Mathematical formulation of quantum mechanics

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. These formulations use parts of functional analysis, especially Hilbert spaces, which are special kinds of linear space. Unlike older physics theories, quantum mechanics uses abstract mathematical structures, like infinite-dimensional Hilbert spaces (L² space mainly) and operators that act on these spaces.

In quantum mechanics, values of physical observables such as energy and momentum are not just simple numbers. Instead, they are understood as eigenvalues or spectral values of linear operators in Hilbert space. These ideas about quantum state and quantum observables are quite different from earlier models of how the physical world works.

These mathematical formulations are still used today. One key feature is that there is a limit to what can be measured at the same time. This was first shown by Heisenberg using a thought experiment, and it is expressed mathematically through the non-commutativity of operators that represent quantum observables. Before quantum mechanics, physics relied mainly on mathematical analysis, starting with calculus and moving up to differential geometry and partial differential equations. Even theories like theories of relativity were built using these older mathematical tools.

History of the formalism

The "old quantum theory" and the need for new mathematics

Main article: Old quantum theory

In the 1890s, Planck discovered that energy could only be exchanged in small, fixed amounts called quanta. This helped explain patterns of light that classical physics could not. In 1905, Einstein suggested that these energy packets, later called photons, were like tiny particles of light.

These ideas challenged existing physics. Scientists like Bohr and Sommerfeld tried to adjust classical mechanics to explain these new findings. However, they struggled with more complex atoms. In 1923, de Broglie proposed that waves and particles were two sides of the same coin, applying not just to light but to all matter.

The "new quantum theory"

In the mid-1920s, scientists developed new mathematical ways to describe these ideas. Heisenberg created matrix mechanics, and Schrödinger developed wave mechanics. Both methods worked well and were later shown to be equivalent. They used advanced math, including Hilbert spaces, to describe the behavior of tiny particles.

These new theories helped explain many mysteries of the atom and laid the foundation for modern quantum mechanics.

Postulates of quantum mechanics

Quantum mechanics uses special mathematical ideas to describe how tiny parts of the universe, like atoms and particles, behave. These ideas help scientists predict things we can’t see with our eyes alone.

The main ideas in quantum mechanics are about three things: the state of a system, how we measure it, and how it changes over time. A system’s state is like a description of everything about it at one moment. In quantum mechanics, this is shown using something called a Hilbert space, which is a special kind of mathematical space. When we measure something, like the spin of an electron, the result we get depends on the state of the system. And over time, the state changes according to rules that scientists can write down as equations.

Mathematical structure of quantum mechanics

Main article: Dynamical pictures

Quantum mechanics uses special math to describe how tiny parts of the world, like atoms, behave. Instead of using simple numbers, it uses something called Hilbert spaces, which are like very fancy math rooms where we can track all the possible states of a particle. This math helps scientists predict things like how light or electrons act in ways that are different from everyday objects.

One important idea in quantum mechanics is how we "see" or picture changes over time. There are different ways to look at these changes, and each way gives us useful hints about linking quantum ideas back to the physics we see in the normal world. For example, there is a way called the phase space formulation that makes it easier to understand how quantum mechanics connects to classic physics. Even though these ideas can get very complex, they all help us understand the hidden rules that control the tiny building blocks of our universe.

Problem of measurement

Main article: Measurement in quantum mechanics

Quantum mechanics differs from older science because it must explain what happens when we measure, or observe, something. When we measure a quantum system, the result is random, but the probabilities can be calculated. For example, if we know the system is in a certain state, we can figure out the chance that a measurement will show a particular value.

After a measurement, the system’s state changes. If we measure something many times in a row, we get the same result each time. This is a key idea in quantum mechanics. There are also more modern ways to think about measurements, which let us describe many different kinds of quantum actions in a single framework.

List of mathematical tools

The book Methods of Mathematical Physics by Richard Courant, based on David Hilbert’s courses at Göttingen University, shows how math was already ready for quantum mechanics before physicists realized it. When Erwin Schrödinger introduced his equation, they found the needed math was already there.

Key math tools for quantum mechanics include: