An electron in a one­-dimensional box has ground ­state energy $2.00$ eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?

Verified step by step guidance

1. Step 1: Understand the problem. The electron is in a one-dimensional box, and its energy levels are quantized. The energy levels are given by the formula: $E_n=\frac{n²h^2}{8m{L}^2}$, where $n$ is the quantum number, $h$ is Planck's constant, $m$ is the mass of the electron, and $L$ is the length of the box.
2. Step 2: Relate the given ground-state energy to the quantum number $n$. The ground-state corresponds to $n=1$, and its energy is given as 2.00 eV. Use this information to determine the constant factor in the energy formula: $E_1=\frac{h^2}{8m{L}^2}$. This constant will help calculate higher energy levels.
3. Step 3: Calculate the energy of the second excited state. The second excited state corresponds to $n=3$. Using the formula for energy levels, substitute $n=3$ into the equation: $E_3=\frac{9h^2}{8m{L}^2}$. Compare this energy to the ground-state energy to find the energy difference.
4. Step 4: Determine the energy of the photon absorbed during the transition. The energy difference between the second excited state ($n=3$) and the ground state ($n=1$) corresponds to the energy of the absorbed photon: $E=E_3-E_1$. Use the values calculated in previous steps to find this energy difference.
5. Step 5: Relate the energy of the photon to its wavelength using the formula: $E=\frac{\mathrm{hc}}{\lambda }$, where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength. Rearrange the formula to solve for $\lambda$: $\lambda =\frac{\mathrm{hc}}{E}$. Substitute the energy of the photon and known constants to find the wavelength.

Key Concepts

Quantum Mechanics and Energy Levels

In quantum mechanics, particles such as electrons occupy discrete energy levels within a potential well, like a one-dimensional box. The energy levels are quantized, meaning that an electron can only exist in specific states with defined energies. The ground state is the lowest energy level, while excited states are higher energy levels. The difference in energy between these levels determines the energy of photons absorbed or emitted during transitions.

Photon Energy and Wavelength Relationship

The energy of a photon is directly related to its wavelength through the equation E = hc/λ, where E is the energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. When an electron transitions between energy levels, it absorbs or emits a photon with energy equal to the difference between the two levels. Thus, knowing the energy change allows us to calculate the corresponding wavelength of the absorbed photon.

Energy Level Calculation in a Particle in a Box

For a particle in a one-dimensional box, the energy levels are given by the formula E_n = n²h²/(8mL²), where n is the principal quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box. The ground state corresponds to n=1, and the second excited state corresponds to n=3. By calculating the energy for these states, we can find the energy difference that corresponds to the photon absorbed during the transition.