Rotational motion

Rotation in two dimensions.

Rotation in three dimensions.

Solution of the angular equation.

Significance of spatial quantization.

Rotation in two dimensions

We consider a circular trajectory. The classical angular momentum is L_z = r x p (r cross p). Since r x p = rp sinq and q = 90^o (so sinq = 1) as shown in the diagram below we ca further state that L_{z<\SUB> = pr. The energy is E = p²/2m, therefore E = L_z²/2I. I is the moment of inertia. If we consider rotation of a single mass about an axis then I = mr², where r is the radius of the mass.}

Using the deBroglie relation p = h/l we also have a condition for quantization of angular motion L_z = hr/l. The wavelength must be a whole number fraction of the circumference for the ends to match after each circuit, l = 2pr/n where n is an integer. These considerations lead to a quantized expression, L_z = nh.

To derive a rotational hamiltonian we can begin with an angular momentum operator. Classically the angular momentum about the z axis is I ¶q/¶t. This operator is analogous to the linear momentum operator.

Linear momentum

p = -ih ¶/¶x

Angular momentum

L_z = -ih ¶/¶f

Again by analogy with the linear momentum problem that we have already discussed we can determine the eigenfunctions for the angular momentum operator.

where m = 0, ±1, ± 2, ... The classical kinetic energy is:

The rotation hamiltonian for this two-dimensional motion is:

The wave function F = (1/2p)^1/2e^imf and the rotational energies are:

Exercise: determine the normalization constant for the wavefunction e^imf.

If there are two masses in two dimensions then we require the reduced mass. Mass 1, m₁ and mass 2, m₂ separated by distance R have a center of mass lying on the line between them as shown in the Figure below.

The reduced mass is:

The moment of inertia is I = mr².

Rotation in three dimensions

The three dimensional spherical polar coordinate systems is shown below.

The extension from the two coordinate system follows the convention that the q angle represents the azimuthal angle or tilt angle from the z axis. The projections of the vector on the three-dimensional cartesian coordinate system are given by

x = r sinq cosf

y = r sinq sinf

z = r cosq

The operator that represents to square of the total angular momentum is:

The eigenvalues of L² are spherical harmonics Y_lm(q,f). The spherical harmonics are given by:

For positive values of m. For negative values of m we can use the identity:

P_lm(cosq) is the associated Legendre polynomial. We make the substitution x = cosq to simplify the following expressions.

The above expressions show the relationship between the Legendre polynomials, P_l(cosq) and the associated Legendre polynomials, P_lm(cosq). Recursion relations are useful for determining higher Legendre polynomials.

Recursion relation for the Legendre polynomials

Recursion relation for the associated Legendre polynomials

Solutions of the angular equation

In this section we demonstrate that the solutions (or eigenvalues) or the spherical harmonics are given by:

We will not derive the spherical harmonics, but rather demonstrate the validity of the above expressions by example.

Example: show that

Solution: first we determine the functional form of Y₁₀(q,f).

We then substitute this functional form into the operator equation (ignoring the normalization constant). Note that while the normalization constant is required in order to interpret the probability distribution, it cancels out in an eigenvalue problem since the wave function appears on both sides of the equation.

Exercise: show that

Significance of spatial quantization

The above solutions show that the total angular momentum of a molecule is

For a given value of l there are 2l + 1 possible orientations. These correspond to angles of

with respect to the z axis. This is illustrated below for l = 2.

The shapes of the wavefunctions give the probability distribution for the rotating molecule. The spherical harmonics have the form of the well-known atomic orbitals for the hydrogen atom. For example, l = 1 corresponds to a p-orbital, l = 2 corresponds to a d-orbital, l = 3 to a f-orbital etc. Thus, we can visualize the molecules rotating with an angular distribution according to the total angular momentum quantum number l and the projection along the z axis, m. You might ask, what is the z axis for a molecule? It is arbitrary, but the point is that once it is chosen, we cannot simultaneously measure the orbital angular momentum along the orthogonal x and y axes. The commutation relations tell us which quantitites can be simultaneously measured. These relations show us that L and L_z can be known simultaneously, but not L_z and L_x or L_y.

One final point. The quantum numbers for rotational orbital angular momentum and the azimuthal quantum number are J and M, respectively. Thus, J corresponds to l and M corresponds to m in the above exposition.