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Pauli exclusion principle states that 'No two electron can exist in same state' or 'No two electron can have same set of quantum numbers'.

But in reality there is no state of electron. The state is of whole system which is composed of many electrons. It is our approximation that each electron is described by a single wave-function which we called spinorbital. So what does that statement even mean in reality?

Quantum numbers is only exact for hydrogen atom. What does quantum number even mean for a electron in a many electron system.

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Wikipedia has the correct definition:

A more rigorous statement is that with respect to exchange of two identical particles the total wave function is antisymmetric for fermions, and symmetric for bosons. This means that if the space and spin co-ordinates of two identical particles are interchanged, then the wave function changes its sign for fermions and does not change for bosons.

The textbook shorthand you quote, 'No two electron can have same set of quantum numbers', only makes sense when you are using the one-electron approximation. As you move beyond that approximation, you also have to move beyond this shorthand definition.

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