The mathematical basis of quantum mechanics is formulated within the framework of complex linear algebra, particularly using a mathematical structure called a Hilbert space. In quantum mechanics, the state of a particle or a physical system is described by a mathematical object known as a wave function (or state vector), usually denoted by the symbol ψ.
The wave function ψ provides information about the probabilities of various outcomes when measurements are made on the system. The square of the absolute value of ψ, denoted by |ψ|^2, gives the probability density distribution associated with the system. The behavior of particles at the atomic scale is described by the wave function evolving over time according to the Schrödinger equation.
The Schrödinger equation is a fundamental equation of quantum mechanics and is expressed as:
iħ ∂ψ/∂t = Hψ
In this equation, ħ (h-bar) represents the reduced Planck's constant, i is the imaginary unit, ∂ψ/∂t denotes the partial derivative of the wave function with respect to time, and H is the Hamiltonian operator. The Hamiltonian operator represents the total energy of the system and includes terms corresponding to the kinetic energy and potential energy of the particles involved.
The solutions to the Schrödinger equation yield the wave functions that describe the system's behavior over time. The wave function can be used to calculate the probabilities of different measurement outcomes. For example, the position of a particle can be described by an operator, and by applying this operator to the wave function, one can obtain the probability distribution of finding the particle at different positions.
The behavior of particles at the atomic scale, such as the phenomenon of wave-particle duality, superposition, and quantum entanglement, is explained through the mathematical formalism of quantum mechanics. These concepts arise from the wave-like nature of particles described by the wave function and the inherent probabilistic nature of quantum measurements.
It is worth noting that while the mathematical formalism of quantum mechanics accurately describes the behavior of particles at the atomic scale, there are different interpretations and debates about the underlying meaning and philosophical implications of these mathematical constructs.