Quantum state

In quantum physics, a quantum state is a description that tells us everything we can know about a tiny part of the world at one moment. It is like a picture that shows the condition of very small particles, such as atoms or photons, which follow special rules called quantum mechanics. Knowing the quantum state helps scientists understand how these particles will act.

Quantum states can be pure or mixed. Pure states are shown using math in a special space called a Hilbert space. Mixed states are mixes of different pure states and are described using something called density matrices. These ways of showing quantum states help scientists solve tricky problems.

Some common quantum states include wave functions, which describe where a particle might be or how fast it is moving. There are also states that describe things like spin, such as the singlet state, and states that describe groups of particles in a Fock space. Learning about quantum states is important for new technologies like quantum computers and advanced sensors.

From the states of classical mechanics

Quantum states started from ideas in classical mechanics. In classical mechanics, we describe things like a moving ball using exact details such as its position and speed. These details follow clear rules that help us know where the ball will go next.

In quantum mechanics, states also change following rules, but they act in a different way. Instead of exact numbers, quantum states use special numbers called complex numbers. These numbers follow special rules and can only show us the chances of different results. For example, the quantum state of an electron shows where we might find it, but not exactly where it will be, like in a double-slit experiment.

Role in quantum mechanics

In quantum mechanics, a quantum state describes a physical system using special variables. When we measure these variables, the system is set in a unique state. This state changes over time in a predictable way.

Unlike in classical physics, where we can know exactly what will happen, quantum mechanics deals with probabilities. Many measurements of the same quantum state show results that match these predicted probabilities.

Measurements

Main article: Measurement in quantum mechanics

In quantum physics, measurements help us learn about tiny systems. When we measure something many times the same way, we usually get the same result. This helps scientists get ready for experiments. But some measurements can change the system. For example, measuring how fast something is moving and then measuring where it is can change the speed the next time. This changing effect is an important idea in quantum mechanics, called the uncertainty principle.

Eigenstates and pure states

See also: Eigenvalues and eigenvectors § Schrödinger equation

After a measurement, a quantum state becomes an eigenstate. This means it matches the result of that measurement. If you do the same measurement again, you will get the same result. Sometimes, doing more measurements can make the state even clearer, leading to a pure state. If a state isn’t pure, it’s called a mixed state.

The solutions to the Schrödinger equation can create pure states, but experiments usually don’t show them exactly. Instead, scientists compare mixtures of these solutions to what they see in the real world.

Representations

The same physical quantum state can be shown in different mathematical ways called representations. One common way is the position wave function, which is often taught first when learning about quantum mechanics. Another is the momentum wave function, which is also a type of wave function. Think of these like different ways to map a place — picking one can make some problems simpler, while others might become harder.

In more advanced quantum mechanics, ideas are built using abstract concepts called 'vector space'. This method helps us understand many ideas in quantum mechanics, even when there is nothing in the everyday world to compare them to.

Wave function representations

Main article: Wave function

In quantum physics, wave functions are special math tools that help us describe quantum states. They show us how tiny particles, like electrons, behave in very small ways. These wave functions can tell us about things such as where a particle is or its momentum.

Pure states are clear descriptions of a quantum system, like knowing exactly where an electron is and how it spins. Mixed states are less certain, where we only know possibilities — like guessing how an electron might spin if we checked it. Both types help scientists predict what might happen with tiny particles.

Formalism in quantum physics

Quantum physics uses math to describe how tiny particles behave. A quantum state is like a plan that shows everything we can know about a small particle or group of particles.

Scientists use a special kind of math, called linear algebra, to study these states. They think of the particle as being in a space with many directions, where each spot shows a possible state. When we know the exact state, it is called a “pure state.”

These states can also blend together in special ways, creating new chances — this is called superposition. Even though the math can look hard, it helps us understand how tiny particles act in surprising but real ways!

Mathematical generalizations

In quantum physics, states can be described using special math tools called observables. These help scientists understand how a system behaves. This approach uses something called a C*-algebra and other advanced math ideas to describe the state of a quantum system.