Schrödinger equation

The Schrödinger equation is a partial differential equation that describes how the wave function of a quantum system changes over time. It was discovered by Austrian physicist Erwin Schrödinger in 1925 and published in 1926. His work helped lay the foundation for modern quantum mechanics, for which he received the Nobel Prize in Physics in 1933.

You can think of the Schrödinger equation as the quantum version of Newton's second law from classical mechanics. Just as Newton's laws predict the motion of objects based on their starting conditions, the Schrödinger equation predicts how the wave function of a quantum system evolves. This wave function gives us information about where a particle might be found.

The Schrödinger equation was inspired by the idea, proposed by Louis de Broglie, that all matter behaves like a matter wave. This idea helped explain why atoms have stable bound states, matching what scientists observed in experiments. While the Schrödinger equation is very important, it is just one way to understand quantum mechanics. Other methods include matrix mechanics, introduced by Werner Heisenberg, and the path integral formulation developed by Richard Feynman.

Definition

The Schrödinger equation is a key idea in quantum mechanics, a part of science that studies very small particles. It helps us understand how these particles behave over time. It was created by Erwin Schrödinger in 1925 and published in 1926.

In simple terms, the equation describes a "wave function." This function gives us information about where a particle might be and how it moves. To use the equation, you need to know about the particle’s mass and the energy around it, like how high up it is or what forces are on it. The equation helps scientists predict the chances of finding a particle in different places.

Properties

The Schrödinger equation is a special kind of mathematical description that helps us understand how tiny particles, like atoms and electrons, behave. It was created by Erwin Schrödinger in 1925 and published in 1926. This equation is very important in quantum mechanics, which is the study of how very small things move and change.

One key feature of the Schrödinger equation is that it is linear. This means that if you have two different solutions, you can combine them in simple ways to get new solutions. This property helps scientists understand how particles can exist in multiple states at once, a phenomenon known as superposition.

Another important aspect is unitarity. This ensures that the total probability of finding a particle somewhere always adds up to one. It means that particles are always somewhere, and their behavior can be predicted in a consistent way over time. These properties make the Schrödinger equation a powerful tool for exploring the quantum world.

Main article: Probability current and Continuity equation

Examples

See also: List of quantum-mechanical systems with analytical solutions

Particle in a box

The particle in a box is a simple example in quantum mechanics. It describes a particle in a region with no potential energy, surrounded by areas where the potential energy is infinite. This setup leads to specific allowed energy levels for the particle.

Harmonic oscillator

The harmonic oscillator is another key example. It describes a particle subject to a restoring force proportional to its displacement, like a mass on a spring. This model helps explain many physical systems, including vibrations in atoms and molecules.

Hydrogen atom

The hydrogen atom is one of the few systems where the Schrödinger equation can be solved exactly. It describes an electron orbiting a proton, leading to specific energy levels and orbitals that explain the structure of atoms.

Semiclassical limit

The semiclassical limit helps us understand how quantum mechanics connects to the physics we see in everyday life. It looks at how the average position and momentum of a quantum particle behave over time and compares them to the motion of objects in classical physics.

One way to make this comparison is by using something called the Ehrenfest theorem. This theorem shows that while the average position of a quantum particle follows rules similar to classical physics, the average momentum sometimes does not exactly match what we would expect from classical laws. However, in special cases like the quantum harmonic oscillator, the averages do match classical behavior perfectly. When a quantum system’s wave function is very focused around a specific point, its averages stay very close to classical paths for a while. This shows how quantum mechanics can approximate classical physics under certain conditions. The Schrödinger equation also has a close relationship with another important physics equation called the Hamilton–Jacobi equation, especially when we consider very small quantum effects.

Main article: Ehrenfest theorem

Main articles: Hamilton–Jacobi equation, Action, Hamiltonian function, Generalized coordinates

Density matrices

Main article: Density matrix

Sometimes, wave functions aren't the easiest way to describe how quantum systems behave. When we don't know everything about a system, or when we're looking at just part of a bigger system, we can use something called a density matrix instead. A density matrix is a special kind of math tool that helps us understand these situations better.

There's also a special math rule, similar to the Schrödinger equation, that tells us how density matrices change over time. This rule is known by a few different names, like the von Neumann equation. It helps scientists predict how quantum systems will act even when things are a bit unclear.

Relativistic quantum physics and quantum field theory

The Schrödinger equation works well for slow-moving particles but does not include effects from special relativity. In relativistic situations, where particles move close to the speed of light, we need different equations.

Two important equations are the Klein–Gordon equation and the Dirac equation. The Klein–Gordon equation describes particles without spin, while the Dirac equation applies to particles with spin, such as electrons. These equations help us understand how particles behave when relativistic effects are important. They are part of a larger framework called quantum field theory, which combines quantum mechanics with special relativity.

History

Following Max Planck's work on light, Albert Einstein suggested that light could behave like particles called photons. This idea helped explain how light carries energy. Later, Louis de Broglie proposed that particles like electrons also have wave-like properties.

Inspired by these ideas, Erwin Schrödinger worked on a mathematical way to describe how these waves behave. In 1926, he published an important equation that shows how the wave-like behavior of particles, such as electrons in an atom, changes over time. This equation helped explain the energy levels in atoms and became a cornerstone of modern physics.

Interpretation

Main article: Interpretations of quantum mechanics

The Schrödinger equation helps us calculate how the wave function of a system changes over time, but it doesn’t tell us exactly what the wave function is. Different interpretations of quantum mechanics explain what the wave function means.

One common view, called the Copenhagen interpretation, says the wave function is a way to store statistical information about the system. According to this view, while the wave function changes in a smooth and predictable way over time, it can change suddenly during a measurement because new information becomes available. Other interpretations, like relational quantum mechanics and QBism, look at the equation in different ways. Some thinkers believe that all possible outcomes described by quantum theory happen at the same time in different parallel universes, an idea known as the many-worlds interpretation. There are also ideas like Bohmian mechanics, which adds extra rules to make everything happen in a determined way.