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Chapter 35: Problem 7

A particle is confined to a two-dimensional box whose sides are in the ratio \(1: 2 .\) Are any of its energy levels degenerate? If so, give an example. If not, why not?

Short Answer

Expert verified
No, there are no degenerate energy levels for a particle confined in a two-dimensional quantum box with sides in the ratio 1:2.

Step by step solution

01

Understand the concept of degeneracy in quantum mechanics

In quantum mechanics, degeneracy refers to the scenario of different states, particularly in quantum systems, having the same energy level. Mathematically, in a quantum system confined in a box (a potential well), the energy levels are usually defined by the quantum numbers that characterize the quantum states. In a three-dimensional quantum box, the energy level is expressed by \(E = (n_x^2 + n_y^2 + n_z^2)E_0\) where \(E_0\) is the ground state energy and \(n_x, n_y, n_z\) are the quantum numbers. Degeneracy can occur when a rearrangement of these quantum numbers yields the same energy level.
02

Apply the principle to a two-dimensional box with different dimensions

However, this system is a two-dimensional quantum box whose sides are in the ratio 1:2. As such, the energy level in this situation could be expressed as \(E = (n_x^2/a^2 + n_y^2/c^2)E_0\) where a and c are lengths of the sides of the box, \(E_0\) is the ground state energy and \(n_x, n_y\) are the quantum numbers. Since the box's sides have different lengths, each quantum number corresponds to different energy value because of the lengths' square in the denominator of their respective terms. Therefore it is impossible to rearrange \(n_x\) and \(n_y\) such that the energy level remains the same as before.
03

Draw conclusions from the energy level equation

From the energy level equation we derived for the two-dimensional quantum box, we can clearly see that due to the squares in the denominators of both terms, no two different states will have the same energy. Therefore, no degenerate energy levels exist for a particle confined in this specific type of two-dimensional quantum box.

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Most popular questions from this chapter

One reason we don't notice quantum effects in everyday life is that Planck's constant \(h\) is so small. Treating yourself as a particle (mass \(60 \mathrm{kg}\) ) in a room-sized one-dimensional infinite square well (width \(2.6 \mathrm{m}\) ), how big would \(h\) have to be if your minimum possible energy corresponded to a speed of \(1.0 \mathrm{m} / \mathrm{s} ?\)

Explain qualitatively why a particle confined to a finite region cannot have zero energy.

Electrons in an ensemble of 0.834 -nm-wide square wells are all initially in the \(n=4\) state. (a) How many different wavelengths of spectral lines could be emitted as the electrons cascade to the ground state through all possible downward transitions? (b) Find those wavelengths. (c) What regions of the electromagnetic spectrum do these spectral lines encompass?

A particle is confined to a 1.0 -nm-wide infinite square well. If the energy difference between the ground state and the first excited state is \(1.13 \mathrm{eV},\) is the particle an electron or a proton?

The solution to the Schr枚dinger equation for a particular potential is \(\psi=0\) for \(|x|>a\) and \(\psi=A \sin (\pi x / a)\) for \(-a \leq x \leq a\) where \(A\) and \(a\) are constants. In terms of \(a,\) what value of \(A\) is required to normalize \(\psi\) ?
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