Introduction

The orbital kinetic moment is a concept of the quantum Mécanique. It is a particular case of kinetic Moment quantum.

Analogies with the traditional Mechanical

The orbital kinetic moment corresponds to the rotation of a particle around a core, like the rotation of an electron around a core in a Atome.

One differentiates the orbital kinetic moment from the intrinsic, interpretable kinetic moment by the rotation of an elementary particle on itself (one speaks about Spin of the electron, for example).

All kinetic Moment is quantified in quantum mechanics (see the quantum article kinetic Moment), i.e. the kinetic moment can take only quite precise discrete values. It is one of the fundamental properties of the quantum theory.

Formulas and quantum formalism

The operator of orbital kinetic moment is noted $\backslash \; hat\; L$ and one defines it by the following relation (similar to that of traditional mechanics):

$\backslash \; hat\; L\; =\; \backslash \; hat\; R\; \backslash \; wedge\; \backslash \; hat\; P$ representing a vector product.

$\backslash \; hat\; R$ is the operator position and $\backslash \; hat\; P$ the operator impulse , who has as Cartesian components in Représentation position:

• $\backslash \; hat\; P\_x=-i\; \backslash \; hbar\; (\backslash \; partial\; \backslash \; partial\; X)$
• $\backslash \; hat\; P\_y=-i\; \backslash \; hbar\; (\backslash \; partial\; \backslash \; partial\; there)$
• $\backslash \; hat\; P\_z=-i\; \backslash \; hbar\; (\backslash \; partial\; \backslash \; partial\; Z)$

In representation position, the Cartesian components of the operator $\backslash \; hat\; R$ are simply:

• $\backslash \; hat\; R\_x\; =\; x$
• $\backslash \; hat\; R\_y\; =\; y$
• $\backslash \; hat\; R\_z\; =\; z$

According to these definition, the Cartesian components of the operator of orbital kinetic moment are written:

• $\backslash \; hat\; L\_x\; =\; \backslash \; hat\; there\; \backslash \; hat\; p\_z\; -\; \backslash \; hat\; Z\; \backslash \; hat\; p\_y\; =\; -\; I\; \backslash \; hbar\; (there\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; Z\}\; -\; Z\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; there\})$
• $\backslash \; hat\; L\_y\; =\; \backslash \; hat\; Z\; \backslash \; hat\; p\_x\; -\; \backslash \; hat\; X\; \backslash \; hat\; p\_z=\; -\; I\; \backslash \; hbar\; (Z\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; X\}\; -\; X\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; Z\})$
• $\backslash \; hat\; L\_z\; =\; \backslash \; hat\; X\; \backslash \; hat\; p\_y\; -\; \backslash hat\; there\; \backslash \; hat\; p\_x=\; -\; I\; \backslash \; hbar\; (X\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; there\}\; there\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; X\})$

One can then calculate the switches of $\backslash \; hat\; L\_x$, $\backslash \; hat\; L\_y$ and $\backslash \; hat\; L\_z$:

• $L\_x,\; \backslash \; hat\; L\_y\; =\; I\; \backslash \; hbar\; \backslash \; hat\; L\_z$
• $L\_y,\; \backslash \; hat\; L\_z\; =\; I\; \backslash \; hbar\; \backslash \; hat\; L\_x$
• $L\_z,\; \backslash \; hat\; L\_x\; =\; I\; \backslash \; hbar\; \backslash \; hat\; L\_y$

Total kinetic moment

The operator of total kinetic moment noted $\backslash \; hat\; J$ is the vectorial sum of the operator of orbital kinetic moment noted $\backslash \; hat\; L$ and of the operator of Spin (intrinsic kinetic moment) noted $\backslash \; hat\; S$.

$\backslash \; hat\; J\; =\; \backslash \; hat\; L\; +\; \backslash \; hat\; S$

See too

• kinetic Moment in traditional mechanics
• quantum kinetic Moment
• Spin

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