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Chapter 35: Problem 6

In terms of de Broglie's matter-wave hypothesis, how does making the sides of a box different lengths remove the degeneracy associated with a particle confined to that box?

Short Answer

Expert verified
Changing the dimensions of the box changes the wavelengths associated with each dimension in accordance with de Broglie's hypothesis. Since the energy of the particle is directly related to the wavelength, the energy levels are no longer equivalent. Therefore, the degeneracy associated with the particle is removed.

Step by step solution

01

Step 1. Understand de Broglie's matter-wave hypothesis

Let's start by understanding the basic principle of de Broglie's matter-wave philosophy. According to de Broglie, particles, like photons or electrons, exhibit properties of both particles and waves. The wavelength of a particle is given by \( \lambda = \frac{h}{p} \) where \( \lambda \) is the de Broglie wavelength, \( h \) is the Planck's constant and \( p \) is the momentum.
02

Step 2. Understanding the energy levels in a box

For a particle confined in a three-dimensional box, the energy levels are determined by the dimensions of the box. When all sides of the box are of equal length, the energy levels of the particles found on the walls of the box would be degenerate (having the same energy level).
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Step 3. Relate the change of box dimension to the degeneracy

If the box's dimensions are changed, meaning the box becomes a rectangular box rather than a cube, it means that the wavelengths associated with each dimension will be different because the wavelengths are inversely proportional to the box's dimensions. Since the energy of the particle is directly related to its wavelength, the energy levels are no longer equivalent. As a result, the degeneracy associated with the particle is removed.

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