Question

Where would a particle in the fist excited state (first above ground) of an infinite well mostly likely be found?

Short answer

The most likely positions to find a particle in the first excited state of an infinite well are at \(x = a/4\) and \(x = 3a/4\), where \(a\) is the width of the box.

Step-by-step solution

Understand the Particle in a Box Model

In the particle in a box model, a particle is confined to a box with infinitely high walls, so that it can't escape from the box. The state of the particle is described by a wave function. This function, squared, gives the probability density, meaning the likelihood of finding the particle at any given location inside the box.

Calculate the Wave Function for the First Excited State

The wave function for a particle in a box is given by \(\Psi_n(x) = \sqrt{2/a}\sin(n\pi x/a)\), where \(a\) is the width of the box, \(n\) is the energy level (or state), and \(x\) is the position. The first excited state corresponds to \(n=2\), so the wave function for this state is \(\Psi_2(x) = \sqrt{2/a}\sin(2\pi x/a)\).

Find the Probability Density Function

The probability density function is the square of the wave function. In this case, it is \(|\Psi_2 (x)|^2 = 2/a \sin^2(2\pi x/a)\). This displays two peaks, indicating the two most likely locations to find the particle when it is in the first excited state.

Identify Most Likely Position

Examining our probability density, the most likely locations to find the particle in the first excited state are at \(x = a/4\) and \(x = 3a/4\), both within the box, where our function peaks.

Key concepts

Quantum Mechanics

Quantum mechanics is the branch of physics that studies the behavior of matter and energy on the atomic and subatomic scales. At these scales, the classical concepts of physics, which rely on deterministic laws, are replaced by probability and wave-particle duality. For example, instead of a particle having a definite position and momentum, quantum mechanics describes how there's a probability to find the particle in various locations and states of motion.

This fundamental aspect of quantum mechanics is illustrated by our particle in a box exercise. The particle is bounded within a finite region, and its behavior can't be described using classical physics. Instead, the quantum state of the particle is represented by a mathematical function known as the wave function, which encapsulates all the information about the system. The wave function leads us to a probability distribution, allowing us to predict where the particle is most likely to be found within the box.

Wave Function

The wave function is a cornerstone of quantum mechanics, representing the state of a quantum system, like our particle in a box. Mathematically, it's a complex function denoted by \(\Psi(x)\) or \(\psi(x)\) and is associated with the properties of particles, such as their position. It might seem abstract at first, but its square (specifically, the square of its magnitude, since it can be complex) contains real physical meaning—it's a probability density.

In the case of the particle in a box, the wave function changes depending on the energy state of the particle—the ground state, excited states, and so forth. A higher energy state means the wave function will have more 'waves' within the confines of the box. However, these waves aren't physical waves; instead, they describe the oscillating probability of finding the particle at a given location.

Probability Density

Probability density is a concept that quantifies how likely it is to find a particle at a specific location. It is important to note that unlike a probability, which is a number between 0 and 1, probability density can be used to calculate the probability of finding a particle within a certain spatial range by integrating over that range.

The square of the magnitude of the wave function, \( |\Psi(x)|^2 \), gives us the probability density. In the case of our particle in the first excited state of an infinite well, the probability density function has specific implications. With its two peaks in the probability density, it suggests there are two specific regions inside the box where the particle is most likely to be found. These two peaks equate to a higher probability of the particle's presence at those locations, compared to others within the box.

Excited State

An excited state refers to a quantum state where the particle has more energy than the ground state, which is the lowest possible energy state. In the analogy of the particle in a box, the ground state represents the basic, most stable form where the particle has the least amount of energy. Conversely, moving to an excited state means the particle's energy increases and usually involves absorbing energy from an external source.

When the particle is in an excited state, the wave function—and thus the probability density—changes. For our exercise, the first excited state corresponds to the second energy level of the system. At this level, we are likely to find the particle at both \(x = a/4\) and \(x = 3a/4\), rather than just in the center of the box, which would be the most expected location in the ground state. These locations signify where the oscillations of the wave function's probability density are at their highest.