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If a system of particles comprises electrons or a system of protons, then the particles are said to be identical when interchanging any of the two particles of the same kind does not change anything about the system. It means the magnitude-squared of the wave function, which gives the probability density, must remain the same when the coordinates of any two electrons are interchanged.

If the wave function involves a pair of particles each in one of two different one-electron states, interchanging the particles (putting each in the state of the other) and doing it again multiplies the wave function with any spin included. This is possible only if a single exchange of particles multiples the wave function or by or.

For electrons, and other particles with half-integer spins, interchanging the quantum numbers of the two electrons always multiplies the wave function.

Symbolically, changing the quantum numbers of the two electrons must change the sign,

${蠄}_{n,n\text{'}}\left(r_{I^2}{r}_2\right)={蠄}_{n\text{'},n}\left(r_{I^2}{r}_2\right)$

Here, denotes the first set of one-electron quantum number and $n\text{'}$the second electron鈥檚 set of quantum numbers. But if the two electrons have all the same quantum numbers, the wave function becomes role="math" localid="1655379141692" ${蠄}_{n,n}\left(r1r2\right)$and the requirements that exchanging quantum numbers,

${蠄}_{n,n\text{'}}(r1,r2)=-{蠄}_{n.n}(r1,r2)$

This requires the impossible condition that ${蠄}_{n,n}\left(r1r2\right)$is identically zero.

Conclusion: Therefore, The two electrons cannot have all quantum numbers the same including spin, and cannot be in the same one-electron states.