A double-slit interference experiment is performed with 2.0...

A double-slit interference experiment is performed with $2.0$ -eV photons. The same pair of slits is then used for an experiment with electrons. What is the kinetic energy of the electrons if the interference pattern is the same as for the photons (that is, the spacing between maxima is the same )?

Key Concepts

de Broglie Wavelength

The de Broglie wavelength concept extends the idea of wave-particle duality to matter, stating that any particle with momentum has an associated wavelength given by ? = h/p. This equation forms the basis for analyzing the interference patterns of electrons, linking their momentum (and thus kinetic energy) to their wave-like behavior.

Wave-Particle Duality

This is the fundamental concept that all particles exhibit both wave-like and particle-like properties. In quantum mechanics, entities such as photons and electrons can interfere and diffract like waves. The double-slit experiment is a classic demonstration of this duality, showing that particles create an interference pattern when not observed, reflecting their wave nature.

Double-Slit Interference

Double-slit interference refers to the formation of an interference pattern due to the coherent superposition of waves passing through two narrow, closely-spaced slits. The fringe spacing in the resulting pattern is directly related to the wavelength of the waves and the geometry of the setup. Understanding this phenomenon is essential in comparing interference patterns produced by different types of particles.

Photon Energy-Wavelength Relationship

For photons, energy and wavelength are inversely related by the equation E = hc/?, where h is Planck’s constant and c is the speed of light. This relationship allows for the determination of a photon's wavelength from its energy, which is critical when comparing the interference patterns produced by light and matter waves.

Kinetic Energy of Particles

For non-relativistic particles such as electrons, kinetic energy is related to momentum by the expression K = p²/(2m), where m is the mass of the particle. By equating the de Broglie wavelength of electrons to that derived from the photon’s energy in a double-slit experiment, one can determine the kinetic energy required for electrons to produce a matching interference pattern.

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