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The mathematical formulation of quantum mechanics (QM) is built upon the concept of an operator. Physical pure states in quantum mechanics are represented as unit-norm vectors (probabilities are normalized to one) in a special complex Hilbert space. Time evolution in this vector space is given by the application of the evolution operator.
The momentum operator can be described as a symmetric (i.e. Hermitian), unbounded operator acting on a dense subspace of the quantum state space. If the operator acts on a (normalizable) quantum state then the operator is self-adjoint. In physics the term Hermitian often refers to both symmetric and self-adjoint operators. [7] [8]
Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms. [2]: 1.1 It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot.
Quantum mechanics is a difficult subject to teach due to its counterintuitive nature. [1] As the subject is now offered by advanced secondary schools, educators have applied scientific methodology to the process of teaching quantum mechanics, in order to identify common misconceptions and ways of improving students' understanding.
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. When the position operator is considered with a wide enough domain (e.g. the space of tempered distributions ), its eigenvalues are the possible position vectors of the particle.
Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral (2nd ed.). World Scientific. ISBN 9789814397735. Sakurai, J. J.; Napolitano, Jim (2017). Modern Quantum Mechanics (2nd ed.). Cambridge University Press. ISBN 978-1-108-42241-3. Leonard I. Schiff (1968) Quantum Mechanics McGraw-Hill Education
An unbounded operator (or simply operator) T : D(T) → Y is a linear map T from a linear subspace D(T) ⊆ X —the domain of T —to the space Y. [5] Contrary to the usual convention, T may not be defined on the whole space X. An operator T is said to be closed if its graph Γ(T) is a closed set. [6]
Classically we have for the angular momentum =. This is the same in quantum mechanics considering and as operators. Classically, an infinitesimal rotation of the vector = (,,) about the -axis to ′ = (′, ′,) leaving unchanged can be expressed by the following infinitesimal translations (using Taylor approximation):