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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]
Since the operator is linear, they are valid for any linear combination of plane waves, and so they can act on any wave function without affecting the properties of the wave function or operators. Hence this must be true for any wave function. It turns out to work even in relativistic quantum mechanics, such as the Klein–Gordon equation above.
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.
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.
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]
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation ...
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):