Quantum Organ – Pauli Exclusion Principle

All quantum states originate from a mirror plane.

We generate the spin states up and down by means of the double arrow shown here. We can generate combined states with two spins by shifting the double arrow to the respective position of the first spin. This shift gives us four combinations: up/up, up/down, down/up, and down/down.

Symmetries under exchange are a key for a deeper understanding of double spin states, and multiple particle systems in quantum physics in general. We have seen such symmetries already when we analyzed classical vibrational states. Let us recall the cup with a handle.

When we mirror at the handle on the plane shown here, the state with the antinode at the handle merges into itself. The state with the node at the handle, however, becomes its own negative.

An exchange of A with B thus results either in symmetry, AB₊, or antisymmetry, AB_–.

Symmetry or antisymmetry under exchange of A and B can be found again as a principle in two-particle states. Combinations of two indistinguishable spins overlap, creating joint vibrational modes that are either symmetric or antisymmetric under exchange of spins A and B.

The three symmetrical spin combinations are called triplets. The antisymmetric combination is known as a singlet.

The entire two-electron state consists of a position and a spin state. The symmetrical spin triplet joins with the antisymmetric position, and the antisymmetric spin singlet joins with the symmetrical position.

This is the Pauli Exclusion Principle: the state of two electrons A and B must be antisymmetric!

However, it also follows from the Pauli principle that each spin state can be occupied only once.

This is due to the ambiguity: a rotation by 360° changes the sign of a spin! The basis state is reached again only after 720°. In case of a single spin, this is irrelevant, because the phase of the spinning wheel cannot be observed.

However, if the same spin state is occupied twice, exchanging the particles A and B or rotating one of the two spins by 360° will result in the exact opposite of the basis state. This is due to the ambiguity.

An identical state, however, can only be its own opposite when it is equal to zero.

So far, we have only analyzed states on one great circle. The geometry of the quantum dimension is much more complex. On the left hand-side, there is the two-dimensional spherical surface in three dimensions, which we are familiar with. On the right hand-side, we draw the corresponding points of the three-dimensional spherical surface in four dimensions. We project them stereographically, that is, we distort them, so that we are able to present them at all.

In fact, the great circle in four dimensions is completed only after it has revolved around the sphere in three dimensions two times. This projection converts the nodal line of the spin state in the quantum dimension to a node on the Bloch sphere. The antipode of the node is the point with the maximum amplitude.

In fact, if we apply this projection, we will loose not only 360° per great circle, but also an entire dimension – the invisible phase. One point on the Bloch sphere corresponds to an entire great circle in the quantum dimension. All invisible phases are hidden in intertwined great circles.

Let us consider a group of points on the Bloch sphere. In the quantum dimension, this group corresponds to a fascinating system of intertwined rings.

This is just the beginning; the dawn of the quantum dimension. May I introduce myself? My name is Omega.