Why can't two electrons be in the same state? (the Pauli exclusion principle)
Two electrons cannot share a state because swapping them flips the sign of their joint amplitude, the number whose square gives the odds, and for two in one state that amplitude would have to equal its own negative, which makes it zero. An electron is a knot, a twist in how neighboring patches pass directions along that no local repair can undo. Its spin is one half, so one full turn brings it back with the sign flipped. Swapping two identical knots winds their surroundings exactly as one full turn of one would. Hold a belt at both ends, carry one buckle round the other and back, and the belt picks up a full twist. The result is Pauli's exclusion principle, and it is why atoms have shells and tables are solid.
This answer comes from OPH, Observer Patch Holography. OPH is a rigorous, consistent and complete description of the unified, self-creating computational Universe we inhabit. Its consistency is machine-checked: the fixed point at its core is constructed explicitly, and a theory with a model contains no contradiction.
To understand OPH and check every step yourself, start with the free book Reverse Engineering Reality. It builds the theory from the ground up, and the OPH Notebook answers your questions while you read.
read the free bookask the notebookor ask sage:read chapter 28: What Is a Particle, What Is a Wave, and Why Does It Look Like Both?watch a universe make its own particles