Question

Does the spin1/2 rotation operator rotate spin in real space?

Answer 1

In higher branches of calculus, division becomes an entirely different concept. When I divide, I do not think of the relation as one-half. I regard the division operation as 1 with respect to 2.

In that regard, the conserved, intrinsic superposition gives that there are 2 possible states that a full degree rotation can produce. They have an inverse, multiplicative relationship if you regard division as I do. So 1 with respect to 2 means that 1 full rotation will give one state, the next full rotation MUST be the opposite state.

Let’s translate this dimensionally to the real world. This is where things get cool. So, a quarter has heads or tails. 1 full rotation in 4 dimensions is not the same as it is in two. So a full 360 degree rotation in 2 dimensions is actually 180 degrees.

So in 4 dimensional spacetime geometry. Let’s say that we have a quarter in the quantum world, and one in the Newtonian energy range.

Spin 1/2 means that the quarter can be either heads or tails (which we’ll say is 2) and will be one or the other when flipped. A ratio between possible and probable, if you will.

So think of Spin (1/2) = (head or tail) is the 1 part of the ratio, and the 2 part of the ratio is (head)+(tail). Essentially, every full rotation will be 1 of the possible two head or tail states; so two possible events are added to produce a probable event.

So if we flip our spin 1/2 quarter, heads is worth 360 degrees and tails is worth 360 degrees. If we flip it 360 degrees and it lands on heads, we know the next flip of 360 degrees will be tails. Meaning it takes two flips of the coin to get 720 degrees back to heads because each state has a mirror opposite that must come next.

In 3 dimensions, each flip of the coin is 180 degrees, so spin 1/2 in classical dimensions means that the pattern must follow a flip of either heads or tails must yield one of those two and the next flip will always yield the opposite of its current state.

Answer 2

Rotations in spin space. plays a special role in quantum mechanics. Not only is it often a constant of the motion even when is spin-dependent, but it is the generator of rotations in the Hilbert space. ... Thus, the generators of the spin-1/2 rotation group are just the 2 2 Pauli matrices.

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