Question

In your research on new solid-state devices, you are studying a solid-state structure that can be modeled accurately as an electron in a one-dimensional infinite potential well (box) of width \(L\). In one of your experiments, electromagnetic radiation is absorbed in transitions in which the initial state is the \(n\) = 1 ground state. You measure that light of frequency \(f\) = 9.0 \(\times\) 10\(^{14}\) Hz is absorbed and that the next higher absorbed frequency is 16.9 \(\times\) 10\(^{14}\) Hz. (a) What is quantum number \(n\) for the final state in each of the transitions that leads to the absorption of photons of these frequencies? (b) What is the width \(L\) of the potential well? (c) What is the longest wavelength in air of light that can be absorbed by an electron if it is initially in the \(n\) = 1 state?

Short answer

(a) \( n = 2 \) and \( n = 3 \) for the transitions; (b) Solve for \( L \) using energy formulas; (c) Find longest wavelength using \( \lambda = \frac{c}{f_1} \).

Step-by-step solution

Understanding the Energy Levels

In a one-dimensional infinite potential well, the energy levels are given by the equation: \( E_n = \frac{n^2 h^2}{8mL^2} \), where \( n \) is the quantum number, \( h \) is Planck's constant, \( m \) is the electron mass, and \( L \) is the width of the well.

Relate Frequency to Energy Transition

When electromagnetic radiation is absorbed, the energy corresponding to the frequency \( f \) is \( E = hf \). Using the energy level formula, for a transition from \( n = 1 \) to \( n = n_f \), the absorbed energy is \( E = hf = \frac{(n_f^2 - 1^2)h^2}{8mL^2} \).

Determine Quantum Numbers for Each Transition

For the first frequency, \( f_1 = 9.0 \times 10^{14} \text{ Hz} \), we solve \( hf_1 = \frac{(n_{f1}^2 - 1)h^2}{8mL^2} \). For the second frequency, \( f_2 = 16.9 \times 10^{14} \text{ Hz} \), we use a similar equation. This forms two equations that we solve simultaneously:1. \( hf_1 = \frac{(n_{f1}^2 - 1)h^2}{8mL^2} \)2. \( hf_2 = \frac{(n_{f2}^2 - 1)h^2}{8mL^2} \).By dividing equation 1 by equation 2, many terms cancel, providing a simpler expression for the ratio of frequencies.

Calculate Quantum Numbers

From dividing the expressions, we get \( \frac{n_{f2}^2 - 1}{n_{f1}^2 - 1} = \frac{f_2}{f_1} \). Plugging in the given \( f_1 = 9.0 \times 10^{14} \text{ Hz} \) and \( f_2 = 16.9 \times 10^{14} \text{ Hz} \), solve the equation to find potential combinations of quantum numbers. We find, \( n_{f1} = 2 \) and \( n_{f2} = 3 \).

Solve for the Well Width \( L \)

Using the result from one of the transitions (e.g., \( n = 1 \) to \( n = 2 \)), we substitute values into \( hf_1 = \frac{(n_{f1}^2 - 1)h^2}{8mL^2} \) to solve for \( L \). This will require rearranging to find \( L = \sqrt{\frac{h}{8mf_1}} \). Using values of \( h \) and \( m \), calculate \( L \).

Determine Longest Wavelength Absorbed

The longest wavelength corresponds to the smallest energy transaction, which for the transition from \( n = 1 \) to \( n = 2 \), we use \( \lambda = \frac{c}{f_1} \) where \( c \) is the speed of light. This gives the longest wavelength of light that can be absorbed for this transition.

Key concepts

Potential Well

In quantum mechanics, a potential well is a concept where a particle, such as an electron, is confined to a region in space called the well. This is important because it dictates where an electron is most likely to be found. In the context of a one-dimensional infinite potential well, which is an idealized version, the walls of the well are considered to be infinitely high. This means the particle cannot escape the well and is confined to a specific region. Understanding the behavior of electrons in potential wells is crucial for studying quantum systems.

• The well is characterized by its width, denoted as \( L \).
• Electrons within the well exhibit behavior governed by quantum mechanics.
• Electrons have specific, quantized energy levels.

Recognizing how potential wells function underpins the understanding of a variety of quantum phenomena, including electron confinement in semiconductors and quantum dots.

Quantum Numbers

Quantum numbers are essential for describing the quantum state of a particle. They arise because electrons confined in a potential well can only occupy certain energy levels. Each energy level corresponds to a unique set of quantum numbers. In an infinite potential well, the most important quantum number is \( n \), which is a positive integer indicating the energy level the electron occupies.

• \( n = 1 \) represents the ground state or the lowest energy level.
• Higher values of \( n \) indicate excited states with more energy.
• The energy of an electron depends directly on its quantum number squared, \( n^2 \).

Quantum numbers allow us to predict the location and energy of electrons around an atom, which is foundational for quantum mechanics.

Energy Levels

Energy levels within a potential well are specific values of energy that an electron can have. These levels are quantized, meaning they can only take on certain discrete values. The energy of an electron in a one-dimensional potential well is given by the formula \( E_n = \frac{n^2 h^2}{8mL^2} \), where \( n \) is the quantum number, \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( L \) is the width of the well.
Important aspects include:

• The ground state corresponds to \( n = 1 \) with the lowest possible energy.
• Excited states, where \( n > 1 \), have higher energy.
• The energy difference between levels is key to understanding electron transitions.

This quantization is integral to explaining phenomena such as electron transitions that result in photon absorption or emission.

Photon Absorption

Photon absorption occurs when an electron absorbs a photon, gaining energy and jumping to a higher energy level. This requires the energy of the photon to match the energy difference between two quantized levels of the electron. For the given potential well, the absorbed photon energy can be calculated using \( E = hf \), where \( E \) is the energy, \( h \) is Planck's constant, and \( f \) is the frequency of the absorbed light.

• Photon absorption triggers transitions between energy levels \( n \).
• Each transition corresponds to a particular photon frequency.
• For example, transitions from \( n = 1 \) to \( n = 2 \) and \( n = 1 \) to \( n = 3 \) can be observed.

Understanding photon absorption is critical for exploring how materials interact with light, influencing technologies such as solar cells and photodetectors.

Wavelength Calculation

Wavelength calculation in the context of a potential well involves finding the wavelength of light absorbed when an electron transitions between different energy levels. The wavelength \( \lambda \) can be calculated using the equation \( \lambda = \frac{c}{f} \), where \( c \) is the speed of light and \( f \) is the frequency of the absorbed light.

• Each electron transition corresponds to specific wavelength absorption.
• Longer wavelengths are associated with transitions involving smaller energy changes.
• The longest wavelength absorbed corresponds to the transition from \( n = 1 \) to \( n = 2 \).

Calculating wavelengths is crucial for understanding spectral properties of materials and enables applications in spectroscopy and optical filters.