Question

Exercises \(94-97\) refer to a bound particle of mass \(m\) described by the wave function $$ \psi(x)=A x e^{-x^{2} / 2 b^{2}} $$ For a total energy of 0 , the potential energy is given in Exercise 96. (a) Given these, to what region of the x-axis would a classical particle be restricted? Is the quantum-mechanical particle similarly restricted? (b) Write an expression for the probability that the (quantum-mechanical) particle would be found in the classically forbidden region, leaving it in the form of an integral. (The integral cannot be evaluated in closed form.)

Short answer

The classical particle would be restricted to the area where the potential energy is equal to or lesser than the total energy. The quantum-mechanical particle isn't subjected to the same restriction due to a phenomenon known as quantum tunneling. The probability of locating this quantum-mechanical particle in a classically restricted area is represented by the integral \(P = \int_{-\infty}^{a}\left| \psi(x) \right|^2 dx + \int_{b}^{\infty}\left| \psi(x) \right|^2 dx\).

Step-by-step solution

Analyzing Energy and Particle Placement

Given the wave function \(\psi(x)=A x e^{-x^{2} / 2 b^{2}}\), it's stated that the total energy is 0. This means kinetic energy and potential energy are equal. The positions where the potential energy equals the total energy determine the boundaries for a classical particle. This is because in classical mechanics a particle can't have more potential energy than total energy.

Differing Particle Placement in Quantum Mechanics

Due to the quantum-mechanical nature of the issue, unlike classical particles, quantum-mechanical particles aren't necessarily restricted to the same boundaries. Quantum mechanics allows for particles to be found in classically restricted regions due to a quantum effect called quantum tunneling.

Probability in Classically Forbidden Region

To write an expression for the probability of finding the quantum particle in the classically forbidden region, we would need to consider the areas where the potential energy exceeds the total energy. The probability can be calculated as the integral of the wave function squared over the classically forbidden region. So, let \(a\) and \(b\) be the boundaries of the classical region, then the probability of finding the quantum-mechanical particle in the classically forbidden region is: \(P = \int_{-\infty}^{a}\left| \psi(x) \right|^2 dx + \int_{b}^{\infty}\left| \psi(x) \right|^2 dx\).

Key concepts

Wave Function

In quantum mechanics, the wave function is a fundamental concept that describes the quantum state of a particle. It provides valuable information about a particle, such as its position, momentum, and energy. The wave function, denoted as \( \psi(x) \), is a mathematical function that, when squared \( |\psi(x)|^2 \), gives the probability distribution of finding a particle at different positions in space. It is important to understand that the wave function itself does not represent a physical observable; rather, it is a tool for calculating probabilities.

For example, consider the wave function from the exercise \( \psi(x)=A x e^{-x^{2} / 2 b^{2}} \). This particular form is not just a set of numbers but encapsulates information about the particle's behavior in space. By examining this wave function, one can deduce the likelihood of locating the particle at various points along the x-axis.

Quantum Tunneling

Quantum tunneling is a phenomenon unique to the realm of quantum mechanics, where particles have the ability to pass through potential barriers that they seemingly should not be able to, given their energy. Imagine a ball rolling up a hill; in classical physics, if the ball lacks the energy to reach the top, it cannot go over to the other side. Quantum particles, however, can 'tunnel' through the barrier, even if their total energy is less than the potential energy of the barrier.

This counterintuitive behavior results from the wave-like properties of particles at the quantum scale, where the wave function representing the particle extends into regions that would be forbidden in classical mechanics. As a result, there is a nonzero probability of the particle being found on the other side of the barrier. Quantum tunneling has real-world applications in devices such as tunnel diodes and the scanning tunneling microscope.

Classically Forbidden Region

The classically forbidden region in quantum mechanics refers to the area where a particle does not have sufficient energy to be, according to classical physics laws. In the context of the given exercise, it is the range of positions wherein the potential energy of the particle exceeds the total energy. Classically, a particle with a total energy of 0 would be confined to regions where its potential energy is also 0 or negative (where allowed to exist).

However, quantum physics allows the wave function to extend into the classically forbidden region, indicating a finite probability for the existence of a particle there, albeit usually low. Thus, while a classical particle would be strictly confined, quantum mechanics allows for the intriguing possibility that the particle might be detected in these unlikely regions due to the probabilistic nature of quantum entities.

Probability Integral

The probability integral in quantum mechanics is a mathematical expression used to quantify the likelihood of a particle's presence within a specified region. Given a wave function, the probability of finding a particle between two points in space is determined by integrating the square of the absolute value of the wave function within those points. This integral essentially sums up the probabilities across a continuum of positions.

In the classically forbidden region, the probability integral takes on special significance as it measures the probability of an event that defies classical expectations—finding the particle where it 'shouldn't' be. This integral, as stated in the exercise, may not always be solvable with simple functions, reflecting the complexity and the non-intuitive nature of quantum phenomena.

Potential Energy

Potential energy in quantum mechanics, much like in classical mechanics, represents the energy stored by a particle due to its position in a field. It is one component of the total energy, the other being kinetic energy. In the quantum context, potential energy influences the shape and behavior of the wave function. Higher potential energy often corresponds to regions where the wave function's amplitude is lower, indicating a lower probability of finding a particle there.

In a given potential energy landscape, such as the one described in the exercise, areas where the potential energy equals or exceeds the total energy become pivotal in understanding where a particle is likely, or unlikely, to be found. The interplay between potential energy and the wave function dictates much of a particle's quantum behavior and is essential for predicting outcomes in quantum systems.

Total Energy

Total energy in a quantum system includes all forms of energy possessed by a particle, typically kinetic plus potential energy. In quantum mechanics, while energy is conserved, the distribution between kinetic and potential energy can vary according to the wave function and the potential landscape. The total energy of a system is intimately connected to the wave function and, by extension, to all quantum mechanical properties, including where a particle can and cannot be found.

As highlighted in the exercise, when the total energy is set to 0, it sets a benchmark to identify classically allowed and forbidden regions. Since the potential energy is given, one can delineate these regions based on the condition that the potential energy cannot be greater than the total energy in classical scenarios. In quantum scenarios, however, the total energy is still a critical indicator of the particle's overall behavior, despite the possibility of tunneling through barriers.