Spin states: triplets and singlets

In a quantum system consisting of two spin 1/2 particles, for example two protons in a hydrogen molecule, the spin of each particle can be “up” (1) or “down” (1), such that there are four possible spin states: (1, 1), (1, 1), (1, 1) and (1, 1). Within these four states, a distinction is drawn between a singlet state in which the spins are opposed and three indistinguishable triplet states in which they are aligned.

The University of York Centre for Hyperpolarisation in Magnetic Resonance has a good diagram on their website for this. Wikipedia contains a more rigorous explanation, including a mathematical viewpoint. This mathematical viewpoint is constructed on the principle of an equation: that the difference between two objects that are the same is zero.

Chiralkine counting provides a viewpoint on singlet and triplet states alternative to the mathematical viewpoint constructed on the principle of an equation. In this viewpoint, the meanings of “up” (1) and “down” (1) are mapped to “same” and “distinct” and oscillate interchangeably according to the two mutually exclusive ways of drawing distinctions in accordance with the logic gates XOR and XNOR.

The singlet state is a like a scalar in chiralkine counting. It is the number of copies of a triplet state that have been generated by counting one or more distinct objects. For example, counting three sheep generates three copies of a triplet state. Counting two of those three copies generates two copies of another triplet state, leaving one behind, and so on. It is conserved through repeated counting.

The singlet state corresponds with a recursive sequence of NOT gates. The truth table for a NOT gate simply has one input and one output. It is symmetric. An oscillation can be thought of as a recursive sequence of NOT operations, each inverting the symbol generated by its predecessor. This sequence forms the column for XOR and XNOR placed one over the other that, running together, generate a simply oscillation containing no triplets. The triplet states correspond with the truth tables for the six logic gates XOR, XNOR, OR, AND, NOR and NAND.