Measurement and Causality part 1

olegt comments:

Techne,

I am not Anon, but I can help by defining what the problem is. We will strip the phenomenon of quantum indeterminacy to its most basic form if we consider a system with the tiniest possible phase space, a Hilbert space with a basis of two states.

The spin of an electron is such a physical system. Prepare an electron in a state with spin pointing along the vertical axis. The spin is in a well-defined state (a pure state in technical terms): if we subsequently measure the spin along that axis, it always points up the axis and never down. We have complete knowledge about the system. Its entropy is zero. There is no more to be learned.

Turn the measuring apparatus and measure the prepared electron spin along a horizontal axis. Now the outcome is undetermined and in fact, completely unpredictable: it can point right or left with equal probabilities.

My question to you is: What caused the spin prepared in a state “up” to choose the specific direction, say “right”? So far as we know, there is no underlying cause. It wasn’t like we had incomplete information prior to the measurement and that we could complete the picture by identifying a missing causal chain. The electron spin was in a pure state, which means that we knew as much about it as physically possible. There was no lack of knowledge.

This case amply illustrates the limited character of classical logic developed on the basis of our interactions with the macroscopic world. Things are quite different in the microscopic world. Like energy, knowledge turns out to be grainy and furthermore, knowing some things makes other things unknowable. Heisenberg’s uncertainty principle is the most famous example of that, but the direction of electron spin, although less known to the lay audience, is the most striking. Bell’s inequalities confirmed experimentally, further confirm the lack of causal chains in such measurements. Events with space-like separation are not causally connected.

Classical logic is simply not equipped to deal with phenomena of this sort.

Before delving into this interesting topic I just want to make sure we are both on the same page and/or we are both talking about the same stuff and have a similar understanding. I will draw a diagram, hopefully it helps.

As far as I understand, the following things are described:
1) An electron in a state with spin pointing along the vertical axis is prepared.
2) Measuring the spin along the vertical axis always results in spin pointing along the vertical axis.
3) Measuring the spin along the horizontal axis always results in spin pointing either the vertical axis or the horizontal axis with equal probabilities.

Now to my understanding, it can be summarised as follows in the following figure.

Experimental setup

If I understand it correctly:
I) At time zero (T0) an electron (X1) in a state with spin pointing along the vertical axis is prepared. If this electron is measured at T=0+x1 with a vertically arranged detector (A) the spin of the electron will always point along the vertical axis.
II) At time zero (T0) an electron (X2) in a state with spin pointing along the vertical axis is prepared. If this electron is measured at T=0+x2 with a vertically arranged detector (B) the spin of the electron will always point along the vertical axis.
III) At time zero (T0) an electron (X3) in a state with spin pointing along the vertical axis is prepared. If this electron is measured at T=0+x1 and T=0+x2 and with the vertically arranged detectors (A and B) the spin of the electron will always point along the vertical axis.

IV) At time zero (T0) an electron (X4) in a state with spin pointing along the vertical axis is prepared. If this electron is measured at T=0+x1 with a horizontally arranged detector (C) the spin of the electron will always point either along the vertical or the horizontal axis with about equal probability.
V) At time zero (T0) an electron (X5) in a state with spin pointing along the vertical axis is prepared. If this electron is measured at T=0+x2 with a horizontally arranged detector (D) the spin of the electron will always point either along the vertical or the horizontal axis with about equal probability.
VI) At time zero (T0) an electron (X6) in a state with spin pointing along the vertical axis is prepared. If this electron is measured at T=0+x1 and T=0+x2 and with the horizontally arranged detectors (C and D) the spin of the electron will always point either along the vertical or the horizontal axis with about equal probability.

VII)At time zero (T0) an electron (X7) in a state with spin pointing along the vertical axis is prepared. If this electron is measured at T=0+x1 with the horizontally arranged detector (C) the spin of the electron will always point either along the vertical or the horizontal axis with about equal probability and when subsequently measured at T=0+x2 with the vertically arranged detector (B) the spin of the electron will always point along the vertical axis.
VIII) At time zero (T0) an electron (X8) in a state with spin pointing along the vertical axis is prepared. If this electron is measured at T=0+x1 with the vertically arranged detector (A) the spin of the electron will always point along the vertical axis and when subsequently measured at T=0+x2 with the horizontally arranged detector (D) the spin of the electron will always point either along the vertical or the horizontal axis with about equal probability.

olegt, is this an accurate way representing understanding the results from real world experiments? Can I use this example for future discussions (part 2)?

UPDATE 1

Experimental setup new

Would it be more accurate if described as above?

Then it would read:

I) At time zero (T0) an electron (X1) with S_z = +1/2 (“a” in the picture) is prepared. If this electron is measured at T=0+x1 with a vertically arranged detector (A) it will still be an electron with S_z = +1/2.
II) At time zero (T0) an electron (X2) with S_z = +1/2 (“a” in the picture) is prepared. If this electron is measured at T=0+x with a vertically arranged detector (B) it will still be an electron with S_z = +1/2.
III) At time zero (T0) an electron (X3) with S_z = +1/2 prepared. If this electron is measured at T=0+x1 and T=0+x2 and measured with the vertically arranged detectors (A and B) the spin of the electron will always point along the vertical axis it will still be an electron with S_z = +1/2.

IV) At time zero (T0) an electron (X4) with S_z = +1/2 (“a” in the picture) is prepared. If this electron is measured at T=0+x1 with a horizontally arranged detector (C) it will be an electron with either S_x = +1/2 (“c” in the picture) or S_x = +1/2 (“d” in the picture).
V) At time zero (T0) an electron (X5) with S_z = +1/2 (“a” in the picture) is prepared. If this electron is measured at T=0+x2 with a horizontally arranged detector (D) it will be an electron with either S_x = +1/2 (“c” in the picture) or S_x = +1/2 (“d” in the picture).
VI) At time zero (T0) an electron (X5) with S_z = +1/2 (“a” in the picture) is prepared. If this electron is measured at T=0+x1 and T=0+x2 and with the horizontally arranged detectors (C and D) it will be an electron with either S_x = +1/2 (“c” in the picture) or S_x = +1/2 (“d” in the picture).

VII) At time zero (T0) an electron (X7) with S_z = +1/2 (“a” in the picture) is prepared. If this electron is measured at T=0+x1 with the horizontally arranged detector (C) it will be an electron with either S_x = +1/2 (“c” in the picture) or S_x = +1/2 (“d” in the picture) and when subsequently measured at T=0+x2 with the vertically arranged detector (B) it will still be an electron with S_z = +1/2.
VIII) At time zero (T0) an electron (X8) with S_z = +1/2 (“a” in the picture) is prepared. If this electron is measured at T=0+x1 with the vertically arranged detector (A) it will still be an electron with S_z = +1/2 and when subsequently measured at T=0+x2 with the horizontally arranged detector (D) it will be an electron with either S_x = +1/2 (“c” in the picture) or S_x = +1/2 (“d” in the picture).

Is this more accurate?

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11 Responses to Measurement and Causality part 1

  1. olegt says:

    IV is wrong. When you measure the projection of the spin onto a given axis you get either a positive or a negative value.

    Let’s call the vertical axis z and the horizontal one x to facilitate the discussion. If you prepare an electron with its spin pointing along the positive z-axis, we say that the projection of the spin onto the z-axis is S_z = +1/2. When you measure S_z in the so prepared state you always get S_z=+1/2 so long as the electron spin was undisturbed.

    If you measure S_z = +1/2, subsequent measurements will yield S_z = +1/2, never S_z = -1/2.

    However, if after preparing an electron spin in the S_z = +1/2 state, and perhaps measuring S_z several time just to be sure that it is sitting there undisturbed, you now decide to measure the projection of spin onto the x-axis. When you measure S_x you get +1/2 or -1/2. If you have a whole battery of electrons in pure S_z=+1/2 states, you will find that measuring S_x yields +1/2 and -1/2 in equal proportions.

  2. olegt says:

    Your statement IV implied that measurements described in the last paragraph of my previous comments reveal the spin to be either in the S_z=+1/2 or in the S_x=+1/2 state. That is incorrect. We are measuring S_x and the measurements yield S_x=+1/2 or S_x=-1/2.

  3. hylemorphist says:

    Thanks, I’ve updated the scheme. Is it more accurate now?

  4. olegt says:

    I see no difference between IV, V, and VI. These are all electron spins prepared in the same initial state with S_z = +1/2 and with S_x measured subsequently once. Why would you like to include three identical cases?

    VII is wrong. If you prepare an electron spin in a state with S_z = +1/2 and leave it undisturbed (its wave function stays unchanged), all subsequent measurements of S_z will yield +1/2. If instead you measure S_x then the outcome is either -1/2 or +1/2. Following that measurement, the electron spin is in the state with S_x = +1/2 (say) and any subsequent measurement of S_x will invariably produce the same result, S_x = +1/2.

    However, now that S_x is well defined, S_z is undeterminate. Its measurement will show either +1/2 or -1/2 (with equal probabilities). Once you have measured S_z, it will again be well defined and then S_x will be entirely random.

  5. hylemorphist says:

    Ok, one more time.
    If at T0: Sz=+½ then Sz will be +½ at T0+a (a=infinity with no other interference)
    If at T0: Sz=+½ then Sx will be ±½ at T1. Sx is indeterminate before T1.

    However:
    If at T0: Sz=+½ and at T1 Sx=-½ (after measurement) then Sx will be -½ at T1+a (a=infinity with no other interference) and Sz will be ±½ at T2. Sz is indeterminate before T2.

    Better now? Does this describe the problem more accurately?

  6. olegt says:

    This seems all right.

  7. hylemorphist says:

    And this is what you would say is a problem for the Scholastic view of causality including the principle of causality as described over @ TT?

  8. olegt says:

    Those were preliminaries.

    Here is the question: What causes S_x to take on the specific value in experiments IV, V, and VI?

  9. hylemorphist says:

    There does not appear to be anything that refutes the Scholastic view of causality or the principle of causality. In other words there is no reason to all of a sudden abandon it. I’ll write a few things about this in part 2 some time during the week.

  10. Pingback: Measurement and Causality part 2 | hylemorphist

  11. Pingback: Quantum Physics vs The Principle of Causality - Telic Thoughts

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