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  1. Pauli exclusion principle means no two identical fermions can be in the same quantum state. Does it mean, two electrons with the same spin cannot be in the same De Broglie Wavelength? Or, more precisely, does Pauli exclusion principle mean, no two electrons with the same spin cannot be in the same state because their wave functions destructively interfere?
  2. Is Pauli exclusion applicable for two electrons with different wavelength? If it is, and if my assumption in the first question is correct, can two electrons with different frequency/wavelength completely cancel each other? If not, how do they interfere?
Nayeem1
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2 Answers2

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"State" means everything about the wavefunction, not including phase or normalization (i.e., a constant factor). The wavefunction includes both the spin and the spatial wavefunction.

A more formal statement of the exclusion principle is either (a) that the combined wavefunction of the two fermions has to flip sign under the interchange of the particles, or (b) that the inner product of the two wavefunctions has to be zero. These are equivalent.

Does it mean, two electrons with the same spin cannot be in the same De Broglie Wavelength?

No, a state doesn't normally have a well-defined wavelength.

Or, more precisely, does Pauli exclusion principle mean, no two electrons with the same spin cannot be in the same state because there wave functions destructively interfere?

No, it has nothing to do with interference.

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Whenever we talk about the Pauli exclusion principle, we must remember that

  • it is a heuristic conclusion from experimental observations and
  • is related to the behaviour of electrons in atoms and molecules.

According to Wikipedia the story was the following (emphasis by me):

In the early 20th century it became evident that atoms and molecules with even numbers of electrons are more chemically stable than those with odd numbers of electrons... (Pauli realized) that the complicated numbers of electrons in closed shells can be reduced to the simple rule of one electron per state, if the electron states are defined using four quantum numbers. For this purpose he introduced a new two-valued quantum number, identified by Samuel Goudsmit and George Uhlenbeck as electron spin.

To your questions

... does Pauli exclusion principle mean, no two electrons with the same spin cannot be in the same state because there wave functions destructively interfere?

The wave function in quantum physics is a probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. In the bundle of possible results of QM calculations an interference of spins is not foreseen. The Principle is so named because it is a rule that has to be followed and so QM calculations could only add the spins of the involved particles (for example calculating helium as an atom with zero electron spin).

Is Pauli exclusion applicable for two electrons with different wavelength?

No and yes. In the meaning, that an excited electron in our example helium has a different wavelength, the Pauli Principle does not hold. The sum of the orientations of the spins of the involved electrons could could be different from zero.
For beryllium two electrons have a different energy content (and in this meaning are off different wavelength), but yes, the Exclusion Principle holds.

If it is, and if my assumption in the first question is correct, can two electrons with different frequency/wavelength completely cancel each other? If not, how do they interfere?

Two electrons in free space do not interfere. For electrons in an atom with the same first three quantum numbers the spin quantum has to be opposite.

One remark. In a first step, not involving QM, it is easy to get an imagination of what happens in atoms by remembering that the spin and the electrons intrinsic magnetic dipole moment are correlated. Simply imaginate, that the electrons show the same behavior like bare magnets.

HolgerFiedler
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