Quantum Mirror – Spectrum of the Rotation Operator

We shall now examine the spectrum of the rotation operator, that is, the complete set of its eigenstates in three dimensions, in more detail. To do this, let us take l =2 as an example.

The most symmetric state, m=0, is completely rotationally invariant. This state does not change at all when it is rotated about the z-axis at any given angle alpha. It is thus obviously an eigenstate with respect to the operator D_z(α). Its eigenvalue is 1.

Is this also true for m=1? This state is rotating about the z-axis. Rotating it additionally by a fixed angle alpha therefore only changes the phase of its rotation. It does not change the state itself. This is another eigenstate of the operator D_z(α).

For rotation by the angle alpha, the eigenvalue is e^iαm. This is true for all m. All states shown here are thus eigenstates of the rotation operator about the z-axis.

What happens when we rotate one of these states about the y-axis? Let us take m = 0. In this case, the state does change. It is thus not an eigenstate with regard to the operator “rotation about the y-axis”. This is also true for all m.

Let’s sum it up. The vibrations on the spherical surface in three dimensions can be classified using the number l of azimuthal nodal lines. This gives us all possible real eigenstates of the rotation operator about the z-axis.

We can only construct eigenstates with regard to rotations about one specific axis. In this example, we have selected the z-axis. These vibrational states on the sphere are therefore not eigenstates with regard to rotations about the x-axis or the y-axis.

However, we can use these two rotation operators, D_x and D_y, to form two new and important operators. Let us call them nodal rotation operators. We shall describe their role in the next slide.