Using the quantum particle in a box model, describe how the possible energies of the particle are related to the size of the box.

Short Answer

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The possible energies of a quantum particle in a box are inversely proportional to the square of the box's size (length) \(L\), as given by the equation \(E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}\), where \(\hbar\) is the reduced Planck constant, \(m\) is the mass of the particle, and \(n\) is the quantum number. This means that when the size of the box increases, the energy levels decrease and vice versa. The energy levels are also proportional to the square of the quantum number \(n\), which indicates the different quantized energy levels within the box, such that higher \(n\) values correspond to higher energy levels.

Step by step solution

01

Understanding the Concept of Quantum Particle in a Box

The quantum particle in a box model is a simplified one-dimensional model to describe the behavior of a particle in a confined space. The model assumes an infinitely high potential at the walls of the box, meaning the particle cannot escape the box. Inside the box, the particle behaves as a quantum mechanical wave with quantized energy levels based on its wave function.
02

Write down the Wave Function for a Particle in a Box

The wave function, which describes the state of the particle, can be written as: \[ \Psi_n(x)=\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}} \] Where \(n\) is the quantum number (an integer), \(x\) is the position along the box, and \(L\) is the length of the box.
03

Calculate the Energy Levels of the Particle

The possible energies \(E_n\) of a quantum particle in a box can be determined using the Schrödinger equation, which gives: \[ E_n=\frac{n^2\pi^2\hbar^2}{2mL^2} \] Where \(\hbar\) is the reduced Planck constant, \(m\) is the mass of the particle, and \(n\) is the quantum number.
04

Discuss the Relationship between Energy Levels and the Size of the Box

From the energy level equation, we can see that the possible energies of a quantum particle in a box are inversely proportional to the square of the box's size (length) \(L\). This means that as the size of the box increases, the energy levels decrease, and vice versa. Additionally, the energy levels are proportional to the square of the quantum number \(n\), which represents the different quantized energy levels within the box. As \(n\) increases, the energy levels also increase.

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