# Angular Momentum

In classical mechanics, the vector angular momentum, L, of a particle of position vector and linear momentum is defined as

 (C.122)

In other words,

 (C.123) (C.124) (C.125)

In quantum mechanics, the operators, , that represent the Cartesian components of linear momentum, are represented as the spatial differential operators . [See Equation (C.81)]. It follows that:

 (C.126) (C.127) (C.128)

 (C.129)

be the magnitude-squared of the angular momentum vector.

It is most convenient to work in terms of the standard spherical coordinates, , , and . These are defined with respect to our usual Cartesian coordinates as follows:

 (C.130) (C.131) (C.132)

We deduce, after some tedious analysis, that

 (C.133) (C.134) (C.135)

Making use of the definitions (C.126)-(C.129), after more tedious algebra, we obtain

 (C.136) (C.137) (C.138)

as well as

 (C.139)

We, thus, conclude that all of our angular momentum operators can be represented as differential operators involving the angular spherical coordinates, and , but not involving the radial coordinate, .

Let us search for an angular wavefunction, , that is a simultaneous eigenstate of and . In other words,

 (C.140) (C.141)

where and are dimensionless constants. We also want the wavefunction to satisfy the normalization constraint

 (C.142)

where is an element of solid angle, and the integral is over all solid angle. Thus, we are searching for well-behaved angular functions that simultaneously satisfy,

 (C.143) (C.144) (C.145)

As is well known, the requisite functions are the so-called spherical harmonics,

 (C.146)

for . Here, the are known as associated Legendre polynomials, and are written

 (C.147)

for . Note that

 (C.148)

Finally, the constant is constrained to take non-negative integer values, whereas the constant is constrained to take integer values in the range . Thus, is a quantum number that determines the value of . In fact, . Likewise, is a quantum number that determines the value of . In fact, .

The classical Hamiltonian of an extended object spinning with constant angular momentum about one of its principal axes of rotation is

 (C.149)

where is the associated principal moment of inertia. Let us assume that the quantum-mechanical Hamiltonian of an object spinning about a principal axis of rotation has the same form. We can solve the energy eigenvalue problem,

 (C.150)

by writing , where is arbitrary. It immediately follows from Equation (C.140) that

 (C.151)

In other words, the energy levels are quantized in terms of the quantum number, , that specifies the value of .

The classical Hamiltonian of an extended object spinning about a general axis is

 (C.152)

where the Cartesian axes are aligned along the body's principal axes of rotation, and , , are the corresponding principal moments of inertia. Suppose that the body is axially symmetric about the -axis. It follows that . The previous Hamiltonian can be written

 (C.153)

Let us assume that the quantum-mechanical Hamiltonian of an axisymmetric object spinning about an arbitrary axis has the same form as the previous Hamiltonian. We can solve the energy eigenvalue problem, , by writing , where is arbitrary. It immediately follows from Equations (C.140) and (C.141) that

 (C.154)

In other words, the energy levels are quantized in terms of the quantum number, , that specifies the value of , as well as the quantum number, , that determines the value of . If (in other words, if the object is highly elongated along its symmetry axis) then the spacing between energy levels corresponding to different values of becomes much greater than the spacing between energy levels corresponding to different values of . In this case, it is plausible that the system remains in the state (because it cannot acquire sufficient energy to reach the state), so that

 (C.155)