Particle interchange and z component of spin

In summary: 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.
  • #1
LostConjugate
850
3
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.
 
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  • #2
If it is [tex]\uparrow\downarrow[/tex], you do get a different state.
Symmetric and antisymmetric combinations are eigenstates of total spin.
 
  • #3
clem said:
If it is [tex]\uparrow\downarrow[/tex], you do get a different state.
Symmetric and antisymmetric combinations are eigenstates of total spin.
So how can the original hypothesis hold. What happens to the z component of spin, it must be transferred to the swapped particle.
 
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  • #4
LostConjugate said:
So how can the original hypothesis hold. What happens to the z component of spin, it must be transferred to the swapped particle.
What is "the original hypothesis"?
When interchanging particles, all properties are interchanged.
 
  • #5
clem said:
What is "the original hypothesis"?
When interchanging particles, all properties are interchanged.

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.
 
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  • #6
LostConjugate said:
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.
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.
 
  • #7
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.
 
  • #8
LostConjugate said:
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.

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.
 
  • #9
SpectraCat said:
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.

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?
 
  • #10
LostConjugate said:
What happens when you only have two particles that are involved in the wave function?

The case I have been describing IS a two particle wavefunction.
 
  • #11
SpectraCat said:
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.

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.
 
  • #12
LostConjugate said:
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.

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".
 
  • #13
SpectraCat said:
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.

That helps clear it up.


SpectraCat said:
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".

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

Related to Particle interchange and z component of spin

1. What is particle interchange?

Particle interchange refers to the process in which particles exchange their positions or properties with each other. This can occur in various physical systems, such as in quantum mechanics where particles can switch places within a multi-particle wave function.

2. What is the z component of spin?

The z component of spin is a quantum mechanical property that describes the intrinsic angular momentum of a particle along the z-axis. It is one of the three components of spin, along with the x and y components, and is measured in units of angular momentum (ħ).

3. How is particle interchange related to the z component of spin?

In quantum mechanics, the z component of spin is conserved during particle interchange. This means that the total z component of spin before and after the interchange remains the same. This conservation law is a fundamental property of particles and is related to their symmetries.

4. Can the z component of spin have any value?

No, the z component of spin can only take on discrete values in quantum mechanics. These values are determined by the spin quantum number, which can be either positive or negative half-integers (1/2, -1/2, 3/2, etc.). The z component of spin is always a multiple of this spin quantum number.

5. How does the z component of spin affect particle behavior?

The z component of spin plays a crucial role in determining the behavior of particles, particularly in magnetic systems. It can influence the orientation and energy levels of particles in a magnetic field, as well as their interactions with other particles. In some cases, the z component of spin can also affect the stability and decay of particles.

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