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In particle physics there are lots of different quantum numbers (e.g. strangeness, isospin, hypercharge etc.) with different conservation laws. I know quantum numbers such as $n, l, m$ in a hydrogen like atom come from solving mathematical equations (e.g. $m$ comes for the requirement of $2\pi$ periodicity). Is the same true quantum numbers in particle physics or do they come about by someone say e.g. 'we will give certain particles these values of quantum number and by looking at the reactions that occur fit conservation laws to that quantum number'.

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The necessity for quantum numbers in elementary particle interactions was obvious from the accumulation of data in high energy physics which displayed symmetries, many more than just isospin symemtries. Does this seem random to you?

octet

The meson octet. Particles along the same horizontal line share the same strangeness, s, while those on the same diagonals share the same charge, q.

decuplet

The baryon decuplet

summary

Baryon supermultiplet using four-quark models and half spin

The data showed symmetries which demanded explanation. Humans are pattern recognition animals, and recognized the patterns. As they also know algebra and groups, they found quantum numbers useful for mapping the patterns.

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As far as I know for many quantum numbers the history was rather different. People didn't notice conservation laws but rather the fact that there exist groups of particles with similar properties. These particles can be therefore arranged in bunches called multiplets. Now suppose that you have $N$ particles labeled $1,..,N$ which are different, but nevertheless have the same properties with respect to some specific type of processes which you are studying at the time. Then states can be written down as $| \psi \rangle = \psi_1 |1 \rangle + ... + \psi_N | N \rangle$. So states correspond to vectors $(\psi_1,...,\psi_N)^T$. But since particles are the same with respect to the process you are studying then $U(\psi_1,...,\psi_N)^T$ with $U$-arbitrary unitary matrix is equivalent. After all it is just composition of mixing particles with each other and multiplying by phases. This means that you have a $SU(N)$ symmetry which is really combinatorial in nature. However, once such symmetry is established it can be used to get plenty of powerful predictions. Nice example is isospin $SU(2)$ symmetry. This is really a combinatorial symmetry arising from similarity of up and down quarks. They have almost the same masses and very similar properties with respect to strong interactions. Therefore elementary particles which are made of quarks as well as atomic nuclei which are made of neutrons and protons can be arranged in multiplets transforming by $SU(2)$. Nuclei and particles in one multiplet will have similar masses and strong decay rates. However, weak interactions are known to transform up and down quarks. This means that transformation $u \to d$ can't possibly commute with weak interaction Hamiltonian. Therefore particles or nuclei in a $SU(2)$ multiplet might have different decay rates if dominating decay channel involves weak interactions. For example take neutron and proton. They have almost the same mass and neither can decay through strong interactions. Strong force inside nucleus between two neutrons is known to be very similar to strong force between two protons. However, neutron can decay to proton by weak interactions. Reverse process has never been observed.

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While not a particular fan of the standard model, I strongly disagree that the terms are 'made up'. As your own detail write-up noted- they were set based on experimental data and the current interpretations at that time. All terms and concepts will be refined as we improve our understanding of particles, via the scientific method.

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