2.6. Rigid rotor
Here, we will derive the solution of the Schrödinger equation for the linear rigid rotor.
(2.101)
Expressed in spherical coordinates, the Hamiltonian is
(2.102)
is the moment of inertia around an axis perpendicular to the molecular axis, and is the norm-squared of the angular-momentum operator ( in Cartesian coordinates), here expressed in spherical coordinates.
2.6.1. Separation of variables
Moving the right-hand side of the Schrödinger equation to the left gives . Dividing by and multiplying by gives
(2.103)
where we have introduced the abbreviation . The first and the third term on the left-hand side depend on only, whereas the second term depends on only. We can therefore use a separation approach and write as a product of a -dependent function and a -dependent function : . Inserting this and dividing by both and gives
(2.104)
The sum of the first and third terms depends only on , and the second term is a function of only. Varying can change only two of the three terms - but since the overall sum must remain constant (zero), the sum of the terms must be constant - let’s denote this constant as . The same reasoning holds for the term, where the constant now has to be the negative of the constant, since they should add up to zero. This yields two equations, one for and one for .
(2.105)
We chose to write the constant as a square, , of another constant, , for future convenience.
Now we have successfully separated the original partial differential equation with two variables into two ordinary differential equations, with one variable each. Next, we solve these two equations.
2.6.2. The equation
The general mathematical solution of the equation is
(2.106)
as can be verified by insertion. In this general mathematical solution, there are no constraints on the constants , , and . However, for a solution to be physical, it has to be single-valued. In particular, a full turn of (the angle in the equatorial plane) brings the orientation back to its original value, and therefore the function also should have the same value:
(2.107)
i.e. it must be periodic. This condition yields
(2.108)
For arbitrary values of and , this is possible only if the exponentials both are equal to 1: . From this it follows that must be an integer:
(2.109)
Since can be positive or negative, we can write the wavefunction, now including normalization, simply as
(2.110)
2.6.3. The equation
Next, we need to solve the equation for . It is
(2.111)
With the substitution (with ranging from to ) and , we obtain
(2.112)
This is a well-known equation, the associated Legendre differential equation, and it can be solved using the Frobenius method. Most of the mathematical solutions are not physical, since they diverge at (corresponding to and ) and are not square integrable. Divergence-free solutions are obtained only if is a product of two consecutive non-negative integers. If we denote them and :
(2.113)
The solutions with integer and are called associated Legendre polynomials.
Finally, this gives the following ranges for the quantum numbers and :
(2.114)
Rearranging the expression for gives
(2.115)
with the rotational constant .