(a) Find the excitation energy from the ground level to the third excited level for an electron confined to a box of width $0.360$ nm.
(b) The electron makes a transition from the $n=1$ to $n=4$ level by absorbing a photon. Calculate the wave­length of this photon.

Verified step by step guidance

1. Step 1: Understand the energy levels of an electron in a one-dimensional box. The energy levels are given by the formula: $E=\frac{n^2}{h_2}8m{L}^2$, where $n$ is the quantum number, $h_2$ is Planck's constant squared, $m$ is the mass of the electron, and $L$ is the width of the box. For part (a), calculate the energy difference between the ground state ($n=1$) and the third excited state ($n=4$).
2. Step 2: Write the expression for the excitation energy: $\mathrm{\Delta E}=E{n}_4-E{n}_1$. Substitute the energy formula for each level: $\mathrm{\Delta E}=\frac{4^2}{h_2}8m{L}^2-\frac{1^2}{h_2}8m{L}^2$. Simplify the expression to find $\mathrm{\Delta E}$.
3. Step 3: For part (b), use the relationship between the energy of a photon and its wavelength: $E=\frac{\mathrm{hc}}{\lambda }$, where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength of the photon. The energy of the photon corresponds to the energy difference between the $n=4$ and $n=1$ levels.
4. Step 4: Substitute the energy difference $\mathrm{\Delta E}$ from part (a) into the photon energy formula to solve for the wavelength: $\lambda =\frac{\mathrm{hc}}{\mathrm{\Delta E}}$. Ensure all constants are in consistent units (e.g., Planck's constant in Joule-seconds, speed of light in meters per second, and energy in Joules).
5. Step 5: Perform the necessary unit conversions for the box width (from nanometers to meters) and calculate the numerical values for the energy levels, energy difference, and wavelength. This will give you the wavelength of the photon absorbed during the transition.

Key Concepts

Quantum Mechanics and Particle in a Box

In quantum mechanics, a particle confined in a one-dimensional box exhibits quantized energy levels. The energy levels are determined by the width of the box and the mass of the particle. For an electron in a box, the energy levels can be calculated using the formula E_n = (n^2 * h^2) / (8 * m * L^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 box.

Energy Transitions and Photon Absorption

When an electron transitions between energy levels, it can absorb or emit a photon. The energy of the photon corresponds to the difference in energy between the two levels, given by ΔE = E_final - E_initial. This relationship is crucial for calculating the wavelength of the absorbed photon using the equation E = hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength.

Wavelength Calculation

The wavelength of a photon can be calculated from its energy using the equation λ = hc/E. This formula shows the inverse relationship between energy and wavelength: as the energy of the photon increases, its wavelength decreases. In the context of electron transitions, this calculation allows us to determine the specific wavelength of light absorbed when an electron moves from a lower to a higher energy level.