NationMaster - Encyclopedia: Angular momentum operator

In quantum mechanics, angular momentum is defined like momentum - not as a quantity but as an operator on the wave function: Fig. ... In physics, momentum is the product of the mass and velocity of an object. ... In mathematics, an operator is some kind of function; if it comes with a specified type of operand as function domain, it is no more than another way of talking of functions of a given type. ... In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared...

$mathbf{L}=mathbf{r}timesmathbf{p}$

where r and p are the position and momentum operators respectively. In particular, for a single particle with no electric charge and no spin, the angular momentum operator can be written in the position basis as Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is generated by the motion of its center of mass about an external point. ...

$mathbf{L}=-ihbar(mathbf{r}timesnabla)$

where is the gradient operator. This is a commonly encountered form of the angular momentum operator, though not the most general one. It has the following properties Del symbol (also known as nabla; used in mathematical physics). ... In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ...

$[L_i, L_j ] = i hbar epsilon_{ijk} L_k$ , $left[L_i, L^2 right] = 0$

and, even more importantly, it commutes with the hamiltonian of such a chargeless and spinless particle For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ... The Hamiltonian, denoted H, has two distinct but closely related meanings. ...

$left[L_i, H right] = 0$ .

Angular momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates. Then, the angular momentum in space representation is: In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

$L^2 = frac{1}{sintheta}frac{partial}{partial theta}left( sintheta frac{partial}{partial theta}right) + frac{1}{sin^2theta}frac{partial^2}{partial phi^2}$

When solving to find eigenstates of this operator, we obtain the following In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

$L^2 | l, m rang = {hbar}^2 l(l+1) | l, m rang$

$L_z | l, m rang = hbar m | l, m rang$

where

$lang theta , phi | l, m rang = Y_{l,m}(theta,phi)$

are the spherical harmonics. In mathematics, the spherical harmonics are an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ...

Category: Quantum mechanics

Results from FactBites:

Angular momentum operator - Wikipedia, the free encyclopedia (299 words)
	In quantum mechanics, the angular momentum operator is an operator that is the quantum analog of the classical angular momentum.
	The third commutation relation states that the angular momentum is a constant of motion, and is a special case of Liouville's equation for quantum mechanics]], or more precisely, of Ehrenfest's theorem.
	Angular momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates.

Angular momentum Summary (2195 words)
	In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in which, the object rotates about the reference point.
	The reason angular momentum is important in physics is that it is a conserved quantity: a system's angular momentum stays constant unless an external torque acts on it.
	Under this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance (As a result, angular momentum isn't conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant).

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