Measurement of two components of electron spin

[starfish99]

Quantum physics texts describe how it is impossible to measure S(z) and S(x) spin components of a single electron at the same time where z represents the z axis,etc.

What would happen if you try to trick nature by using a correlated pair of electons, emitted from an ion, moving in opposite directions.
First measure the z component of the spin of electron 1. Because of correlation we now know the z component of spin of electron 2.
We now measure the x component of the spin of electron 2
For electron 2 we know its z component and x component of spin , having made only one measurement on it.

Is this possible?
Does quantum physics have a way of forbidding this to happen?

 

[arkajad]

There is nothing in quantum theory that forbids making simultaneous measurements of two different spin components. It is only that the standard QM has nothing to say about the pattern of the results of such measurements. For this you need to look somewhere else.

 

[zonde]

Quantum physics texts describe how it is impossible to measure S(z) and S(x) spin components of a single electron at the same time where z represents the z axis,etc.

What would happen if you try to trick nature by using a correlated pair of electons, emitted from an ion, moving in opposite directions.
First measure the z component of the spin of electron 1. Because of correlation we now know the z component of spin of electron 2.
We now measure the x component of the spin of electron 2
For electron 2 we know its z component and x component of spin , having made only one measurement on it.

Is this possible?
Does quantum physics have a way of forbidding this to happen?

First I would advise to discuss photon polarization measurement if you want to talk about entangled particles. Because photon polarization entanglement experiments are performed routinely but electron spin entanglement experiment has never been performed.

So if we speak about measuring polarization for entangled photons then by measuring one photon you know the polarization of other photon only if you measure it in the same base as it was created (say horizontal/vertical). Then if you measure other photon in base that is non-commuting with creation base (say horizontal+45°/horizontal-45°) then you know nothing about polarization of other photon. However in that case if you perform phase measurement after polarization measurement then you know something about possible behavior of first particle if such complex measurement would be performed with it.
If you look at descriptions of actual experiments then you would notice that different polarization modes should be indistinguishable. That is characteristic requirement for interference i.e. phase measurement. If you do not fulfill that requirement then observed result reduces to product state that gives correlated results for polarization measurements in creation base (H/V) and uncorrelated results for polarization measurements in that other base (+45/-45).

QM formalism however treat both measurements in different bases equally as they produce correlated results just the same (if indistinguishability condition is fulfilled).

 

[DrChinese]

Quantum physics texts describe how it is impossible to measure S(z) and S(x) spin components of a single electron at the same time where z represents the z axis,etc.

What would happen if you try to trick nature by using a correlated pair of electons, emitted from an ion, moving in opposite directions.
First measure the z component of the spin of electron 1. Because of correlation we now know the z component of spin of electron 2.
We now measure the x component of the spin of electron 2
For electron 2 we know its z component and x component of spin , having made only one measurement on it.

Is this possible?
Does quantum physics have a way of forbidding this to happen?

Yours is the argument originally made (in a slightly different form) by EPR in 1935.

That logic assumes that the entangled particles are independent, a fact which is subject to dispute. In fact, experiments (a la Bell, Aspect) show that they are not. Accordingly, the experimental setup you propose does not supply any more information about a particle than the Heisenberg Uncertainty Principle allows.

You cannot beat the HUP!

 

[arkajad]

You cannot beat the HUP!

HUP, if you carefully and without prejudices follow its mathematical derivation, does not forbid such measurements. It tells us only what will be the product of root mean square deviations for an infinite series of measurements, without specifying whether the measurements are simultaneous or not. What is required is that the state is the same in each of these measurement. All the rest is interpretation and extrapolation of the principle that some physicists follow and some not.

 

[Fredrik]

(The particles I'm talking about in this post are all spin-1/2 particles).

I don't know what it would even mean to measure two components of spin simultaneously. Consider a Stern-Gerlach apparatus. The silver atom passes between two magnets, which produce an inhomogeneous magnetic field. If we insert two more magnets, identical to the first pair, but rotated 90 degrees and placed on the left and on the right instead of above and below the particle's path, the result isn't a device that measures both components simultaneously. The fields would add up, and the result would be a device that might be able to measure the spin component in the x+z direction, but it certainly wouldn't be able to measure the x or z components.

This is what it means for two observables to be incompatible. The corresponding measuring devices would interfere with each other, and at best end up measuring one observable which isn't either of the two that the devices were supposed to measure.

It's probably true that the uncertainty relation doesn't forbid a simultaneous measurements of these two observables (I haven't really thought about it), but that doesn't mean that QM allows it. The uncertainty relation is just a theorem about one aspect of QM. An eigenstate of S_x is a superposition of the two eigenstates of S_z. So if a particle has a well-defined value of S_x, it doesn't have a well-defined value of S_z. A measurement of an observable Q with result q, will by the axioms of QM, leave the system in a state such that an immediate second measurement of Q will have result q with probability 1 (assuming that the system wasn't destroyed by the first measurement). So a "simultaneous measurement" of S_z and S_x with result "+1/2 and +1/2" must leave the system in a state such that a subsequent measurement of either of these two observables yields +1/2 with probability 1. But there is no such state.

 

[unusualname]

Quantum physics texts describe how it is impossible to measure S(z) and S(x) spin components of a single electron at the same time where z represents the z axis,etc.

What would happen if you try to trick nature by using a correlated pair of electons, emitted from an ion, moving in opposite directions.
First measure the z component of the spin of electron 1. Because of correlation we now know the z component of spin of electron 2.
We now measure the x component of the spin of electron 2
For electron 2 we know its z component and x component of spin , having made only one measurement on it.

Is this possible?
Does quantum physics have a way of forbidding this to happen?

The 2nd measurement is not simultaneous with the first, so what's the problem? After the 2nd measurement the first one is no longer valid. Similarly, you could just measure a particle's momentum and then say "now if I measure its position I've beaten HUP", but you haven't because after measuring the position the momentum measurement is no longer valid.

 

[starfish99]

So according to Dr. Chinese you can't fool nature:

Measurement of the z component of the spin of electron 1 affects the z component of the spin of electron 2 in a correlated pair of electrons. Subsequent measurement of the x component's spin of electron 2 is trying to "cheat" ,i.e the electron 2 spin z component was already "measured", although this measurement was made indirectly. Is this correct?

 

[DrChinese]

So according to Dr. Chinese you can't fool nature:

Measurement of the z component of the spin of electron 1 affects the z component of the spin of electron 2 in a correlated pair of electrons. Subsequent measurement of the x component's spin of electron 2 is trying to "cheat" ,i.e the electron 2 spin z component was already "measured", although this measurement was made indirectly. Is this correct?

Yes, exactly. And when you do something like measure z of Alice (electron 1) and x of Bob (electron 2), and THEN measure x of Alice and z of Bob: these measurements will NOT be matching the earlier pair any more than to a random degree. This shows you did not really accomplish anything by that action. No net new information gained as you might have classically expected.