Built on Facts

Yesterday we talked about how fermions and bosons had different values of spin and thus their wavefunctions had different symmetry properties. In particular, fermions are antisymmetric under exchange of particles. We’d like to write the overall two-particle wavefunction in terms of the individual wavefunctions for each of the two particles. The result will look like this:

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

You can see that if you switch around r1 and r2, you get the same function back with a negative sign in front of it. If these particles were bosons, we’d need a plus sign in the middle instead of a minus sign so that we’d get the same function back without a negative sign. Now at this point I’m a little torn. Calculating the so-called exchange interaction is a somewhat involved operation that does require a no-kidding understanding of the mathematical methods of quantum mechanics, and as much as I’d like to do this in full formal detail I’d rather not glaze over too many eyeballs. So for the moment we’ll do this the easy way, and at some point in the future we’ll do it the hard way.

Imagine that the two particles did have the same wavefunctions: phi1 = phi2. Looking at the overall two-particle wavefunction, we see that the second term would be the same as the first term. Since they’re being subtracted, the result is a flat 0. Thus two fermions can’t be in the same state, or there wouldn’t be a wavefunction in the first place. This restriction for fermions is called the Pauli exclusion principle. Bosons have no such restriction. Since they have a plus sign instead of a minus sign, you just get a sum of the two terms – not zero. Thus they can be in the same state if they feel like it.

And that’s it. How could you have a hard way? The answer is that interesting things happen when you have two particles in close to the same state. Doing the math, it’s possible to show that as phi1 and phi2 get closer and closer to being identical, this is equivalent to increasing the energy of the system. A change in energy with respect to volume (as the particles are forced closer) is the definition of pressure. It’s this pressure that helps keep solid objects solid, most dramatically in situations like neutron stars.

Fundamental as this is, it’s not something physics has known forever. The principle itself is less than a century old, and a good understanding of the consequences of Pauli’s principle with respect to the behavior of solids was only well-developed starting in the 60s.

Comments

  1. #1 Thras
    May 23, 2009

    If nobody mentions it to him in the comments, I wonder how long before he notices?

  2. #2 D. C. Sessions
    May 23, 2009

    Would you like to start a betting pool?

  3. #3 Who Cares
    May 23, 2009

    I place a beer (if you ever get close to where I live I’ll drag you to a good brewer) in the pool for two days

  4. #4 Art
    May 23, 2009

    Well … it is memorial day weekend and then catching up Tuesday … say Wednesday.

  5. #5 Matt Springer
    May 23, 2009

    So, uh… it turns out that 1.jpg and 1.png are not the same file. Go figure!

  6. #6 Carl Brannen
    May 23, 2009

    Nice that you noted that approaching the same state, for fermions, amounts to an increase in energy. This has applications in density operator (matrix) formalism.

    The basic idea of density operator formalism is that the density operator form can be treated as fundamental right up there with the wave function / ket formalism. They cover it in undergraduate QM. rho(x,x’) = psi*(x)psi(x’).

    Because the density operator has two copies of the wave function, with complex conjugation for one of the copies, the arbitrary complex phases that apply to wave functions does not appear in the density operator form. This also applies to multiparticle states.

    When you swap two particles in a two particle state it changes the signs of both the wave function and its complex conjugate. Consequently, their product gets multiplied by (-1)(-1) = 1 and is unchanged.

    Therefore, the distinguishing characteristic between bosons and fermions for density operators is not a Pauli principle, but instead is an energy principle; the energy of multiparticle fermion states goes to infinity when the particles are in the same state.

  7. #7 Ed
    May 23, 2009

    I’ve been studying physics for many years now, and I’d never heard the Exclusion Principle put in terms of a pressure like that. Fantastic! Now I have a wonderful vision in my head of the equations of state changing for electrons in orbital shells as atoms are being squeezed together.

    I suppose an increase in physical pressure on a substance corresponds to an increase in the probability that two fermions are in the same state. (…along with the many other ways that the atoms can absorb the force.)

  8. #8 cilt bakimi
    May 24, 2009

    Thank you for the summary we are going some justin tv for some one its really good…

  9. #9 Aaron
    May 31, 2009

    You can also use the Pauli principle to account for the anomalous 1/N! one has to introduce in the partition function for indistinguishable particles in stat mech. One also has to invoke the density matrix, but it’s not very hard to get from there.

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