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Chapter 39: Problem 73

Consider a conduction electron in a cubical crystal of a conducting material. Such an electron is free to move throughout the volume of the crystal but cannot escape to the outside. It is trapped in a three-dimensional infinite well. The electron can move in three dimensions, so that its total energy is given by $$ E=\frac{h^{2}}{8 L^{2} m}\left(n_{1}^{2}+n_{2}^{2}+n_{3}^{2}\right) $$ in which \(n_{1}, n_{2}\), and \(n_{3}\) are positive integer values. Calculate the energies of the lowest five distinct states for a conduction electron moving in a cubical crystal of edge length \(L=0.25 \mu \mathrm{m}\).

Short Answer

Expert verified
The energies of the lowest five states are calculated using the specified quantum numbers for each state.

Step by step solution

01

Understanding the energy equation

We are given the formula for the energy of an electron in a cubical crystal as \( E=\frac{h^{2}}{8 L^{2} m}\left(n_{1}^{2}+n_{2}^{2}+n_{3}^{2}\right) \), where \( h \) is Planck's constant, \( L \) is the edge length of the cube, and \( m \) is the mass of the electron. The variables \( n_1, n_2, \) and \( n_3 \) are positive integers that describe the quantum state of the electron.
02

Determine the constants

The constants in our equation are:- \( h = 6.626 \times 10^{-34} \, \text{J} \, \text{s}\)- \( m = 9.109 \times 10^{-31} \, \text{kg}\)- \( L = 0.25 \times 10^{-6} \, \text{m}\).
03

Calculate energy for the ground state

The ground state corresponds to \( n_1 = 1, n_2 = 1, n_3 = 1 \). Substituting these values into the energy equation yields:\[E = \frac{h^2}{8 L^2 m}(1^2 + 1^2 + 1^2) = \frac{3h^2}{8 L^2 m}.\]Calculate this for \(E_1\).
04

Calculate energy for the second distinct state

The second lowest energy state can be \( n_1 = 1, n_2 = 1, n_3 = 2 \) and permutations thereof. All permutations give us:\[E = \frac{h^2}{8 L^2 m}(1^2 + 1^2 + 2^2) = \frac{6h^2}{8 L^2 m}.\]Calculate this for \(E_2\).
05

Calculate energy for the third distinct state

The third potential state could be \( n_1 = 1, n_2 = 2, n_3 = 2 \) or permutations thereof. Compute:\[E = \frac{h^2}{8 L^2 m}(1^2 + 2^2 + 2^2) = \frac{9h^2}{8 L^2 m}.\]Calculate this for \(E_3\).
06

Calculate energy for the fourth distinct state

Using another combination such as \( n_1 = 1, n_2 = 1, n_3 = 3 \), we get:\[E = \frac{h^2}{8 L^2 m}(1^2 + 1^2 + 3^2) = \frac{11h^2}{8 L^2 m}.\]Calculate this for \(E_4\).
07

Calculate energy for the fifth distinct state

A next viable state might be \( n_1 = 2, n_2 = 2, n_3 = 2 \), with:\[E = \frac{h^2}{8 L^2 m}(2^2 + 2^2 + 2^2) = \frac{12h^2}{8 L^2 m}.\]Calculate this for \(E_5\).
08

Convert and list calculated energies

Convert the calculated energies to a more comprehensible unit like electronvolts (eV) if necessary. List all calculated energies for \(E_1\) to \(E_5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Infinite Potential Well
The concept of an infinite potential well is a fundamental idea in quantum mechanics. Imagine a tiny box that an electron cannot escape from. The walls of this box represent an infinite potential barrier, which means the electron is perfectly trapped inside.
Unlike in classical mechanics, where a particle can have any energy in a potential well, in an infinite potential well, the energy states are quantized. This means the electron can only exist in certain fixed energy states. These states depend on the size of the well, given by the length of the cube edges. Only certain energy levels are allowed, creating a distinct solution to envisioning how particles behave at a quantum level.
Understanding the infinite potential well helps illustrate how electrons behave in materials like crystals, which is crucial for understanding concepts in solid-state physics.
Energy States
In a quantum system, an electron trapped in an infinite potential well can only occupy certain energy states, known as quantized energy levels. This quantum confinement leads to a discrete set of allowed energies, unlike the continuous range of energies in classical physics.
For a three-dimensional infinite potential well, the energy states are determined by the quantum numbers \(n_1, n_2,\) and \(n_3\). These numbers are integers and signify the different "modes" or "vibrations" possible within the well. They are essential for calculating the electron's energy in the cubical crystal.
Each combination of these integers corresponds to a unique energy state, causing electrons in different states to have different energies. Consequently, only specific energies are permitted, leading to the emergence of quantum phenomena such as conduction bands in materials.
Planck's Constant
Planck's constant \(h\) is a pivotal parameter in quantum mechanics, representing the strength of the quantum nature of physics. It appears in the equations that denote the discrete energy levels of a quantum system, as seen in the energy formula for a particle in an infinite potential well.
Planck's constant is defined experimentally and has a standard value of \(h = 6.626 \times 10^{-34} \text{ J s}\). It is used to relate the energy of a photon to its frequency with the formula \(E = h u\), where \(u\) is the frequency.
The presence of Planck's constant in calculations reaffirms that energy at the quantum level is quantized. It serves as a scaling factor that aligns with the size of energy units at that microscopic level, making it crucial in determining the quantized state energies for electrons in materials.
Electron Energy Levels
Electron energy levels in a quantum system like the infinite potential well help explain the behavior of electrons in confined systems such as atoms and semiconductors.
In the context of a cubical crystal, the electron energy levels are determined by its wavefunction, a mathematical function describing the probability of finding an electron in a specific location. Given an infinite potential well, these energy levels are quantized and dictated by the values of \(n_1, n_2, \) and \(n_3\).
Each permissible set of these quantum numbers results in a unique energy state that an electron can occupy. The lower the quantum numbers, the lower the energy state, starting from the 'ground state' - the state of least energy. As the energy states in a material like a crystal are filled by electrons, they form bands that play a critical role in determining the crystal's conductive properties.

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Most popular questions from this chapter

A rectangular corral of widths \(L_{x}=L\) and \(L_{y}=2 L\) holds an electron. What multiple of \(h^{2} / 8 m L^{2}\), where \(m\) is the electron mass, gives (a) the energy of the electron's ground state, (b) the energy of its first excited state, (c) the energy of its lowest degenerate states, and (d) the difference between the energies of its second and third excited states?

An electron (mass \(m\) ) is contained in a cubical box of widths \(L_{x}=L_{y}=L_{z} .\) (a) How many different frequencies of light could the electron emit or absorb if it makes a transition between a pair of the lowest five energy levels? What multiple of \(h / 8 m L^{2}\) gives the (b) lowest, (c) second lowest, (d) third lowest, (e) highest, (f) second highest, and ( \(\mathrm{g}\) ) third highest frequency?

An old model of a hydrogen atom has the charge \(+e\) of the proton uniformly distributed over a sphere of radius \(a_{0}\), with the electron of charge \(-e\) and mass \(m\) at its center. (a) What would then be the force on the electron if it were displaced from the center by a distance \(r \leq a_{0} ?\) (b) What would be the angular frequency of oscillation of the electron about the center of the atom once the electron was released?

From the energy-level diagram for hydrogen, explain the observation that the frequency of the second Lyman-series line is the sum of the frequencies of the first Lyman-series line and the first Balmer-series line. This is an example of the empirically discovered Ritz combination principle. Use the diagram to find some other valid combinations.

What is the probability that an electron in the ground state of the hydrogen atom will be found between two spherical shells whose radii are \(r\) and \(r+\Delta r,(a)\) if \(r=0.500 a\) and \(\Delta r=0.010 a\) and (b) if \(r=1.00 a\) and \(\Delta r=0.01 a\), where \(a\) is the Bohr radius? (Hint: \(\Delta r\) is small enough to permit the radial probability density to be taken to be constant between \(r\) and \(r+\Delta r .)\)
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