Quantum Mirror – Rotation Operator

Simple rotation operations allow us to generate all kinds of vibrational modes on the two-dimensional spherical surface. Using suitable rotating operations, we can convert the total number l of azimuthal nodal lines into m nodal lines that rotate clockwise or anti-clockwise. The sphere we are considering here has a fixed radius. This is why we can leave radial nodal lines out.

Let’s now take a closer look at the rotation operator. An operator manipulates properties of a state. A state is generally a complex object with a variety of properties. It could be a banana, for example. Let us express this state with B. The state “banana” has many different properties, such as a shape, a colour, and a taste. Of course, we can also rotate the banana.

Three-dimensional objects have three different axes of rotation: the x-axis, the y-axis, and the z-axis. The rotation operator D manipulates only the banana’s rotation property. The banana will not change its taste just because you rotate it, for example, by 90° about its z-axis, right?

We can also perform several rotations one after another. For example, we can first rotate the banana by 90° about its z-axis, and then by 90° about its y-axis. The banana is now lying on its back.

Rotation operators have a curious feature. To demonstrate this, let us perform the same rotation operations in reverse order. We first rotate the banana about its y-axis, and then about its z-axis. The final state of the banana is different, although the initial state was the same as before. The banana is not lying on its back now. It has been turned to the side. Rotating operations do not commute. It means that the order in which they are applied plays a crucial role.

Let us now consider a slightly different state: a banana that is rotating about its z-axis. Let us express this state with R_zB. Applying a rotation operator along the rotational axis of the banana does not change its state. This banana has a so-called eigenstate. Generally speaking, the eigenstate does not change when the corresponding operator is applied.

Be careful, though. The only rotation operator that does not change the state is the one that rotates it along the specified axis. Rotation about an incorrect axis will surely change the state.

Therefore, Bob is an eigenstate only with respect to rotation about his z-axis … or rather… he was an eigenstate …