"The mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space."
Discussion of the mathematical tools used in quantum mechanics, including linear algebra and Hilbert spaces.
Linear Algebra: Linear algebra is one of the most fundamental topics in mathematics that plays a significant role in quantum mechanics. It helps in the understanding of the algebraic structures that underpin quantum mechanics.
Complex Analysis: Complex analysis is a branch of mathematics that deals with the study of functions of complex variables. This topic is essential in quantum mechanics since its formalism involves complex numbers.
Probability Theory: Probability theory is a branch of mathematics that deals with the analysis of random phenomena. In quantum mechanics, probability theory is used to predict the probabilities of the outcome of a quantum measurement.
Mathematical Logic: Mathematical logic is the study of the principles and methods used in mathematics. It is essential in Quantum mechanics because it ensures that the mathematical structures used in quantum mechanics are consistent.
Fourier Analysis: Fourier analysis is used to study the properties of periodic and continuous functions. It is also used in quantum mechanics to understand the properties of wave functions.
Geometry: Geometry plays a critical role in quantum mechanics since the wave functions of particles are typically represented by geometric structures.
Information Theory: Information theory deals with the quantification, storage, and communication of information. In quantum mechanics, the principles of information theory are used to develop quantum computation and quantum cryptography.
Topology: Topology is a branch of mathematics that deals with the study of geometric shapes and spaces. It is essential in quantum mechanics since the properties of quantum systems can be described using topological methods.
Stochastic Processes: Stochastic processes deal with the analysis and modeling of random processes. In quantum mechanics, stochastic processes are used to model the behavior of quantum systems.
Group Theory: Group theory is the study of the mathematical structures that underpin a group of objects. In quantum mechanics, group theory is used to study the symmetries of quantum systems.
Linear Algebra: The mathematical foundation for quantum mechanics, involving the description of quantum states as vectors and observables as operators.
Calculus of Variations: Used in deriving the variational principle in quantum mechanics and in the calculation of scattering amplitudes.
Complex Analysis: Used in the development of wave functions, quantum states, and the calculation of the probability amplitudes.
Differential Geometry: Used in the development of curved spacetime and the connection between gravity and quantum mechanics.
Measure Theory: Used in the calculation of probabilities in quantum mechanics.
Topology: Used in the development of topological phases of matter and quantum field theories.
Functional Analysis: Used in the study of Hilbert spaces and the construction of the Schrödinger equation.
Group Theory: Used in the study of symmetries in quantum mechanics.
Information Theory: Used in the development of quantum information theory and quantum computation.
Representation Theory: Used in the study of the behavior of quantum states under symmetry operations.
Category Theory: Used in the development of topological quantum field theory.
Algebraic Geometry: Used in the study of algebraic structures in quantum mechanics.
Statistical Mechanics: Used in the development of quantum statistical mechanics and the treatment of many-body systems.
Functional Integration: Used in the development of path integrals and the calculation of transition amplitudes.
Number Theory: Used in the study of the arithmetic properties of quantum states and in the development of quantum cryptography.
"Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces and operators on these spaces."
"In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space."
"At the heart of the description are ideas of quantum state and quantum observables, which are radically different from those used in previous models of physical reality."
"While the mathematics permits calculation of many quantities that can be measured experimentally..."
"...there is a definite theoretical limit to values that can be simultaneously measured."
"This limitation was first elucidated by Heisenberg through a thought experiment..."
"...and is represented mathematically in the new formalism by the non-commutativity of operators representing quantum observables."
"Prior to the development of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of formal mathematical analysis, beginning with calculus..."
"Probability theory was used in statistical mechanics."
"...accordingly, theories of relativity were formulated entirely in terms of differential geometric concepts."
"The phenomenology of quantum physics arose roughly between 1895 and 1915..."
"...physicists continued to think of quantum theory within the confines of what is now called classical physics..."
"The most sophisticated example of this is the Sommerfeld–Wilson–Ishiwara quantization rule..."
"...which was formulated entirely on the classical phase space."