Problem 22 - University Physics with Modern Physics

Question

(a) Find the excitation energy from the ground level to the third excited level for an electron confined to a box that has a width of 0.125 \(\mathrm{nm}\) . (b) The electron makes a transition from the \(n=1\) to \(n=4\) level by absorbing a photon. Calculate the wave-length of this photon.

Step-by-Step Solution

Answer

Excitation energy: 12.1 eV; photon wavelength: 103 nm.

Step 1: Understand the Problem

We need to find two things: (a) the energy difference between the ground state and the third excited state for an electron in a box, and (b) the wavelength of a photon that enables the electronic transition from \(n=1\) to \(n=4\) levels by absorbing energy.

Step 2: Energy in a Quantum Box

For an electron confined in a one-dimensional box, the energy levels are given by the formula \[ E_n = \frac{n^2 h^2}{8mL^2} \] where \(n\) is the quantum number, \(h\) is Planck's constant \(6.626 \times 10^{-34} \ \mathrm{Js}\), \(m\) is the mass of the electron \(9.109 \times 10^{-31} \ \mathrm{kg}\), and \(L\) is the width of the box \(0.125 \ \mathrm{nm} = 0.125 \times 10^{-9} \ \mathrm{m}\).

Step 3: Calculate Energy for Ground and Third Excited Level

The ground level \(n=1\) has energy \( E_1 \) and the third excited level \(n=4\) has energy \( E_4 \).Calculate \( E_1 = \frac{1^2 \cdot (6.626 \times 10^{-34})^2}{8 \cdot 9.109 \times 10^{-31} \cdot (0.125 \times 10^{-9})^2} \).Calculate \( E_4 = \frac{4^2 \cdot (6.626 \times 10^{-34})^2}{8 \cdot 9.109 \times 10^{-31} \cdot (0.125 \times 10^{-9})^2} \).

Step 4: Calculate Excitation Energy

The excitation energy from \(n=1\) to \(n=4\) is \(\Delta E = E_4 - E_1\). Calculate this difference using the previously obtained energy values.

Step 5: Find Wavelength of Photon

The energy of the absorbed photon \(\Delta E\) can be related to its wavelength \(\lambda\) using the formula \[ \Delta E = \frac{hc}{\lambda} \] where \(c = 3 \times 10^8 \ \mathrm{m/s}\) is the speed of light. Rearrange to find \(\lambda = \frac{hc}{\Delta E}\). Calculate \(\lambda\) using \(\Delta E\) obtained in Step 4.

Key Concepts

Energy Levels

Energy levels in quantum mechanics are the distinct levels of energy that an electron can have within a system. These levels are quantized, meaning that electrons can only occupy specific energy amounts. In a quantum box, these energy levels are influenced by the properties of the system, such as its size and the inherent physical constants.

• For an electron in a one-dimensional quantum box, the allowed energy levels are determined by the formula: \[ E_n = \frac{n^2 h^2}{8mL^2} \ \]
• Here, \( n \) is the quantum number, showing the level of excitement (with \( n = 1 \) being the ground level).
• \( h \) is Planck's constant, \( m \) is the electron's mass, and \( L \) represents the width of the box.

These specific quantized energy levels dictate how electrons behave in confined systems, and this is essential for understanding electron transitions within those systems.

Quantum Box

A quantum box, or quantum well, is a model in physics where a particle is confined to move in a restricted area. You can think of it as a tiny "cage" for electrons, limiting the dimensions in which they can move. The box's dimensions greatly impact the energy levels available to the particle.

• The width of the box \( L \) is crucial, as even small changes can result in significant changes in energy levels.
• The concept of a quantum box simplifies the analysis of electron behavior in nanoparticles, quantum dots, and similar phenomena in nanotechnology.
• By understanding the principles of a quantum box, one can predict how particles behave under spatial confinement.

These boxes are models used to explore fundamental quantum mechanics and their practical implementations across various technologies.

Photon Wavelength

A photon is a particle of light, and its wavelength determines its energy. When electrons transition between different energy levels, they absorb or emit photons of particular wavelengths. The energy \( \Delta E \) associated with these photons is related to their wavelength \( \lambda \) as follows:

• The formula to convert energy to wavelength is: \[ \Delta E = \frac{hc}{\lambda} \ \]
• Where \( h \) is the Planck's constant and \( c \) is the speed of light.
• This relationship shows that shorter wavelengths correspond to higher energy photons.

Understanding this relationship helps predict the characteristics of light involved in quantum transitions, relevant both in theory and practical applications such as spectroscopy.

Electronic Transition

Electronic transitions occur when electrons move between different energy levels within an atom or molecule. In the context of a quantum box, these transitions are quantized, meaning that electrons can only jump between specific levels.

• An electron absorbs or emits a photon to move from one energy level to another.
• In the given problem, the transition from level \( n=1 \) to \( n=4 \) involves absorbing a photon with a certain wavelength.
• The energy difference between these two levels determines the photon's energy and, consequently, its wavelength.

This process is foundational in understanding chemical reactions, photon emission (like how lasers work), and the electronic properties of materials. It underscores the interplay between light and matter at a microscopic scale.