Quantum mechanics is a mathematical framework that describes the behavior of particles at the microscopic scale. At its core, it is based on the principles of linear algebra and probability theory. The fundamental mathematical description of quantum mechanics is given by a vector space called the Hilbert space.
In this mathematical formalism, the state of a quantum system is represented by a vector, known as the state vector or wave function, which belongs to the Hilbert space. The state vector contains all the information about the system's observable properties.
The evolution of a quantum system is described by the Schrödinger equation, which is a partial differential equation. It determines how the state vector changes over time. The Schrödinger equation incorporates the Hamiltonian operator, which represents the total energy of the system.
When a measurement is performed on a quantum system, the mathematical formalism provides the probabilities of obtaining different outcomes. The outcomes are associated with the eigenvalues of the corresponding observable. The measurement process "collapses" the state vector onto one of the eigenstates of the observable, and the probability of obtaining a particular eigenvalue is determined by the square of the amplitude of the corresponding component of the state vector.
Mathematically, observables in quantum mechanics are represented by Hermitian operators, and their eigenstates form a complete orthonormal set in the Hilbert space.
Additionally, there are mathematical operations called operators that correspond to physical quantities such as position, momentum, and spin. These operators act on the state vector to yield information about the system's properties.
Overall, quantum mechanics provides a mathematical framework that describes the probabilistic nature of microscopic phenomena and allows for the calculation of observable quantities based on the state of the system.