Angular momentum

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.

Angular momentum in quantum mechanics

In quantum mechanics, angular momentum is defined like momentum - not as a quantity but as an operator on the wave function:

Related Topics:
Quantum mechanics - Momentum - Operator - Wave function

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:mathbf{L}=mathbf{r} imesmathbf{p}

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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

Related Topics:
Electric charge - Spin

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:mathbf{L}=-ihbar(mathbf{r} imes abla)

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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

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: = i hbar epsilon_{ijk} L_k, left = 0

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and even more importantly commutes with the hamiltonian of such a chargeless and spinless particle

Related Topics:
Commutes - Hamiltonian

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:left = 0.

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Angular Momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates. Then, the angular momentum in space representation is:

Related Topics:
Spherical symmetry - Spherical coordinates

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:: L^2 = rac{1}{sin heta}rac{partial}{partial heta}left( sin heta rac{partial}{partial heta} ight) + rac{1}{sin^2 heta}rac{partial^2}{partial phi^2}

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When solving to find eigenstates of this operator, we obtain the following

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:: L^2 | l, m ang = {hbar}^2 l(l+1) | l, m ang

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:: L_z | l, m ang = hbar m | l, m ang

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where

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:: lang heta , phi | l, m ang = Y_{l,m}( heta,phi)

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are the spherical harmonics.

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