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1-D Quantum Mechanics with Atomic Ions

In most metals, the atomic ions form a regular arrangement called a crystal lattice. The conduction electrons in the sea of electrons move through this lattice. The figure below is a one-dimensional model of a crystal lattice. The ions have mass m, charge e, and an equilibrium separation b.

a) Suppose the middle charge is displaced a very small distance (x << b) from its
equilibrium position while the outer charges remain fixed. Show that the net electric force
on the middle charge is give approximately by:

F = (-e^2 / b^3 pi e_0) x see attachment for better formula representation

In other words, the charge experiences a linear restoring force.

b) Suppose this crystal consists of aluminum ions with an equilibrium spacing of 0.30
nm. What are the energies of the four lowest vibrational states of these ions?

c) What wavelength photons are emitted during quantum jumps between adjacent energy
levels? Is this wavelength in the infrared, visible, or ultraviolet portion of the spectrum?

Attachments

Solution Summary

This solution shows that the net electric force on the middle charge experiences a linear restoring force, the energies of the four lowest vibrational states of the ions, and what part of the electromagnetic spectrum does the wavelength of the photons belong to. Step-by-step calculations and explanations are included for further topic understanding.

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