Wavefunction collapse and antisymmetry

In summary: Cannot we have up up> result sometime after?No, because that would violate the Pauli exclusion principle.In summary, the conversation is about the concept of quantum entanglement and the implications of measurement on the wave function of a system. There is a discussion about the singlet state and the possibility of obtaining an "up-up" result after measurement, with the conclusion that this would violate the Pauli exclusion principle. The conversation also touches on the accuracy of information presented in a book on the philosophy of science.
  • #1
Amok
256
2
Hey guys,

I was reading a book about the philosophy of science, and in the chapter about QM the author uses a well known example in order to explain quantum entaglement and illustrate the non-separability of individual system in QM. He describes a system composed of two spin-1/2 particles. Said system is in a singlet state:

[tex] \psi_{12} = \frac{1}{\sqrt{2}} (\psi_{1}^{+} \otimes \psi_{2}^{-} - \psi_{2}^{+} \otimes \psi_{1}^{-})[/tex]

The subscripts indicate particle 1 or 2, and the subscipts spin up or down. This is clearly an antysymmetric. Now suppose these two particles are emitted from the same source, but go in different directions. When they are spatially separate, we do an experiment to measure the spin of one electron, and it comes up as plus, and we'd automatically know the spin of the other electron. So far so good. The author then goes on to say that in this case the wavefunction would collapse into:

[tex]
(\psi_{1}^{+} \otimes \psi_{2}^{-})
[/tex]

Which is really weird, because this wavefunction is not antisymmetric. How do you conciliate wavefunction collapse with the antisymmetric requirement of the w.f.?
 
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  • #3
Wow, took me quite a few minutes to understand that post, but that's great. So what you're saying is that we cannot really know whether it is electron one or two that is localized on the right side (or left). What we do know is that a certain orbital (spatial w.f.) is more localized on the right or the left. So we have to pair up the orbital on the right with with spin up, if we did indeed measure the rightmost electron to have spin up. So in my case, the initial w.f. is:

[tex] (\varphi_R(r_1) \otimes \varphi_L(r_2) + \varphi_L(r_1) \otimes \varphi_R(r_2)) [\alpha(s_1) \otimes \beta(s_2)-\beta(s_1) \otimes \alpha(s_2)][/tex]

Distributing:

[tex] \varphi_R(r_1)\alpha(s_1) \otimes \varphi_L(r_2)\beta(s_2) -
\varphi_R(r_1)\beta(s_1) \otimes \varphi_L(r_2)\alpha(s_2) +
\varphi_L(r_1)\alpha(s_1) \otimes \varphi_R(r_2)\beta(s_2)-
\varphi_L(r_1)\beta(s_1) \otimes \varphi_R(r_2)\alpha(s_2)[/tex]

So measuring that the electron on the right has up (alpha) spin, means the w.f. will collapse to:

[tex] \varphi_R(r_1)\alpha(s_1) \otimes \varphi_L(r_2)\beta(s_2) -
\varphi_L(r_1)\beta(s_1) \otimes \varphi_R(r_2)\alpha(s_2)[/tex]

i.e. we keep only the first and last term, because the pair up R with alpha. This w.f. is still antisymmetric. I think the statement in the book is actually inaccurate and leads to confusion, but maybe it's just used and example to discuss quantum entaglement (without considering the exclusion principle; it's not a physics book after all).

Thanks for replying :D
 
Last edited:
  • #4
Before measurement the system was in an antisymmetric singlet | J M> = |0 0>.
(the total spin is always null)

Is it still in an antisymmetric singlet after? (entangled?)
Cannot we have up up> result sometime after?
 
  • #5
This is most definitely NOT my area of expertise - but as I understand it, a singlet state is characterised by a complete lack of information about the spin of each particle, as opposed to a simply 'unmeasured' state where the spin has a value (which may already be known to another party).

After the collapse, the latter condition prevails so the electrons are no longer entangled.
 
  • #6
naima said:
Before measurement the system was in an antisymmetric singlet | J M> = |0 0>.
(the total spin is always null)

Is it still in an antisymmetric singlet after? (entangled?)
Cannot we have up up> result sometime after?


I don't know if you can label it a singlet, but you'd still have S=0 and Sz = 0. You can't have and up-up result unless you flip the spin of one of the electrons (i.e. go to one of the triplet states).
 
  • #7
look at a Clebsch Gordan table for 1/2 * 1/2 coupling: here
we see that up down > is in a superposition of the initial antisymmetric singlet and of the |1 0> triplet with equal probabilities.
I think this means that entanglement (|J M> =|0 0>) stops after measurement.
So your book was right.
I would be happy to read Demistifier's explanation
 
  • #8
naima said:
Before measurement the system was in an antisymmetric singlet | J M> = |0 0>.
(the total spin is always null)

Is it still in an antisymmetric singlet after?
Yes. As I have already explained (see post #2), you are missing the particle-position part of the wave function.

naima said:
(entangled?)
Entanglement may be destroyed by measurement.
 

Related to Wavefunction collapse and antisymmetry

1. What is wavefunction collapse?

Wavefunction collapse is a phenomenon in quantum mechanics where the superposition of multiple possible states of a system collapses into a single definite state when the system is observed or measured. This is sometimes referred to as the "observer effect."

2. How does wavefunction collapse relate to antisymmetry?

In quantum mechanics, the wavefunction of a system is described by a mathematical function that contains information about the possible states of the system. The principle of antisymmetry states that the wavefunction of a system containing identical particles must change sign when the positions of any two particles are exchanged. This is important because it results in the exclusion of certain states and affects the behavior of the particles.

3. Can wavefunction collapse be predicted or controlled?

No, wavefunction collapse is a random and unpredictable event in quantum mechanics. It cannot be controlled or predicted with certainty, only the probability of it occurring can be calculated using mathematical equations.

4. What are the implications of wavefunction collapse and antisymmetry on our understanding of reality?

The concept of wavefunction collapse and antisymmetry challenges our traditional understanding of reality, as it suggests that the behavior of particles is inherently probabilistic and unpredictable. It also raises questions about the role of observation and measurement in shaping our perception of reality.

5. Are there any practical applications of wavefunction collapse and antisymmetry?

Yes, wavefunction collapse and antisymmetry play a crucial role in the functioning of technologies such as transistors and lasers, which are based on the principles of quantum mechanics. They also have applications in fields such as quantum computing, cryptography, and superconductivity.

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