# 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. Oct 25, 2014 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). Oct 25, 2014 at 17:27
• Have a look at this page hyperphysics.phy-astr.gsu.edu/hbase/pauli.html Oct 25, 2014 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). Oct 25, 2014 at 20:24
• @RobJeffries Hmm, do you not know what fermions means? half integer spin? Oct 26, 2014 at 4:23

Is it not possible for two electrons, even if they are at different locations, to have the same momentum and the same spin directions?

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.

• This sounds correct. Conversely if you knew two electrons were extremely close to each other, they must have very different momentum states, so I don't see that the PEP forbids this. Oct 25, 2014 at 17:19
• @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, 2014 at 17:22
• I'm sure you're right but now I'm getting a headache. If a particle is localised by a delta function at one instant, then in the next instant it can be anywhere in the universe. Perhaps I should ask another question for someone to clearly explain the last sentence of the quote. How close could they be? Oct 25, 2014 at 17:37
• @RobJeffries: How close could they be? The question assumes that the distance between them is known with infinite precision, which isn't normally the case.
– user4552
Oct 25, 2014 at 18:58
• As I understood it, you can get them as close as you like providing you give them sufficiently different momenta. Though I concede, not in precisely the same spot. Oct 25, 2014 at 19:06