Question

A ball is thrown straight up at \(25 \mathrm{~m} / \mathrm{s}\). Someone asks, "Ignoring air resistance, what is the probability of the ball tunneling to a height of \(1000 \mathrm{~m}\) ?" Explain why chis is not an example of tunneling as discussed in this chapter, even if the ball were replaced with a small fundamental particle. (The fact that the potential energy varies with position is not the whole answer - passing through nonrectangular barriers is still tunneling.)

Short answer

The ball cannot reach a height of 1000 meters and the scenario described doesn't align with quantum tunneling. Even if the ball was replaced with a small fundamental particle, it would still not be tunneling because the requirements for tunneling (particle's wave function and potential barrier) are not met in this case.

Step-by-step solution

Clarifying Tunneling

Quantum tunneling refers to the quantum mechanical phenomenon where a wavefunction can propagate through a potential barrier. This is a principle of quantum mechanics that states that particles can exist in multiple states simultaneously and move in ways that classical physics would deem impossible. It's a microscopic phenomenon that can only be directly observed on a very small scale, typically when dealing with elementary particles (e.g. electrons).

Determining Initial Conditions

In this case, a ball is thrown straight up at a velocity of 25 m/s. If there is no friction (air resistance), the height it would reach can be obtained using the equation of motion: \[H = V^2 / (2g)\], where \(V = 25 m/s\) is the initial velocity and \(g = 9.8 m/s^2\) is the acceleration due to gravity. Solving this will give a maximum height of approximately 31.88 meters.

Comparing With Tunneling

The question of whether this ball can reach a height of 1000 meters is not about quantum tunneling for two reasons. First, quantum tunneling applies to quantum particles, while this ball is macroscopic, composed of large amount of atoms. Second, quantum tunneling depends on both the size of the barrier (in this case, it is the energy required to reach 1000 meters) and the wave properties of the particle, which in this case does not apply as the ball neither displays wave properties nor is a quantum particle.

Concluding Remarks

So, it can be concluded that the regular upward motion of a thrown ball, even in the absence of air resistance, doesn't qualify as quantum tunneling. The conditions related to the wave-particle duality and the scale of the scenario do not match the prerequisites for quantum tunneling. Thus, even if the ball were replaced with a small fundamental particle, the situation still wouldn't amount to tunneling, as the entire setup isn't relevant to quantum mechanics.

Key concepts

Quantum Mechanics

Quantum mechanics is a pillar of modern physics that explains the behavior of particles at the nanoscopic scale. It departs from classical mechanics by describing nature in terms of probabilities rather than definitive outcomes.

At the heart of quantum mechanics is the idea that particles are not always particles, nor are waves always waves. This is because particles like electrons exhibit properties of both particles and waves, depending on the situation. These properties are described by the wave function, a mathematical formula that provides the probabilities of finding a particle in various places.

One of the most mysterious and interesting phenomena predicted by quantum mechanics is quantum tunneling. In this process, particles have a finite probability of crossing through a potential energy barrier they classically shouldn't surmount. This manifests because of the particle's wave-like nature, allowing it to exist in multiple states and traverse regions that would be off-limits in a classical world.

Wave-Particle Duality

Wave-particle duality is a fundamental concept in quantum mechanics, which tells us that every particle or quantic entity may be partly described in terms not only of particles, but also of waves. This concept is key to understanding phenomena like interference and diffraction, which are wave behaviors, as well as collisions and emissions, which are particle behaviors.

It was French physicist Louis de Broglie who proposed that just as light has wave and particle characteristics, electrons and other particles also have wave properties. This wave nature is significant because it describes the probability of locating a particle in a certain position in space. It's the wave aspect of particles that allows for the tunneling effect, where a particle penetrates a barrier without having the energy to overcome it in classical terms.

Understanding the dual nature of matter is crucial when solving problems in quantum mechanics, as it guides the application of the appropriate models and equations, such as the Schrödinger equation, which is used to determine the wave function of a particle.

Potential Energy Barrier

In quantum mechanics, a potential energy barrier is often visualized as a hill or a wall that a particle encounters. According to the laws of classical physics, a particle must have enough energy to surmount the barrier; if it doesn't, it will be reflected back. However, quantum mechanics introduces a subtler and far less intuitive aspect.

In the quantum world, particles behave in probabilistic ways, allowing for the possibility of tunneling through a barrier without having the classical energy required. The height and width of the potential energy barrier are key factors determining the likelihood of a particle tunneling through it. The probability of tunneling decreases exponentially with the thickness of the barrier and the mass of the particle.

This property is exploited in many modern technologies, such as semiconductor electronics in which electron tunneling is key to the operation of components like transistors and diodes. Understanding potential energy barriers is also crucial in fields such as chemistry, where tunneling plays a role in nuclear fusion and chemical reactions at very small scales.