Schrödinger equation

The Schrödinger equation is a partial differential equation that shows how the wave function of a tiny system changes over time. It was found by Austrian scientist Erwin Schrödinger in 1925 and shared in 1926. His work helped start modern quantum mechanics, and he won the Nobel Prize in Physics in 1933.

You can think of the Schrödinger equation like the tiny version of Newton's second law from the big world of classical mechanics. Just like Newton's rules can guess how things will move, the Schrödinger equation can guess how the wave function of a tiny system will change. This wave function tells us where a tiny piece might be.

The Schrödinger equation was inspired by an idea from Louis de Broglie that everything tiny acts like a matter wave. This idea helped explain why atoms stay together, just like scientists saw in tests. The Schrödinger equation is very useful, but it is just one way to understand quantum mechanics. Other ways include matrix mechanics from Werner Heisenberg, and the path integral formulation made by Richard Feynman.

Definition

The Schrödinger equation is an important 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 where a particle might be found.

Properties

The Schrödinger equation is a special math rule that helps us understand how tiny parts of nature, like atoms and electrons, behave. It was made by Erwin Schrödinger in 1925 and shared in 1926. This equation is very important in quantum mechanics, which is the study of how very small things move and change.

One key part of the Schrödinger equation is that it is linear. This means if you have two different answers, you can mix them in simple ways to get new answers. This helps scientists learn how particles can be in many places or states at once.

Another important part is unitarity. This makes sure that the chance of finding a particle somewhere always adds up to one. It means that particles are always somewhere, and we can guess how they will act over time. These parts make the Schrödinger equation a useful tool for learning about 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 shows a particle in an area with no energy cost, but the edges have very high energy cost. This creates special energy levels the particle can have.

Harmonic oscillator

The harmonic oscillator is another important example. It shows a particle pulled back to its starting point by a force that depends on how far it moves, like a mass on a spring. This model helps explain how atoms and molecules vibrate.

Hydrogen atom

The hydrogen atom is one of the few systems where the Schrödinger equation can be solved exactly. It shows an electron moving around a proton, creating special energy levels and shapes that explain how atoms are built.

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 movement of a tiny particle behave over time and compares them to how objects move 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 tiny particle follows rules similar to classical physics, the average movement sometimes does not exactly match what we would expect. However, in special cases, 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 are not 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 particles that move slowly, but it does not include effects from special relativity. When particles move very fast, 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, and the Dirac equation applies to particles with spin, such as electrons. These equations help us understand how particles behave when effects from special relativity are important. They are part of a larger framework called quantum field theory, which combines quantum mechanics with special relativity.

History

After Max Planck studied light, Albert Einstein suggested that light could act like tiny particles called photons. This idea helped us understand how light carries energy. Later, Louis de Broglie proposed that tiny particles like electrons also have wave-like properties.

Erwin Schrödinger used these ideas to create a math formula. In 1926, he shared his equation. This equation shows how the wave-like behavior of particles, such as electrons in an atom, changes over time. It helped explain the energy levels in atoms and became very important in physics.

Interpretation

Main article: Interpretations of quantum mechanics

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

One common idea, called the Copenhagen interpretation, says the wave function is a way to keep information about the system. According to this idea, the wave function changes smoothly over time, but it can change suddenly during a measurement when new information appears. Other ideas, 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 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 set way.