How is cosine squared used in measuring particle properties?

[thenewmans]

I have a few questions: What’s that formula called? The correlation formula? Is cos^2 the formula for only particles with spin=1 like photons? If so, what’s used for electrons (1/2 spin). Also, cos^2 seams funny to me since it returns 1 for 0 and 180 degrees. Zero I can understand since you should expect complete correlation if you measure from the same angle each time. But I would expect something like a -1 correlation from the opposite side. In that case, if one measurement is spin up the other is spin down.

 

[muppet]

Sorry... it's really unclear what you're asking here. You appear to be talking about the probabilities associated with the measurement of the spins of entangled particles? Nothing I can immediately get my hands on mentions cos squareds I'm afraid.

 

[thenewmans]

Yeah, I thought there might have been a problem. For background, I have read a few times here about using cos^2(theta) to predict how often the measurements of Alice and Bob are the same for entangled photons. If they take their measurements from the same angle, the correlation should be quite high. But if the interments are offset by a specific angle, then you can use that angle in this formula to predict the correlation. I only just found some http://en.wikipedia.org/wiki/Local_hidden_variable_theory#Bell_tests_with_no_.22non-detections.22" but they don't look like cos^2 and I don't see any formula as nice and neat as that either.

 

[DrChinese]

This basic cos^2(theta) formula for light was discovered around 1807 by Malus. For electrons, the formula is cos(theta).

 

[muppet]

And that would be why I couldn't find such a cos-squared forumula for Aspect -type experiments...
Thanks DrChinese.

 

[QuantumBend]

Cos squares theta come from quantum matrix in bras and kets. Mallus classically described results by fit equation from practicals. If you know math you do it.

 

[JesseM]

And that would be why I couldn't find such a cos-squared forumula for Aspect -type experiments...
Thanks DrChinese.

But see some old comments by DrChinese here, there are important differences between the classical and quantum versions of Malus' law. The classical version of Malus' law compares the polarization angle of the light emitted by the source with the angle of the polarizer, and it gives you the fraction of the energy that makes it through the polarizer. On the other hand, the quantum version compares the angles of two different polarizers measuring two entangled particles in an Aspect type experiment, and it tells you the probability they will both give the same spin result when measured with their respective polarizers (both spin-up or both spin-down). It would be impossible to replicate the quantum version in a local realist universe.

 

[thenewmans]

DrChinese and JesseM (or anyone else listening in), please tell me if I understand this correctly. I realize now that these formulas depend on what property you’re trying to measure. For light, that’s polarization. And I can see how you’d get a 1 or a 0 but not a -1 by imagining an experiment. Let’s say a detector dings 100 times a minute from a light source that is polarized vertically. I can verify this by adding a polarizer aligned vertically and it still dings 100 times. Now I turn the polarizer 30 degrees and the detector dings 75 times a minute (cos(30)^@=.75). That to me is like 75 ones and 25 zeros. At 60 degrees, it dings 25 times (cos(60)^2=.25).

Question:
For electrons, I don’t know what we’re detecting. I mean I know it’s spin but I don’t know what that physical property is that’s like polarization. Still, I know I get a 1 (spin up) or a -1 (spin down). And that’s why the correlation coefficient is cos(theta). I keep imagining a TV tube with an electron gun in it.

 

[JesseM]

Question:
For electrons, I don’t know what we’re detecting. I mean I know it’s spin but I don’t know what that physical property is that’s like polarization. Still, I know I get a 1 (spin up) or a -1 (spin down). And that’s why the correlation coefficient is cos(theta). I keep imagining a TV tube with an electron gun in it.

For an electron you're detecting whether it's deflected upward or downward by a Stern-Gerlach device oriented at a particular angle--there's a helpful explanation here. In this case cos^2 gives you the probability that a pair of entangled electrons will be deflected in opposite directions by their respective SG devices. So if both SG devices are at the same angle, they'll be deflected in opposite directions with probability 1, indicating opposite spins at that angle. If the angle between the devices is 60 degrees then there's an 0.25 chance they'll be deflected at opposite angles, and so forth.

 

[DrChinese]

DrChinese and JesseM (or anyone else listening in), please tell me if I understand this correctly. I realize now that these formulas depend on what property you’re trying to measure. For light, that’s polarization. And I can see how you’d get a 1 or a 0 but not a -1 by imagining an experiment. Let’s say a detector dings 100 times a minute from a light source that is polarized vertically. I can verify this by adding a polarizer aligned vertically and it still dings 100 times. Now I turn the polarizer 30 degrees and the detector dings 75 times a minute (cos(30)^@=.75). That to me is like 75 ones and 25 zeros. At 60 degrees, it dings 25 times (cos(60)^2=.25).

Keep in mind that for photons, you can also use a Polarizing Beam Splitter, which is somewhat analogous to the S-G apparatus for electrons. With a PBS, the photon comes out in one of 2 spots depending on its orientation. By labelling these arbitrarily as +1 or -1, you have the same basic results as an electron.