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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. [1]
If one chooses the (generalized) eigenfunctions of the position operator as a set of basis functions, one speaks of a state as a wave function ψ(r) in position space. The familiar Schrödinger equation in terms of the position r is an example of quantum mechanics in the position representation. [5]
In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator .
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.
Since the angular momenta are quantum operators, they cannot be drawn as vectors like in classical mechanics. Nevertheless, it is common to depict them heuristically in this way. Depicted on the right is a set of states with quantum numbers ℓ = 2 {\displaystyle \ell =2} , and m ℓ = − 2 , − 1 , 0 , 1 , 2 {\displaystyle m_{\ell }=-2,-1,0 ...
In quantum mechanics, a translation operator is defined as an operator which shifts particles and ... is an eigenvector of the position operator ^ with ...
In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. 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.
Since the only wavefunction that can have lowest position-momentum uncertainty, , is a gaussian wavefunction, and since the coherent state wavefunction has minimum position-momentum uncertainty, we note that the general gaussian wavefunction in quantum mechanics has the form: (′) = ^ (′ ^ ) (′ ^ ).
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