Similarly, the 2p quantum states form a 3-dimensional irrep ( called " spin-1 " ), and the components of the position operator also form the 3-dimensional " spin-1 " irrep.
22.
For example, in one dimension, the spatial variable of, say, a particle, can only be measured by the quantum mechanical " position operator " at the cost of losing information about the momentum of the particle.
23.
Where the operator acting on a wave function yields that wave function multiplied by the value, in an analogous fashion to the way that the position operator acting on a wave function yields that wave function multiplied by the value " x ".
24.
If one chooses the eigenfunctions of the position operator as a set of basis functions, one speaks of a state as a wave function \ psi ( "'r "') in position space ( our ordinary notion of space in terms of length ).
25.
Where is the particle's mass, is the angular frequency of the oscillator, is the position operator ( given by ), and is the momentum operator, given by \ hat p =-i \ hbar { \ partial \ over \ partial x } \,.
26.
The Heisenberg group is a central extension of such a commutative Lie group / algebra : the symplectic form defines the commutation, analogously to the canonical commutation relations ( CCR ), and a Darboux basis corresponds to canonical coordinates in physics terms, to momentum operators and position operators.
27.
For example, if a particle is in state | \ psi \ rangle, then the expectation value of position is \ langle \ psi | \ mathbf { \ hat r } | \ psi \ rangle, where \ mathbf { \ hat r } is the position operator.
28.
Where, the initial state is on the right and the final one on the left . The position operator "'r "'has three components, and the initial and final levels consist of 2 ! + 1 and 2 ! 2 + 1 degenerate states, respectively.
29.
The connection between this representation and the more usual wavefunction representation is given by the eigenstate of the position operator representing a particle at position, which is denoted by an element | x \ rangle in the Hilbert space, and which satisfies \ hat { X } | x \ rangle = x | x \ rangle.
30.
The key result is the Stone von Neumann theorem, which, informally stated, says that ( with certain technical assumptions ) every representation of the Heisenberg group H 2 " n " + 1 is equivalent to the position operators and momentum operators on "'R " "'n ".
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