C. David Sherrill

School of Chemistry and Biochemistry

Georgia Institute of Technology

November 1999

School of Chemistry and Biochemistry

Georgia Institute of Technology

November 1999

We start from the classical expression for angular momentum,
, to obtain the quantum mechanical version
, where , , and are all three-dimensional vectors. This definition
leads immediately to expressions for the three components of :

(1) | |||

(2) | |||

(3) |

From these definitions, we may easily derive the following commutators

(4) | |||

(5) | |||

(6) |

where the indices can be , , or , and where the coefficient is unity if form a cyclic permutation of [i.e., (), (), or ()] and -1 for a reverse cyclic permutation [(), (), or ()]. The final commutator indicates that we cannot generally know , , and simultaneously except if we have an eigenstate with eigenvalue 0 for each of these.

Classically, any component of the angular momentum must be less than or
equal to the
magnitude of the overall angular momentum vector. Quantum mechanically,
the *average* value of any component of the angular momentum must be
less than or equal to the square root of the expectation value of dotted with itself:

(7) |

Since commutes with any component , we *can* have simultaneous eigenfunctions of and a given component
. We usually pick the axis, since the expression for is the easiest of the three when we work in spherical polar
coordinates:

(8) | |||

(9) | |||

(10) |

Of course it is also possible to express in terms of the unit
vectors for spherical polar coordinates,

(11) | |||

(12) | |||

(13) |

Here,

(14) |

(15) |

The simultaneous eigenfunctions of and are
called the *spherical harmonics*,
, where
is the total angular momentum quantum number, and is the so-called
magnetic quantum number. The spherical harmonics are defined as

(16) |

(17) |

(18) | |||

(19) |

It can be useful to define *ladder operators* for angular momentum. The
following ladder operators work not only for straight angular momentum , but also for combined angular momenta such as
. If

(20) |

(21) |

One can also show that in spherical polar coordinates

(22) |

(23) |

we can see that the Hamiltonian can be written as

(24) |

Clearly commutes with the kinetic energy term, has no dependence. Likewise, if , then commutes with the whole Hamiltonian. Hence, for problems where the potential depends only on (central force problems), we can find simultaneous eigenfunctions of , , and .