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The operators used in quantum mechanics are collected in the table below (see for example [1] [4]). The bold-face vectors with circumflexes are not unit vectors, they are 3-vector operators; all three spatial components taken together.
The relativistic mass-energy relation: = + where again E = total energy, p = total 3-momentum of the particle, m = invariant mass, and c = speed of light, can similarly yield the Klein–Gordon equation: ^ = ^ + ^ = ^ + where ^ is the momentum 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]
Canonical commutation rule for position q and momentum p variables of a particle, 1927.pq − qp = h/(2πi).Uncertainty principle of Heisenberg, 1927. The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics.
Matrix mechanics easily extends to many degrees of freedom in a natural way. Each degree of freedom has a separate X operator and a separate effective differential operator P, and the wavefunction is a function of all the possible eigenvalues of the independent commuting X variables.
In quantum mechanics, an observable is an operator, or gauge, where the property of the quantum state can be determined by some sequence of operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value.
The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum problems involving rotational symmetry. Being an observable, its eigenfunctions represent the distinguishable physical states of a system's angular momentum, and the corresponding eigenvalues the observable experimental values.
Multiplying by the operator [S], the formula for the velocity v P takes the form: = [] + ˙ = / +, where the vector ω is the angular velocity vector obtained from the components of the matrix [Ω]; the vector / =, is the position of P relative to the origin O of the moving frame M; and = ˙, is the velocity of the origin O.