Particle interchange and z component of spin

LostConjugate

The one thing that is NOT identical for different particles in a state is the z component of spin, having a probability of being positive or negative.

So when interchanging particles how do you handle the z component of the spin, it simply can't be interchanged along with the position and total spin as you could end up with a different state.

clem

If it is [tex]\uparrow\downarrow[/tex], you do get a different state.
Symmetric and antisymmetric combinations are eigenstates of total spin.

LostConjugate

So how can the original hypothesis hold. What happens to the z component of spin, it must be transfered to the swapped particle.

clem

What is "the original hypothesis"?
When interchanging particles, all properties are interchanged.

SpectraCat

His point was that if the spin directions are opposite, then the particles are not identical. However of course that is not an issue, because any fermionic wavefunction must be properly (anti)symmetrized. So you never have a state like, [tex]\uparrow\downarrow[/tex] rather you have states like [tex]\frac{1}{\sqrt{2}}(\uparrow\downarrow - \downarrow\uparrow)[/tex] which are properly symmetrized.

[EDIT] that is a very simple example of how the Pauli exclusion principle is applied, and also why it is required.

matonski

Everything is interchanged. Indeed, naively, you could end up with a different state. However, those states are just not allowed. The (anti) symmetrization requirements of identical particles severely restricts the possible multiple particle states in the Hilbert Space.

LostConjugate

Oh I do see here that not all states satisfy the interchange hypothesis. Only combinations of states. I still do not see how this just makes it OK to swap particles when there is some probability that they have different spin up/down states, physically it is not logical.

SpectraCat

You are not thinking about it correctly then ... the whole process of symmetrizing the states is to create wavefunctions where you CANNOT tell when the particles have been exchanged. If you have an equal mix of |up>|down> and |down>|up>, and you flip the spins, you will have and equal mix of |down>|up> and |up>|down> ... nothing has changed.

LostConjugate

Ok, so the idea is to carefully select only wave functions where particles can be exchanged.

An equal mix of up/down states for a large number of particles may work well.

What happens when you only have two particles that are involved in the wave function?

SpectraCat

The case I have been describing IS a two particle wavefunction.

LostConjugate

This puts each particle in a perfect superposition with equal probabilities which then makes "swapping" them non-trivial. Isn't that just a mathematical loop hole? Physically each particle could have opposite spins if measured.

SpectraCat

The "mathematical loophole" I described demonstrates how a wavefunction for a two-particle state with paired spins can conform to the Pauli exclusion principle, which we know from experimental evidence must be true. Identical fermions obey Fermi-Dirac statistics (another experimental fact) ... therefore writing a wavefunction for a fermion state that doesn't conform to those experimental facts doesn't make any sense. I cannot prove to you that the properly symmetrized wavefunction has any experimental reality .. I can only say that it is consistent with available experimental results, and has never been shown to be wrong.

For that example, they *will* have opposite spins when measured ... what does that have to do with anything? Once you measure the state, you "collapse" the wavefunction, so only one of the two possible components of the superposition can be observed. That is why we describe fermions in states like the one I gave as "spin-entangled".

LostConjugate

That helps clear it up.


I suppose because it is uncomfortable knowing that even without measurement you could be swapping spins that are not identical.