Question

A photon is emitted when an electron in a threedimensional cubical box of side length 8.00 $\times$ 10${}^{-11}$ m makes a transition from the n${}_X$ = 2, n${}_Y$ = 2, n${}_Z$ = 1 state to the n${}_X$ = 1, n${}_Y$ = 1, n${}_Z$ = 1 state. What is the wavelength of this photon?

Short answer

The wavelength of the emitted photon is approximately 3.63 pm.

Step-by-step solution

Understand the Energy Levels in Quantum Box

The energy levels of an electron in a cubical box are determined by the quantum numbers $n_X$, $n_Y$, and $n_Z$ using:$$E=\frac{h^2}{8m{L}^2}(n_X^2+n_Y^2+n_Z^2)$$where $h$ is Planck's constant, $m$ is the electron's mass, and $L$ is the side length of the box.

Calculate Initial and Final Energy Levels

For the initial state $(n_X=2,n_Y=2,n_Z=1)$, the energy $E_i$ is calculated as:$$E_i=\frac{h^2}{8m{L}^2}(2^2+2^2+1^2)=\frac{h^2}{8m{L}^2}\times 9$$For the final state $(n_X=1,n_Y=1,n_Z=1)$, the energy $E_f$ is:$$E_f=\frac{h^2}{8m{L}^2}(1^2+1^2+1^2)=\frac{h^2}{8m{L}^2}\times 3$$

Determine the Change in Energy

The change in energy when the electron transitions is $\mathrm{\Delta }E=E_i-E_f$:$$\mathrm{\Delta }E=\frac{h^2}{8m{L}^2}\times (9-3)=\frac{6h^2}{8m{L}^2}=\frac{3h^2}{4m{L}^2}$$

Find the Wavelength of the Photon

The energy of the photon emitted is related to its wavelength by the equation $\mathrm{\Delta }E=\frac{hc}{\lambda }$. Solve for $\lambda$:$$\lambda =\frac{hc}{\mathrm{\Delta }E}=\frac{hc\times 4m{L}^2}{3h^2}=\frac{4mc{L}^2}{3h}$$Plug in the known values, $m=9.11\times {10}^{-31}$ kg, $L=8.00\times {10}^{-11}$ m, $c=3.00\times {10}^8$ m/s, and $h=6.626\times {10}^{-34}$ Js to find $\lambda$.

Calculate Numerical Value for Wavelength

Substitute the given values into the equation for $\lambda$:$$\lambda =\frac{4\times 9.11\times {10}^{-31}\times 3.00\times {10}^8\times (8.00\times {10}^{-11}{)}^2}{3\times 6.626\times {10}^{-34}}$$Perform the calculations to find $\lambda$ which will give the wavelength of the emitted photon.

Key concepts

Photon Emission

When an electron transitions between energy levels, it can emit a photon. A photon is a particle of light, carrying energy that correlates to the change in the electron's state. This process of photon emission is fundamental in quantum mechanics because it describes how atoms interact with light. In the given scenario, the electron moves from a higher energy state to a lower one, causing the emission of a photon with a specific energy and wavelength.

• The energy of the emitted photon is equal to the difference in energy between the two levels.
• The wavelength of the photon can be determined using the energy-wavelength relationship.

This principle is crucial because it explains phenomena such as the color of the light emitted by LEDs or atoms in gas tubes. It also underpins technologies like lasers and fluorescent lights.

Quantum Numbers

Quantum numbers are integers that describe the quantized states of particles, like electrons in an atom or a box. These numbers are fundamental to understanding the behavior of quantum systems. In a three-dimensional box, three quantum numbers $n_X$, $n_Y$, and $n_Z$ are used to describe the state of an electron.

• Each quantum number represents the state along one dimension of the box.
• The numbers determine the energy levels of the electron based on their values.

These quantum numbers are essential for solving problems in quantum mechanics, as they provide a way to quantify the complex, probabilistic nature of electron behavior. Understanding them helps explain why electrons occupy discrete energy levels, leading to the emission or absorption of photons during transitions.

Energy Transition

An energy transition occurs when an electron moves between different energy states. The change in energy $\mathrm{\Delta }E$ is matched by the energy of an emitted or absorbed photon. This phenomenon is a cornerstone of quantum mechanics, supporting the concept of quantization.

• Electrons lose energy when they fall to a lower energy level, emitting a photon.
• The energy difference between initial and final states determines the photon's characteristics.

In the context of a particle in a box, transitions depend on the initial and final quantum numbers. The energy levels in a box are determined by the sum of the squares of these quantum numbers, resulting in specific energy differences and corresponding photon wavelengths.

Knowing how these transitions occur is crucial for fields such as spectroscopy, which uses photon emissions and absorptions to study materials and their properties.

Particle in a Box

The particle in a box model is a fundamental concept in quantum mechanics, used to simplify and understand the behavior of particles confined to a small space. It provides an essential framework for studying quantum systems.

• In this model, an electron behaves as a particle trapped within an infinitely high potential box.
• The energy levels are quantized and depend on the size of the box and the quantum numbers.

The sides of the cube, denoted by the length $L$, determine the spacing of the energy levels. Using this model makes it easier to calculate the properties of particles, including their energy and movement between levels.
This concept not only applies to electrons in atoms but also has implications for understanding nanotechnology and semiconductor physics, where the confinement of particles results in distinct quantum behaviors.