Question

Given the same particle energy and barrier height and width, which would tunnel more readily: a proton or an electron? Is this consistent with the usual rule of thumb governing whether classical or nonclassical behavior should prevail?

Short answer

Given the same particle energy and similar barrier parameters, an electron would tunnel more readily than a proton. This conclusion aligns with the usual rule of thumb stating that lighter particles are more prone to demonstrate nonclassical behavior, such as tunneling.

Step-by-step solution

Understand Quantum Tunneling

The probability of a particle tunneling through a barrier is given by the formula: \(T = e^{ - 2 k a}\) where \(k = √{2m(V-E)/ ℏ}\) , V is the barrier height, E is the energy of particle, a denotes the barrier width, m denotes the mass of the particle and ℏ is the reduced Planck constant. The heavier the particle, the lower the probability of tunneling.

Analyze Particle Mass

Given the same energy and barriers, it's clear that the mass of the particle would play the deciding role as all other factors are equal. In this case, we are comparing protons and electrons. Protons have a significantly larger mass than electrons (roughly 1836 times more).

Determine Tunneling Probability

Substituting the different masses into the tunneling formula, it becomes evident that the exponent, -2ka, would be larger for the proton (due to its more substantial mass). This leads to a smaller tunneling probability. Therefore, an electron would tunnel more readily than a proton when given equal energy and barrier parameters.

Assessing Rule of Thumb

According to the rule of thumb, non-classical behavior (like tunneling) is more likely for light particles and less likely for heavier ones when other parameters are equal. In our case, this rule holds because the lighter electron exhibits more non-classical behavior (tunneling) than the heavier proton.

Key concepts

Particle Physics

Particle physics is the branch of physics that studies the nature of particles that are the constituents of what is usually referred to as matter or radiation. In this field, particles are often understood as being excitations of quantum fields and the fundamental interactions between them. The proton and electron, pivotal in the exercise, are classic examples of such particles.

Protons are composite particles, meaning they are made up of smaller particles known as quarks, bound together by the strong force. Contrarily, electrons are elementary particles and are one of the primary components of atoms, interacting primarily through electromagnetic forces. The masses of these particles vastly differ, with the proton being approximately 1836 times more massive than the electron. This fundamental property of mass influences their behavior in quantum phenomena like tunneling, which the exercise explores.

Quantum Mechanics

Quantum mechanics is the fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. One of the most intriguing and non-intuitive aspects of quantum mechanics is the concept of quantum tunneling, which allows particles to pass through potential barriers that would be insurmountable according to classical physics.

The tunneling phenomenon can be observed in various quantum systems and is fundamental in technologies such as tunnel diodes and quantum computing. It's important to note that quantum tunneling is a probabilistic event, and its likelihood decreases with increasing mass and barrier specifications, as noted in the exercise solution. This feature of quantum mechanics is crucial to understanding why an electron, being much less massive than a proton, would more readily tunnel through the same potential barrier.

Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle is a fundamental principle of quantum mechanics that states it is impossible to simultaneously know with perfect accuracy both the position and momentum of a particle. This principle has far-reaching implications for the behavior of particles on the quantum scale and directly influences the phenomenon of quantum tunneling.

In the context of tunneling, the uncertainty principle implies that there is a non-zero probability that a particle can be found on the other side of a barrier, even if its energy would not classically allow for such passage. This arises because the precise energy and position of the particle cannot be definitely known. As a result, particles like electrons can 'borrow' energy from their uncertain quantum state, allowing them to appear on the opposite side of a barrier which is well described by the tunneling probability formula given in the exercise solution.