Question

How should you answer someone who asks, "in tunneling through a simple barrier, which way are particles moving in the three regions before, inside, and after the barrier?"

Short answer

In the context of quantum tunneling through a simple barrier, particles move towards the barrier before it, through the barrier inside it (due to the quantum phenomenon of tunneling), and away from the barrier after it.

Step-by-step solution

Particle Motion Before the Barrier

Before the barrier, particles generally show classical behavior, moving towards the barrier. However, due to quantum mechanics, the particle doesn't necessarily need to have enough energy to overcome the barrier.

Particle Motion Inside the Barrier

Inside the barrier - this is where quantum tunneling happens. Even if a particle doesn't technically have enough energy to overcome the barrier, it can nonetheless 'tunnel' through it. This is described by a wave function, which exponentially decays within the barrier. Hence, the particle is moving through the barrier, despite it seemingly being impossible in classical physics.

Particle Motion After the Barrier

After the barrier, the particle shows once again classical behavior, moving away from the barrier. The probability of finding the particle in this region is given by the transmission coefficient. The most fascinating part is that the particle appears on the other side of the barrier as if it had enough energy to overcome it, although it didn't.

Key concepts

Particle Motion in Quantum Mechanics

Understanding particle motion in quantum mechanics requires a departure from the classical notion of particles moving along definitive trajectories. Unlike their classical counterparts, quantum particles are not always described in terms of specific positions or velocities. Instead, quantum mechanics introduces the concept of probability distributions.

Whereas a classical particle approaching a barrier would simply reflect back if it lacks the necessary energy to overcome it, a quantum particle has a non-zero probability of appearing on the other side of the barrier. This is possible even if the particle's energy is less than the potential energy of the barrier, a phenomenon known as quantum tunneling.

In the region before the barrier, the quantum particle is represented by a wave function, which describes a spread-out wave of possibilities rather than a distinct point-like particle. While it may seem counterintuitive, this concept is a cornerstone of quantum mechanics that explains a wide range of phenomena.

Quantum Mechanics Wave Function

At the heart of quantum mechanics is the wave function, symbolized as $\psi$, which is used to describe the quantum state of a particle. The wave function contains all the information about a particle's position and momentum. However, it does not provide these as definite values but as a probability cloud which illustrates where the particle is likely to be found upon measurement.

This wave function evolves according to the Schrödinger equation, a fundamental equation within quantum mechanics. The probability of finding a particle in a particular location is given by the square of the absolute value of the wave function at that point, commonly referred to as the probability density $|\psi {|}^2$.

Inside the barrier, a particle's wave function does not stop abruptly but instead, decreases exponentially, highlighting the non-zero chance of the particle tunneling through to the other side. The concept of a wave function is an indispensable tool in explaining the quantum behavior of particles, especially when dealing with phenomena that have no classical analog, such as tunneling.

Transmission Coefficient

The transmission coefficient, often denoted as $T$, is a vital quantity in describing quantum tunneling. It represents the probability that a particle will tunnel through a barrier rather than being reflected by it. The value of this coefficient ranges from 0 to 1, with 0 indicating no transmission (complete reflection) and 1 indicating complete transmission.

The coefficient is determined by the properties of the barrier, such as its width and height, as well as the energy of the particle. Calculating the transmission coefficient involves solving the Schrödinger equation for the particular potential barrier and using boundary conditions to link the wave function inside and outside the barrier.

A higher transmission coefficient suggests a greater likelihood of finding the particle on the other side of the barrier post-tunneling. This number is crucial for understanding the probabilistic outcomes of tunneling events in quantum systems and has applications across various fields, including electronics and chemistry.