The Repeatablility of Quantum Mechanical Measurements and Wavefunction Collapse.

[eucleia]

Blarghlksfla... This is the typed out version of a hand-written essay I wrote for Quantum Mechanics class at uni. Believe it or not, the hand-written version took less time. D8 I'm sticking this on because it will hopefully explain some very interesting aspects of Physics to you without going into too much lingo. That is, you can all impress your friends by understanding and being able to explain to them what that paradox of Schodinger's Cat really pertains to. :3

 

The Repeatability of Quantum Mechanical Measurements and Wavefunction Collapse.

 

            It has long been a problem and puzzle in physics to equate the really big with the really small – the macroscopic and microscopic worlds. Indeed this problem is yet to be solved in a consistent manner, but there was a time the very small could barely be understood at all.

            Quantum Mechanics was initially developed to provide a better explanation of the atom, the “quantum” (term being derived from Latin, meaning “how much”) theory of the atom being developed as an explanation for the electron’s staying in it’s orbital, which was failed to be explained using the Classically based Newton’s Laws of Motion, and Maxwell’s Theory of Electromagnetism.

            Already the Quantum and Classical world can be seen to be very different, but one notable difference in particular is that the act of measurement itself may change or distort the value of the outcome, unlike in Classical Mechanics. After all, where in Classical the macroscopic objects are unaffected (to any degree that would render the disturbances negligible) by the bombardment of photons required to measure the parameters, in Quantum the photons themselves may be comparable to the size of the objects in question and thus may change the value being measured.

            As strange as the Quantum Mechanical world may seem, there appear to be plausible explanations, or at least attempts, to explain Quantum phenomena. While x and p (the positioning and momentum of an object respectively) in the Quantum world cannot be observed or known with certainty simultaneously i.e. noncommutable, a fact which gives rise to what is known as the Heisenberg Uncertainty Principle, one may obtain a probability Ψ as to how likely a particle may be found in a particular location at a certain time. So while the wavefunction describing a system/event/particle may be meaningless and unobservable, the probability density |Ψ|² can be obtained.

            Tied into the ideas of probability and how measurement itself may be capable of altering a result is the phenomenon known as “wavefunction collapse”. Bearing in mind that a wavefunction Ψ is a description or mathematical representation of a ‘mixed’ state system, wavefunction collapse occurs when all other terms in the expansion of Ψ have vanished/collapsed into nothing.

            As one of two methods by which Quantum systems may evolve, wavefunction collapse can also be brought about by the measurement of a system. In this case, the act of measurement brings about a change of state in the system, which collapses from a mixed state to a ‘pure’ one. In the case of a particle in which a mathematical representation is used to calculate the probability for it to be found in a location or state of motion, and the act of measurement causes the calculated set of probabilities to collapse. The case is referred to as the Copenhagen Interpretation. The nature of Copenhagen Interpretation can be demonstrated by the paradox of “Schrödinger’s Cat”.

            Suppose a cat is put in a box with some generic radioactive substance and a radiation detector e.g. a Geiger counter. Since one can never explicitly tell when a radioactive particle will decay, we can only have a “half-life”, which is the period of time in which there is a 50% chance that the particle will decay, and a particle will be emitted (and thus detected). If a particle is detected, poisonous gas will be released and the cat killed. Since the cat is inside the box and the observer cannot see inside it (assume that said observer cannot look at the Geiger counter) then they have no idea whether the cat is dead or alive at a given time, only a probability, the wavefunction being a reflection of the observers knowledge of said system as per Copenhagen’s Interpretation.

            By this interpretation only opening the box will tell whether the cat is alive or dead (or by looking at the Geiger counter and checking to see if it has detected the decay). In the event of opening the box and finding the cat still alive, well then it is clearly not dead to the observer and the possibility of it being dead disappears. Perhaps more clearly, if one were to look at the Geiger counter or open the box and find the cat dead, then the act of knowing with certainty that the cat will be dead inside the box effectively collapses it (meaning Ψ) into a pure state in which the cat is dead. Previously, until guaranteed detection, the observer only had a probability. So in essence, the act of detection collapsed Ψ into a pure state, after which it remains in that state. After all, a cat that is dead can not come back to life. Unless the laws of biology have somehow been violated.

            This leads us into the repeatability of Quantum Mechanical measurements. Again taking Schrödinger’s Cat as an example; a dead cat does not come back to life, and once the decay has been detected (essentially measured) we know that the cat is dead (or will be soon, considering the poison). That in mind, no matter how many more times you measure the system of the cat in the box with the decaying particle, you will always have the same results – because the cat is already dead and we know it to be so. The collapsed wavefunction remains in the pure state into which it collapsed. Then again, the wavefunction being and remaining in a pure state assumes that you know the particle has decayed already, and thus the cat being dead. Repeatability therefore implies you know the history of the system in question.

            The reality of wavefunction collapse however, has always been widely debated. Whether it is a fundamental physical phenomenon in its own right or merely an epiphenomenon of another process. Whether measurement actually determines the state i.e. whether the event of collapse is “real” or perceived.

            Many Worlds Theory, for example, denies the objective reality of wavefunction collapse, instead explaining the subject of wavefunction collapse with the mechanism of quantum decoherence, which I will not go into here (as it would require another essay). Many Worlds Theory claims to resolve all paradoxes of Quantum theory since every possible outcome to every event defines or exists in its own history or “world”. In essence, that wavefunction collapse is only perceived by the observer, but that the other possible outcomes continue to exist in another “world”.

            In the end, there are many other arguments, and what started out as physics could quickly slide into the realm of philosophical debate. Perhaps in another world or maybe this one (pardon the pun), that is why they call a PhD a “Doctor of Philosophy”.

 

References:

-         http://en.wikipedia.org/wiki/Wave_function_collapse

-         http://en.wikipedia.org/wiki/Many_worlds

-         Michel Bitbol , Schrödinger’s Philosophy of Quantum Mechanics [p.113]

-         Peter Mitterstaedt , The Interpretation of Quantum Mechanics and the Measurement Process [p.35]

 

Comments

[nanani.livejournal.com]

８D

Mai phoenix is so smart *feathers your brains*

I think you did an awesome job on it! Though I must say your essay writing "voice" is a bit weird. Probably because I am only used to seeing normal phoenix rather than serious-business phoenix.

GOOD JOB!

[midgar-skyline.livejournal.com]

:3 Why sankyuu. <3

It was mainly thanks to you that I did it as well as I did. I had no idea where to start reading let alone research - thanks to your enlightening speed-lecture I knew what to read up on, and from that bash out an essay plan in roughly one hour, thus easing the essay-writing itself to only two hours or so at most.

:DD Weird Phoenix, eh? XDDD But of course, because writing essays is SRZ BUSINESS! <33 ILU! :3 *squishyhug* I suppose with all the spazzing with each other we do, it's been a good while since you've heard "that" voice of mine; ever since the uber-long e-mails. Er... but even then I was doing a fair amount of spazzing. Prouu~ prouuuu~~~!! :D