Angular Momentum

Quantum Numbers

Quantum Numbers

The four quantum numbers , , , and specify the complete and unique quantum state of a single electron in an atom:

• : principal quantum number;
• : orbital quantum number, azimuthal quantum number;
• : magnetic quantum number;
• : spin quantum number.

Orbital Angular Momentum

Def Orbital Angular Momentum Operator Orbital angular momentum operators are: Or in spherical polar coordinates: In matrix representation under condition :

Proposition

Prop The angular momentum commutator is given byThus the components of angular momentum are not compatible.

Remark

This means we cannot make simultaneous measurements of the components of the angular momentum with arbitrary precision.

Proposition

Prop The commutator between a particular component of angular momentum and total angular momentum squared is zero:

Lemma In a central potential, we have the following holds:Thus and share simultaneous eigenfunctions.

Ladder Operators

Def Orbital Ladder Operators Define the ladder operators as the following: These are called the raising and lowering operators respectively.

Prop Ladder operators commutes with .

Prop Ladder operators act on the wavefunction to produce new wavefunctions which are also eigenstates of belonging to the eigenvalue but which have higher or lower .

Thrm Thrm Suppose is the eigenvalue associated with and is the eigenvalue associated with , then we have the following relation holds: And we usually label states of definite angular momentum with and : . e.g. .

Corollary The number of states with different observables for are .

Prop We have following operator actions:

Def Spherical Harmonics We denote as the eigenfunction of , called spherical harmonics. And thus we have the following formula: The solutions to the spherical harmonics are given by: which is the eigenfunction with the lowest value of . All other eigenfunctions can then be generated by successive application of the ladder operator . Table of spherical harmonics can be found here.