An electron confined in a one-dimensional box is observed, at different times, to have energies of 12 eV, 27 eV, and 48 eV. What is the length of the box?

Verified step by step guidance

1. Step 1: Recognize that the problem involves the quantum mechanics concept of a particle in a one-dimensional box. The energy levels of the electron are quantized and 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: Identify the quantum numbers corresponding to the given energy levels. Since the energy levels are proportional to $n²$, compare the ratios of the energies: $\frac{27}{12}$ = $2.25$ and $\frac{48}{12}$ = $4$. This indicates the quantum numbers are $n=2$, $n=3$, and $n=4$, respectively.
3. Step 3: Use the energy formula for the lowest energy level ($n=2$) to solve for the length of the box. Rearrange the formula to isolate $L$: $L=\sqrt{\frac{n²h^{²}}{8m{E}_n}}$. Substitute $n=2$, $E_2=12\mathrm{eV}$, and convert the energy to joules ($1\mathrm{eV}=1.602\times 10⁻19J$).
4. Step 4: Substitute the known constants into the formula. Use Planck's constant $h=6.626\times 10⁻34Js$ and the mass of the electron $m=9.109\times 10⁻31\mathrm{kg}$. Perform the substitution into the rearranged formula for $L$.
5. Step 5: Simplify the expression to find the length of the box. Ensure all units are consistent (e.g., joules, kilograms, meters) and calculate the square root to determine $L$. This will give the final length of the box in meters.

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 wave-particle duality and quantization of energy, which are essential for understanding how particles like electrons behave in confined spaces, such as a one-dimensional box.

Particle in a Box Model

The particle in a box model is a fundamental concept in quantum mechanics that describes a particle free to move in a small space with infinitely high potential walls. The energy levels of the particle are quantized, meaning the particle can only occupy specific energy states, which are determined by the length of the box and the mass of the particle.

Energy Quantization

Energy quantization refers to the phenomenon where a particle can only have certain discrete energy levels rather than a continuous range. For a particle in a one-dimensional box, the energy levels are given by the formula E_n = n^2 * (h^2 / (8mL^2)), where n is a quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box. This relationship is crucial for determining the length of the box based on the observed energy levels.