Quantum Organ – Quantum Nodes

An electron state combines position and spin state. Both components can be derived from the spectrum of a guitar string. Let us analyze the fundamental frequency and the first overtone. All other overtones follow the same model.

The key lies in the geometry of space, in which the state swings. Let’s start with a simple circle, like we have done in U2 station 5 slide 5.

If we connect the starting point with the endpoint, we obtain this vibrational state on a circular line. Such a vibration with a “twist”, however, cannot be stable. Hence, this state does not exist.

Let us now consider the first overtone. When we wind this vibrational state on a circle by connecting the starting point with the endpoint, we obtain a familiar result. This state has one nodal line, thus l=1.

In the quantum dimension, all vibrational states on a spherical surface are in at least four dimensions. Some large circles in four or higher dimensions can be divided into 720°, instead of 360°. This is how the fundamental and harmonic frequencies look like in the quantum dimension. Everything is doubled!

Let us wind this state on a circle in three-dimensional space. All vibrational modes are doubly wound.

What implications does that have?

For the fundamental frequency, the double winding results in the following vibrational state with one node. The “twist” has disappeared. This state is, in fact, possible.

For the state l=1, the double winding makes no difference. In this case, it does not matter whether we project the state from four dimensions, or just analyze it directly in three dimensions.

The double-wound fundamental mode has one node. It represents the spin state. The first overtone has one nodal line, as usual, and represents the state l=1.

Let us now analyze the spin state s with one node in more detail. The antipode of the node is the point with the maximum amplitude. On the Bloch sphere, this corresponds to the exact direction of the spin vector. In this projection, the spin “up” state is an exact mirror image of the spin “down” state. The distance between the states is specified by ℏ. The distance to the mirror plane is thus s=+1/2 and s=-1/2 respectively, in ℏ units.