An electron confined in a one-dimensional box emits a 200 nm photon in a quantum jump from n = 2 to n = 1. What is the length of the box?

Verified step by step guidance

1. Step 1: Understand the problem. The electron is confined in a one-dimensional box, which means its energy levels are quantized according to the particle-in-a-box model. The problem involves a quantum jump from n=2 to n=1, emitting a photon of wavelength 200 nm. We need to find the length of the box.
2. Step 2: Recall the energy levels for a particle in a one-dimensional box. The energy of the nth level is given by the formula: $E_n=\frac{n^2{h}^2}{8m{L}^2}$, 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.
3. Step 3: Calculate the energy difference between the two levels (ΔE). The energy difference is given by: $\mathrm{\Delta E}=E_2-E_1$. Substitute the formula for energy levels into this expression: $\mathrm{\Delta E}=\frac{2^2{h}^2}{8m{L}^2}-\frac{1^2{h}^2}{8m{L}^2}$.
4. Step 4: Relate the energy difference to the emitted photon. The energy of the photon is given by: $E=\frac{\mathrm{hc}}{\lambda }$, where h is Planck's constant, c is the speed of light, and λ is the wavelength of the photon (200 nm). Set this equal to ΔE: $\frac{\mathrm{hc}}{\lambda }=\mathrm{\Delta E}$. Substitute the expression for ΔE from Step 3.
5. Step 5: Solve for the length of the box (L). Rearrange the equation to isolate L: $L=\sqrt{\frac{8m\mathrm{hc}\lambda }{h^2(2^2-1^2)}}$. Substitute the known values for h, c, m (mass of the electron), and λ (200 nm) to calculate L.

Key Concepts

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. It introduces concepts such as quantization, where certain properties, like energy, can only take on discrete values. This framework is essential for understanding phenomena like electron transitions in atoms and the emission of photons.

Particle in a Box Model

The particle in a box model is a fundamental quantum mechanics concept that describes a particle confined to a one-dimensional potential well. The energy levels of the particle are quantized, and the length of the box determines these levels. This model helps in calculating the energy associated with different quantum states, which is crucial for solving the given problem.

Wavelength and Energy Relationship

The relationship between the wavelength of a photon and its energy is described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. This relationship allows us to calculate the energy difference between quantum states based on the emitted photon's wavelength, which is key to determining the length of the box in the problem.