I am trying to understand the Pauli exclusion principle. Here is an except from Feynman Lectures on Physics
It just isn’t possible at all for two Fermi particles—such as two electrons—to get into exactly the same state. You will never find two electrons in the same position with their two spins in the same direction. It is not possible for two electrons to have the same momentum and the same spin directions. If they are at the same location or with the same state of motion, the only possibility is that they must be spinning opposite to each other.
http://www.feynmanlectures.caltech.edu/III_04.html [emphasis added]
I don't understand about "It is not possible for two electrons to have the same momentum and the same spin directions." Is it not possible for two electrons, even if they are at different locations, to have the same momentum and the same spin directions?
Best Answer
A particle that is in a pure momentum state has a wavefunction that is a sinusoidal plane wave. Therefore its position is infinitely uncertain. You can also see this in the Heisenberg uncertainty relation, $\Delta p \Delta x \gtrsim h$; if $\Delta p=0$, then $\Delta x$ blows up to infinity.
I'm in Los Angeles, and let's assume that you're in Chicago. Obviously if I manipulate an electron here, it can have no effect on an electron that you're manipulating there. But I cannot prepare an electron in Los Angeles in a state of pure momentum. If I wanted to do that, I would have to prepare it in pure sine-wave state that extended to infinity in all directions, and it would therefore not be localized to Los Angeles or Chicago. The whole universe can only hold one such electron.