A course in consciousness
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Interesting :)
4.2. Schrödinger’s cat paradox
This thought experiment was originally created by Schrödinger in an attempt to show the possible absurdities if quantum theory were not confined to microscopic objects alone. (Since then, nobody has succeeded in showing that quantum theory actually is absurd.) Schrödinger thought the wave properties of the microworld could be transmitted to the macroworld if the former is coupled to the latter.
Imagine a closed box containing a single radioactive nucleus and a particle detector such as a Geiger counter (see drawing above). We assume this detector is designed to detect with certainty any particle that is emitted by the nucleus. The radioactive nucleus is microscopic and therefore can be described by quantum theory. Suppose the probability that the source will emit a particle in one minute is 1/2=50%. The period of one minute is called the half-life of the source. (See the animation of the radioactive decay of "Balonium" at http://www.upscale.utoronto.ca/PVB/Harrison/Flash/Nuclear/Decay/NuclearDecay.html.)
Since the wavefunction of the nucleus is a solution to the Schrödinger equation and must describe all possibilities, after one minute it consists of a wave with two terms of equal amplitude, one corresponding to a nucleus with one emitted particle, and one corresponding to a nucleus with no emitted particle, both measured at the same point in space:
y = y1 (particle) + y2 (no particle)
where, for simplicity, we again assume the wavefunctions are real rather than complex. Now, y12 is the probability that a measurement would show that a particle was emitted, and y22 is the probability that it would show that no particle was emitted. (We shall see below that the interference term 2y1y2 in y2 does not contribute to the final observed result.)
The remaining items in the box are all macroscopic, but because they are nothing more than collections of microscopic particles (atoms and molecules) that obey quantum theory, we assume they also obey quantum theory.
[Technical note: If macroscopic objects do not obey quantum theory, we have no other theory to explain them. For example, classical physics cannot explain the following semi-macroscopic and macroscopic phenomena: 1) Interference fringes (Section 4.1) have been directly produced with buckminsterfullerenes ("buckyballs") consisting of 60 carbon atoms and 48 fluorine atoms (C60F48, http://arxiv.org/PS_cache/quant-ph/pdf/0309/0309016v1.pdf). 2) A superconducting quantum interference device (SQUID) containing millions of electrons was made to occupy Schrödinger's cat states (http://www.sciencemag.org/cgi/content/full/287/5462/2395a). 3) Ferromagnetism, superconductivity, and superfluidity all are quantum phenomena which occur in macroscopic systems. 4) The period of inflation in the early history of the universe is thought to be quantum mechanical in origin (see the excellent lectures in cosmology at http://abyss.uoregon.edu/~js/cosmo/lectures/).]
Hence, we assume the Geiger counter can also be described by a wavefunction that is a solution to the Schrödinger equation. The combined system of nucleus and detector then must be described by a wavefunction that contains two terms, one describing a nucleus and a detector that has detected a particle, and one describing a nucleus and a detector that has not detected a particle:
y = y1(detected particle) + y2(no detected particle)
Both of these terms must necessarily be present, and the resulting state y is a superposition of these two states. Again, y12 andy22 are the probabilities that a measurement would show either of the two states.
Put into the box a vial of poison gas and connect it to the detector so that the gas is automatically released if the detector counts a particle. Now put into the box a live cat. We assume that the poison gas and cat can also be described by the Schrödinger equation. The final wavefunction contains two terms, one describing a detected particle, plus released gas and a dead cat; and one describing no detected particle, no released gas, and a live cat. Both terms must be present if quantum theory can be applied to the box’s contents. The wavefunction must describe both a dead cat and a live cat:
y = y1(detected particle, dead cat) + y2(no detected particle, live cat)
After exactly one minute, you look into the box and see either a live cat or a dead one, but certainly not both! What is the explanation?
Schrödinger considered the possibility that until there is an observation, there is no cat, live or dead! There is only a wavefunction. The wavefunction merely tells us what possibilities will be presented to the observer when the box is opened. The observation itself manifests the reality of either a live cat or a dead cat (this is called observer created reality).
Now we must ask why the observer him/her self is not included in the system described by the Schrödinger equation, so we put it in the following equation:
y = y1(detected particle, observer sees dead cat) + y2(no detected particle, observer sees live cat)
If we square this expression, as in Eq. 1, we obtain
y 2 = (y1 + y2) 2 = y1 2 + 2y1y2 + y2 2
We know that the observer observes only a live or a dead cat, not a superposition. That means that the interference term 2y1y2does not contribute to the observation. Why doesn't it? Schrödinger did not have the benefit of extensive theoretical research done in the last few decades. This has shown that, because in practice it is impossible to isolate any macroscopic object from its environment, we must include its effects in this equation. Environmental effects include all of the interactions between the rest of the universe and everything in the experiment including the detector, the poison gas bottle, the cat, the box, and the observer. When such effects are included and averaged over, the interference term gets averaged out, leaving only
y 2 = (y1 + y2) 2 = y1 2 + y2 2 (Eq. 2)
Without the interference term, Eq. 2 no longer describes the superposition of a dead cat and a live cat. Superficially, it is similar to the description of classical objects like bullets as was discussed above Fig. 2. In the classical case, before an observation the cat is real but either alive or dead. The probabilities represent only our ignorance of the actual case. However, in the quantum case, before an observation there is no cat, live or dead. There is only a wavefunction that represents the possibilities that will be manifested when an observation is made.