Measurement in quantum mechanics

In quantum physics, a measurement is when we test or change a physical system to get a number result. One big idea in quantum theory is that its predictions are probabilistic, meaning they tell us the chances of different outcomes, not exactly what will happen.

To find these chances, we combine a quantum state, which describes the system, with a math rule called the Born rule. For example, a tiny particle like an electron can be described by numbers called probability amplitudes. Using the Born rule on these numbers gives the chances of finding the electron in different places when we look for it. The theory can't say for sure where the electron will be, only the chances.

Measuring a quantum system usually changes its quantum state. This idea is both tricky and important in quantum mechanics. The math for predicting measurement results and how states change was developed in the 20th century using linear algebra and functional analysis. Quantum physics has been very successful and useful in many areas.

There are also ongoing discussions about what the idea of measurement really means. Different interpretations of quantum mechanics try to solve what is known as the measurement problem.

Mathematical formalism

Main article: Canonical quantization

Further information: Dirac–von Neumann axioms

In quantum physics, a measurement is a way to test or change a system to get a number result. One key idea in quantum theory is that it makes predictions that are chances, not certainties.

To find these chances, we combine a special description of the quantum system (its quantum state) with a math rule for the measurement we want to do. This helps us understand what might happen when we measure something in the quantum world.

History of the measurement concept

The "old quantum theory"

Main article: Old quantum theory

The old quantum theory is a group of ideas from 1900 to 1925, before modern quantum mechanics. It was not complete, but it tried to fix problems in classical mechanics. Important work included Max Planck’s study of blackbody radiation, Albert Einstein’s work on the photoelectric effect, and Niels Bohr’s model of the hydrogen atom.

The Stern–Gerlach experiment, done in 1922, showed that quantum measurements can have only certain results. In this experiment, silver atoms were passed through a magnetic field, which bent their path. The atoms landed in specific spots on a screen, showing that their spin was “quantized.”

Transition to the "new" quantum theory

In 1925, Werner Heisenberg published a paper that helped shape modern quantum physics. He focused on what could be observed, like the light frequencies atoms absorb or emit.

The uncertainty principle began here. It says we cannot know both an electron’s position and speed at the same time exactly. Later, others gave this principle a precise mathematical form.

From uncertainty to no-hidden-variables

Main articles: EPR paradox, Bell's theorem, and Bell test

Some wondered if quantum mechanics was just an approximation and if there were “hidden variables” that would let us predict more. Bell’s theorem, published in 1964, showed that certain hidden-variable ideas do not match what we see in experiments. Tests since then have supported quantum mechanics and shown these hidden variables are not likely.

Quantum systems as measuring devices

The Wigner–Araki–Yanase theorem shows that saving energy limits how well we can measure some things in quantum systems.

Decoherence

Main article: Quantum decoherence

When a quantum system interacts with its surroundings, it can lose its special quantum features. This is called decoherence. It is important in quantum computing, where keeping systems isolated helps maintain their quantum properties.

Quantum information and computation

Quantum information science studies how we use quantum physics to handle and use information in new ways. Understanding how we measure things in quantum physics is very important for this area.

Quantum circuits are a way to do calculations using quantum physics. In these circuits, special steps called quantum gates are used, followed by measurements to get the results. These circuits work with tiny parts called qubits.

Measurement-based quantum computation is another way to do calculations, where the answer is found by measuring the system itself. Quantum tomography is a method to figure out the state of a quantum system by looking at the results of many measurements. Quantum metrology uses quantum physics to make very precise measurements, like in experiments that detect very small changes.

Laboratory implementations

In the early days of quantum physics, scientists used simple tools like watching light patterns, seeing flashes of light, and listening to clicks from special counters to study tiny particles. These early methods helped them understand how particles behave.

One famous experiment is the double-slit experiment, where scientists shine light or send electrons through two narrow openings. This shows how particles can act like waves. Modern experiments use very sensitive tools to detect single particles, helping scientists learn more about the strange behavior of the very small.

Interpretations of quantum mechanics

Main article: Interpretations of quantum mechanics

Even though scientists agree that quantum physics works well, they still argue about what it really means. These arguments often focus on how we understand measuring something in quantum physics. One big question is whether the random results we see when measuring are truly random or if there is a hidden, fixed process causing them. Different ways of thinking about these questions are called "interpretations" of quantum mechanics.

Some interpretations try to explain measurement as just an approximation of a deeper, fixed process. Others see quantum states as information about systems, meaning sudden changes in these states are just updates in our knowledge. There is no agreement yet on which approach is best.