Spin – Spin

The figure with a candle and a mirror, which serves to illustrate states and operators, can be manipulated by changing the distance between the states. Another way to manipulate it is by changing the position of the mirror plane. The mirror should reflect states onto further states, and the distance between the states should always be equal.

This can only be achieved in two ways. We already know one solution: if a state is its own mirror image, the number of states will always be an odd number.

If the mirror is positioned exactly in the middle between two states, the number of states will be even, and no state will be its own mirror image.

The simplest scenario thus includes two states. The transition to quantum physics is performed by scaling the distance between the two states by ℏ. These two states have an important function. They can, for example, describe the spin of the electron, spin up and spin down, with plus one half and minus one half in ℏ units.

What modes of vibration correspond to these two states? Those cannot be vibrations on the two-dimensional spherical surface, since they have been fully classified already.

Something new is happening here. It is possible to demonstrate mathematically that, in the simplest scenario, those are vibrations in four dimensions on a three-dimensional spherical surface.

This spherical surface is a complex geometrical object. In quantum physics, such a four-dimensional state is possible because it cannot be directly observed, and only has an indirect impact.

In classical physics, such a state is simply not possible. This is because four-dimensional vibrational modes that can be directly observed do not exist.

The spin is the first and the most important example of a state that cannot exist in real space, and therefore represents a purely quantum mechanical phenomenon. What we are able to see is, so to speak, a projection into the real three dimensions. Let’s define the z-axis from the bottom up, as usual. In relation to the z-axis, the spin up state points upwards, and the spin down state points downwards.

We can project the three-dimensional spherical surface onto a conventional, two-dimensional spherical surface, whose north and south poles correspond to the states spin up and spin down.
These two basic spin states exist alongside the potential superposition states of these vibrations, such as “spin up + spin down”, or “spin up – spin down”.

The geometric representation would be a spin pointing in the direction of +y or –y. By way of analogy, the spin can also point in the direction +x or –x.

The spin in the three-dimensional space can thus point in any direction. This is indicated here by the direction of the large red arrow.

Each point on the sphere corresponds to the spin state which points in that spatial direction. All potential spin orientations of this so-called qubit yield a two-dimensional spherical surface, the so-called Bloch sphere.