# Can two electrons have the same momentum and spin directions?

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.

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?

• I must admit I don't fully understand this quote. If you know the two electrons are at a particular position then they could have any momentum. The Pauli exclusion principle forbids two electrons from occupying the same quantum state in a system, it doesn't refer to their position. – Rob Jeffries Oct 25 '14 at 16:11
• In words, a particle with a definite momentum is not located (or more precisely, localized) anywhere. Put another way, the 'spread' in location is inversely proportional to the 'spread' in momentum. Localizing momentum (position) necessarily means delocalizing position (momentum). – Alfred Centauri Oct 25 '14 at 17:27
• Have a look at this page hyperphysics.phy-astr.gsu.edu/hbase/pauli.html – anna v Oct 25 '14 at 19:24
• @annav Not really impressed with that description. It doesn't mention spin, says several times that no two fermions can exist in the same energy state (when often exactly two can!) and doesn't spell out why the initial postulated wavefunction is no good for indistinguishable particles (i.e. that $\psi^2$ only stays the same after swapping the particle identities if $\psi_1^{2}(a)=\psi_1^{2}(b)$ etc., which cannot be true in general). – Rob Jeffries Oct 25 '14 at 20:24
• @RobJeffries Hmm, do you not know what fermions means? half integer spin? – anna v Oct 26 '14 at 4:23

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.
• @RobJeffries: If you know that two electrons are both at a certain position, say $x=0$, then each electron's wavefunction is a Dirac delta function $\delta(x)$. This violates the Pauli exclusion principle, because they are both in the same state. – user4552 Oct 25 '14 at 17:22