Problem 16 - University Physics with Modern Physics

Question

An electron in a one-dimensional box has ground-state energy 1.00 eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?

Step-by-Step Solution

Answer

The wavelength of the absorbed photon is approximately 155 nm.

Step 1: Understanding the Energy Levels

In a one-dimensional box, the energy levels of an electron are given by the formula \( E_n = \frac{n^2h^2}{8mL^2} \), where \( n \) is the energy level, \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( L \) is the length of the box. The ground state (\( n=1 \)) energy is given as 1.00 eV.

Step 2: Energy of the Second Excited State

For the second excited state, \( n=3 \). Thus, the energy of the second excited state \( E_3 = \frac{3^2}{1^2} \times 1.00 \, \text{eV} = \frac{9}{1} \times 1.00 \, \text{eV} = 9.00 \, \text{eV} \).

Step 3: Calculate the Energy Difference

The electron absorbs a photon to make a transition from the ground state to the second excited state. The energy of the absorbed photon is the difference between the energy of the second excited state and the ground state: \( \Delta E = E_3 - E_1 = 9.00 \, \text{eV} - 1.00 \, \text{eV} = 8.00 \, \text{eV} \).

Step 4: Convert Energy Difference to Wavelength

The energy of a photon is related to its wavelength by the equation \( \Delta E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength. Rearranging for \( \lambda \), we get \( \lambda = \frac{hc}{\Delta E} \). Substituting \( h = 4.135667696 \times 10^{-15} \, \text{eV}\cdot\text{s} \), \( c = 2.998 \times 10^8 \, \text{m/s} \), and \( \Delta E = 8.00 \, \text{eV} \), \( \lambda = \frac{4.135667696 \times 10^{-15} \times 2.998 \times 10^8}{8.00} \approx 1.55 \times 10^{-7} \, \text{m} = 155 \, \text{nm} \).

Key Concepts

Particle in a Box

The concept of a "particle in a box" is a fundamental example in quantum mechanics often used to illustrate quantum behaviors. Imagine an electron trapped in a one-dimensional box with perfectly rigid walls where it can't escape. The electron's movements are confined to this box, making them quantized, meaning the electron can only occupy specific energy levels. These energy levels are not continuous but discrete. The box represents an idealized system used to explore how particles behave at a quantum scale, reflecting principles such as uncertainty and quantization of energy. In simple terms, it's like a sandbox for quantum particles, where the rules of classical physics do not apply.

Energy Levels

In the context of a one-dimensional box, each energy level of a quantum particle can be described using the formula \( E_n = \frac{n^2h^2}{8mL^2} \). Here, \( n \) represents the quantum number of an energy level: 1 for the ground state, 2 for the first excited state, and so on.
This formula tells us that energy levels increase with increasing \( n \), and are inversely proportional to the square of the box length \( L \). The ground state, with \( n=1 \), is the minimum energy the particle can have. Higher energy levels (\( n=3, 4, \, etc. \)) represent states where the particle has more energy. Thus, moving from \( n=1 \) to \( n=3 \) (as the second excited state) means the particle absorbs energy, indicating it has moved to a higher energy level in this quantized system.

Photon Absorption

Photon absorption occurs when a quantum particle, like an electron, absorbs a photon's energy, allowing it to move to a higher energy level. In our case of the particle in a box, the electron in the ground state absorbs a photon to jump to the second excited state.
The energy of the absorbed photon equals the difference between the higher and lower energy levels, quantified in electron volts (eV). This transition exemplifies how energy is absorbed at specific quantized amounts, a fundamental characteristic of quantum mechanics. In short, the electron "jumps" to a new quantum state by absorbing a photon's energy.

Wavelength Calculation

The wavelength of the absorbed photon can be determined using its energy, thanks to the relationship between a photon's energy and its wavelength. The formula used here is \( \Delta E = \frac{hc}{\lambda} \), where \( \Delta E \) is the energy difference, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.
After rearranging this formula, we can solve for the wavelength: \( \lambda = \frac{hc}{\Delta E} \). Given that the energy difference \( \Delta E \) is 8.00 eV, Planck's constant \( h = 4.135667696 \times 10^{-15} \, \text{eV}\cdot\text{s} \), and the speed of light \( c = 2.998 \times 10^8 \, \text{m/s} \), you can plug in these values to find that the wavelength is approximately 155 nm. This result shows the direct relationship between a photon's energy and its wavelength, emphasizing the concept of quantization at the quantum level.