Question

Think about what happens to wave functions for a particle in a square infinite potential well as the quantum number \(n\) approaches infinity. Does the probability distribution in that limit obey the correspondence principle? Explain.

Short answer

Answer: As the quantum number n approaches infinity in a square infinite potential well, the wave function rapidly oscillates, causing the probability density function to concentrate around the classical locations of the particle. This behavior is consistent with the correspondence principle, as the quantum description of the system starts to resemble the classical description when the quantum numbers become very large.

Step-by-step solution

1. Definition of the wave function for a square infinite potential well

The wave function for a particle in a square infinite potential well of width L is given by the following function: \(\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)\) Where \(n\) is the quantum number.

2. Calculate the probability density

To find the probability density, we square the wave function of the particle: \(P_n(x) = |\psi_n(x)|^2 = \frac{2}{L}\sin^2\left(\frac{n\pi x}{L}\right)\) The probability density provides information about the likelihood of finding the particle at a specific location in the well.

3. Analyze the behavior as n approaches infinity

As the quantum number n approaches infinity, the term inside the sine function \(\left(\frac{n\pi x}{L}\right)\) becomes very large. Consequently, the sine function oscillates more and more rapidly between +1 and -1 as \(n\) increases. When \(n\) is large, these oscillations lead to a very rapid change in the value of the probability density function. This rapid change results in a density function that is concentrated around the classical locations of the particle (i.e., locations where the particle has a greater probability of being found).

4. Check correspondence principle

The correspondence principle states that the quantum description of a physical system should resemble the classical description as the quantum numbers become very large. In the case of a particle in a square infinite potential well, as n approaches infinity, the probability density function concentrates around the classical locations of the particle. Thus, the behavior of the wave function and its probability distribution in this limit obeys the correspondence principle as the quantum system resembles the classical description when the quantum number \(n\) approaches infinity.

Key concepts

Wave Function

The concept of a wave function is central to quantum mechanics. It is a mathematical description of the quantum state of a system. In simple terms, it can be likened to a set of instructions that details all the possibilities of where a particle might be and with what properties. For example, the wave function for a particle in an infinite potential well is given by
\[\begin{equation}\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)\end{equation}\]
where \(L\) is the width of the well and \(n\) is a positive integer known as the quantum number, which describes the energy level of the particle. The sine function shows that the particle's presence varies sinusoidally within the well and also reflects the wave-like nature of the particle, as postulated by quantum theory.

Probability Density

Probability density is the likelihood of finding a particle in a certain position within its range. To determine this, we square the absolute value of the wave function, creating a probability density function:
\[\begin{equation}\left|\psi_n(x)\right|^2 = \frac{2}{L}\sin^2\left(\frac{n\pi x}{L}\right)\end{equation}\]
For a particle in an infinite potential well, this becomes particularly significant. As the quantum number increases, the probability density shows more prominent oscillations between positions, essentially outlining where the particle is most likely to be found. These peaks and troughs in the probability density give us an in-depth picture of the particle's behavior within the well.

Infinite Potential Well

An infinite potential well is a theoretical construct used to simplify the daunting complexities of quantum systems. It's as if we confine a particle within impenetrably high walls, having zero potential energy inside and infinite potential energy outside. In practice, the particle can't be found outside the well because it cannot have infinite energy.
The beauty of this model lies in its boundary conditions. They force the wave function and therefore the particle to have a discrete set of energy levels. This restriction helps students grasp how quantum mechanics differs fundamentally from classical mechanics—where energy can vary continuously. Within an infinite potential well, the particle's wave function takes the form of standing waves, beautifully illustrating the quantization of energy levels in quantum systems.

Correspondence Principle

The correspondence principle is an important bridge between quantum mechanics and classical physics. It dictates that the behavior of quantum systems should converge with classical expectations as the quantum numbers become very large. In plain language, when you look at large scales or high energies—the macroscopic world—you should recover the familiar laws of classical physics.
In the context of an infinite potential well, the principle manifests when the quantum number \(n\) is large. Here, the probability density starts mimicking a classical particle's behavior. The particle in quantum mechanics, with its wave functions and probabilistic nature, begins to resemble a classical particle, bounded by the laws of Newtonian physics. This reconciliation is integral to our understanding of how quantum mechanics evolves into classical mechanics and ensures consistency across the scales of physics.