{B} = \nabla \times \vec{A}

In the Aharonov–Bohm effect, the potentials V and \vec{A} affect the charged particle’s phase, even if the fields \vec{E} and \vec{B} are zero.

The Quantum Phase Shift

The Aharonov–Bohm effect is generally observed in the context of electron interference in a two-slit experiment. In the presence of the electromagnetic potential, the wave function of the electron acquires a phase shift.

This phase shift \Delta \phi for a charged particle moving in an electromagnetic potential is given by:

\Delta \phi = \dfrac{q}{\hbar} \displaystyle\int_{C} \left(\vec{A} \cdot d\vec{l} - V dt\right)

where q is the charge of the particle, \hbar is the reduced Planck’s constant, C denotes the path of the particle, \vec{A} is the magnetic vector potential, V is the electric potential, and dt is the time interval.

Aharonov–Bohm Solenoid Experiment

In the classic two-slit experiment demonstrating the Aharonov–Bohm effect, a solenoid (or magnetic flux tube) is placed between the slits, such that the electron paths outside the solenoid are free from magnetic fields. However, the magnetic vector potential \vec{A} exists along the path.

For an electron moving around the solenoid, the phase shift is:

\Delta \phi = \dfrac{q}{\hbar} \displaystyle\int_{C} \vec{A} \cdot d\vec{l}

where the integral is along the path C of the electron around the solenoid.

This phase shift causes an observable shift in the interference pattern.

Significance and Applications

The Aharonov–Bohm effect highlights the profound difference between classical and quantum physics. It has been experimentally verified multiple times and is a key aspect in understanding quantum interference, quantum rings, and the behavior of superconducting quantum interference devices (SQUIDs).

Conclusion

The Aharonov–Bohm effect is a significant quantum phenomenon that validates the role of electromagnetic potentials in quantum physics. Understanding this effect allows for a deeper comprehension of quantum mechanics and its fundamental differences from classical physics.

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