Square Well Potential in Quantum Mechanics

Fundamental theory in physics describing the properties of nature on an atomic scale.

In quantum mechanics, potential wells are a key concept that helps us understand how particles behave under certain conditions. One of the simplest and most illustrative examples of this is the square well potential.

Introduction to Potential Wells

In classical physics, a particle can't exist in a region where its energy is less than the potential energy. However, in quantum mechanics, due to the probabilistic nature of particles, they can exist in regions where classically they should not, a phenomenon known as quantum tunneling. A potential well is a region where a particle is trapped due to the surrounding potential barriers.

The Square Well

A square well is a potential well defined by a piecewise constant potential function. The simplest form of a square well is a particle trapped in a one-dimensional box with infinitely high walls. This is also known as the particle in a box problem.

Solving the Schrödinger Equation for a Particle in a Square Well

To solve the Schrödinger equation for a particle in a square well, we need to consider two regions: inside the well and outside the well. Inside the well, where the potential energy is zero, the Schrödinger equation simplifies to a second-order differential equation whose solutions are sine and cosine functions. Outside the well, where the potential energy is infinite, the solution is simply zero, indicating that the particle cannot exist outside the well.

Bound States and Scattering States

The solutions to the Schrödinger equation for a square well represent different physical states of the particle. Bound states are solutions where the particle is trapped inside the well. These states have discrete energy levels, similar to the energy levels of an atom. Scattering states, on the other hand, represent solutions where the particle has enough energy to escape the well.

Tunneling Effect

One of the most fascinating implications of the square well potential is the tunneling effect. Even when a particle doesn't have enough energy to overcome a potential barrier, quantum mechanics allows for a non-zero probability that the particle can still be found on the other side of the barrier. This is known as quantum tunneling, and it's one of the key differences between classical and quantum physics.

In conclusion, the square well potential provides a simple yet powerful model for understanding some of the most fundamental concepts in quantum mechanics. It illustrates the probabilistic nature of particles in quantum mechanics and introduces key concepts such as bound states, scattering states, and quantum tunneling.