An electron confined to a one-dimensional box has a...

An electron confined to a one-dimensional box has a ground-state energy of $40.0 \mathrm{eV} .$ (a) If the electron makes a transition from its first excited state to the ground state, what is the wavelength of the emitted photon? (b) If the box were somehow made twice as long, how would the photon's energy change for the same transition (first excited state to ground state)?

Key Concepts

Dependence of Energy on Box Length

The energy levels in a confined system are inversely proportional to the square of the confinement length (L²). Therefore, if the size of the box is modified, the energies of the quantum states change accordingly. For instance, doubling the box length reduces the energy levels by a factor of four, and this scaling directly affects the energy difference between states during a transition.

Photon Emission and Transition Energy

When a quantum system transitions from a higher energy level to a lower energy level, it emits a photon whose energy equals the difference between the two quantized states. The relationship between the photon energy and its wavelength is given by the equation E = hc/?, where h is Planck's constant and c is the speed of light. This principle is fundamental to understanding spectroscopic transitions.

Particle in a One-Dimensional Box

This concept describes a fundamental quantum mechanical system in which a particle, such as an electron, is confined to move within a finite region of space bounded by infinitely high potential barriers. The confinement leads to discretized energy levels, as the particle's wavefunction must satisfy specific boundary conditions at the walls of the box. This model serves as an idealized system that illustrates the quantization of energy due to spatial confinement.

Quantized Energy Levels

In the one-dimensional box model, the allowed energy states are quantized and depend on an integer quantum number n. The energies are given by a formula proportional to n² and inversely proportional to the square of the box length (L²). This means that each energy level is distinct and the difference in energy between successive levels increases with n.

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