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The gauge-invariant angular momentum (or "kinetic angular momentum") is given by = (), which has the commutation relations [,] = (+ ()) where = is the magnetic field. The inequivalence of these two formulations shows up in the Zeeman effect and the Aharonov–Bohm effect .
These commutation relations are relevant for measurement and uncertainty, as discussed further below. In molecules the total angular momentum F is the sum of the rovibronic (orbital) angular momentum N, the electron spin angular momentum S, and the nuclear spin angular momentum I.
The commutation relation between the cartesian components of any angular momentum operator is given by [,] =, where ε ijk is the Levi-Civita symbol, and each of i, j and k can take any of the values x, y and z.
The angular momentum of m is proportional to the perpendicular component v ⊥ of the velocity, or equivalently, to the perpendicular distance r ⊥ from the origin. Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a ...
Creation and annihilation operators. Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. [1] An annihilation operator (usually denoted ) lowers the number of particles in a given state by one.
In quantum mechanics, the Holstein–Primakoff transformation is a mapping from boson creation and annihilation operators to the spin operators, effectively truncating their infinite-dimensional Fock space to finite-dimensional subspaces. One important aspect of quantum mechanics is the occurrence of—in general— non-commuting operators ...
Conjugate variables. Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals, [1][2] or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty relation—in physics called the Heisenberg uncertainty principle —between them.
Stone–von Neumann theorem. In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named after Marshall Stone and John von Neumann. [1][2][3][4]