An electron is confined in a one-dimensional box of length...

An electron is confined in a one-dimensional box of length $L$. Another electron is confined in a box of length $2 L$. Both are in the ground state. What is the ratio of their energies $E_{2 L} / E_{L} ?$

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Key Concepts

Particle in a Box Model

This model is a fundamental quantum mechanics problem that illustrates the behavior of a particle confined in an infinite potential well. It helps in understanding how the restrictions on a particle's position due to boundary conditions lead to quantized energy levels that are not continuous but discrete.

Energy Quantization

In quantum systems like the particle in a box, the energy of a particle is confined to certain discrete levels rather than being able to vary in a continuous range. This concept emerges due to the wave nature of particles and the requirement that the wavefunction satisfies specific boundary conditions at the walls of the box.

Dependence on Confinement Length

The energy levels of a confined particle are inversely related to the square of the length of the box. This relationship means that an increase in the size of the confining region leads to a reduction in the energy levels, emphasizing the scale-dependence of quantum systems.

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