# Exploring EPR Spin & the Paradox of Measurement

• aaaa202
In summary, the epr paradox is that if you have two particles in a state where one has spin up and the other has spin down, measuring either spin does not always result in the particles' spins being antisymmetric.
aaaa202
The epr paradox is usually explained as something like:

Suppose you have two electrons in the singlet state (+=spin up, -=spin down):

lψ>= l+>l-> - l->l+>

Now if you measure the spin on the first electron the explanation is (I think) that this collapses one electron onto l+> or l-> such that the wave function becomes:

lψ>= l+>l-> (for measuring +)

But I don't understand this. How is this form of wave function consistent with the antisymmetrization requirement for the wavefunction for electrons?
Actually I'm also generally confused by how measurement in quantum mechanics is precisely defined. Suppose I want to measure the total spin. How does one do this? Is measurement of a quantity defined as the interaction of the quantity with an object large enough to exhibit classical behaviour.

aaaa202 said:
How is this form of wave function consistent with the antisymmetrization requirement for the wavefunction for electrons?

##\hat{P}_{12}|\psi \rangle = \frac{1}{\sqrt{2}}(|- \rangle |+\rangle - |+ \rangle |- \rangle ) = -|\psi \rangle##

aaaa202 said:
Is measurement of a quantity defined as the interaction of the quantity with an object large enough to exhibit classical behaviour.

Yes so in other words a classically described measuring apparatus that can always take on definite measurement values. In the case of spin we could use a position measurement apparatus that records the position of the deflected electron on the screen. However if you allow for the measuring apparatus itself to be described by a quantum state, such as ##\{|z_0 \rangle , |z_{\uparrow} \rangle, |z_{\downarrow} \rangle \}## where in order this would represent the measuring apparatus in the pre-measurement state, measured spin up state (position corresponding to upwards deflection), and measured spin down state (position corresponding to dowwnards deflection), then the measurement interaction gets rather interesting when the deflected electron is not in an eigenstate of say ##\hat{S}_z## when we make a measurement of ##S_z##.

Aaaa202, were you referring to the second expression rather than the first?

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Yes, I don't understand how the second expression fulfills the antisymmetrization requirement.

I thought that this requirement only applied to a system of indistinguishable particles.

But aren't electrons that?

Not when you have measured them and made a distinction between them. (i.e. This one is spin up and the other one is spin down).

aaaa202 said:
The epr paradox is usually explained as something like:

Suppose you have two electrons in the singlet state (+=spin up, -=spin down):

lψ>= l+>l-> - l->l+>

Now if you measure the spin on the first electron the explanation is (I think) that this collapses one electron onto l+> or l-> such that the wave function becomes:

lψ>= l+>l-> (for measuring +)

But I don't understand this. How is this form of wave function consistent with the antisymmetrization requirement for the wavefunction for electrons?

It is the total wave function that must be anti-symmetric, not just the spin part.

If you have two experimenters, and one particle is localized near one experimenter, and the other particle is localized near the other experimenter, then the total wave function can be factored into a spatial part and a spin part. If the spin part is anti-symmetric, then the spatial part must be symmetric.

The spin part of the joint wave function is anti-symmetric under particle exchange:
$| + \rangle | - \rangle \ -\ | - \rangle | + \rangle$

The spatial part of the joint wave function is symmetric:
$| \psi_A \rangle | \psi_B \rangle \ +\ |\psi_B \rangle | \psi_A \rangle$

where $\psi_A$ might be a spatial wave function localized near one experimenter (Alice), and $\psi_B$ might be a spatial wave function localized near the other experimenter (Bob).

So the total wave function would look like this:

$|\psi_A, + \rangle |\psi_B, - \rangle + |\psi_B, + \rangle |\psi_A, - \rangle - |\psi_A, - \rangle |\psi_B, + \rangle - |\psi_B, - \rangle |\psi_A, + \rangle$

Either particle can have spin-up or spin-down, and either particle can be localized at Bob, or at Alice.

Now, if Alice measures a spin-up electron, then the "collapse" would reduce the total wave function to:

$|\psi_A, + \rangle |\psi_B, - \rangle - |\psi_B, - \rangle |\psi_A, + \rangle$

Now, there is a correlation between the spatial part and the spin part. If the first particle is spin-up, then it is localized at Alice. If it is spin-down, it is localized at Bob, and similarly for the second particle. The total wave function is still anti-symmetric under particle exchange.

The rule of thumb, "if the particles are distinguishable, then the wave function does not need to be anti-symmetric" is sort of a short hand. If the particles are distinguishable through their spatial part, then you can always make a total wave function that is anti-symmetric. So there is no constraint that the spin parts alone be anti-symmetric. So it's as if they are distinguishable particles, and not subject to Fermi statistics. But that's a short cut. Fermi statistics still applies, but it no longer makes any observable constraints on spin.

1 person

## 1. What is EPR spin and how does it relate to quantum mechanics?

EPR spin, also known as Einstein-Podolsky-Rosen spin, is a phenomenon in quantum mechanics where two particles that are entangled have a correlated spin, regardless of how far apart they are. This means that if one particle's spin is measured, the other particle's spin will have a corresponding value, even if they are separated by large distances.

## 2. What is the paradox of measurement in EPR spin?

The paradox of measurement in EPR spin refers to the seemingly contradictory relationship between quantum mechanics and classical mechanics. In quantum mechanics, particles can exist in multiple states simultaneously, while in classical mechanics, particles have definite properties at all times. This paradox arises when trying to measure the spin of entangled particles, as it is impossible to predict the outcome of the measurement without collapsing the wavefunction and determining a definite state.

## 3. How is EPR spin being explored in current research?

Scientists are currently exploring EPR spin in various ways, including testing the Bell inequality and conducting experiments using entangled photons. Additionally, researchers are using advanced techniques such as quantum teleportation to better understand the nature of EPR spin and its implications for quantum mechanics.

## 4. What is the significance of EPR spin in quantum technology?

EPR spin has significant implications for quantum technology, particularly in the development of quantum computers. The ability to entangle particles and measure their spin could potentially lead to faster and more complex computing capabilities. It also has potential applications in secure communication systems, as any attempt to intercept the entangled particles and measure their spin would be immediately detectable.

## 5. Is there a resolution to the paradox of measurement in EPR spin?

There is ongoing debate and research surrounding the paradox of measurement in EPR spin. Some scientists believe that it may be possible to resolve the paradox by incorporating elements of both quantum mechanics and classical mechanics. Others suggest that the paradox may indicate a need for a new theory that can reconcile the two conflicting perspectives.

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