Is science sometimes in danger of getting tunnel vision? Recently published ebook author, Ian Miller, looks at other possible theories arising from data that we think we understand. Can looking problems in a different light give scientists a different perspective?

Wave Function Collapse

Perhaps my challenge, "what is the most illogical thing you can associate with standard quantum mechanics? " was not to everybody's taste. Sorry, but I cannot help feeling that ignoring problems is not the way to enlightenment. Also, part of what follows is critical to my question, where do you expect violations of the Woodward Hoffmann rules?
 
Consider the following:
(1)  You cannot know something about a phenomenon until you observe it.
(2)  Prior to observation, a state is a superposition of all possible wave functions.
What I consider illogical is that it asserted that the superposition of states is a reality and when an observation is made, there is a physical event known as the "collapse of the wave function". Such a superposition of states can never be observed, because by definition, when it is observed, it collapses to one state. The electron either is or is not, there. So, why does this assertion persist? (I have no idea on this one. Anyone, please help.)
 
The most well-known example is the Schrodinger cat paradox, in which a box contains a cat and radioactive particles associated with a device that emits hydrogen cyanide. If no particle is emitted, the cat is alive; if it is emitted, the cat is dead. Before we observe, we do not know whether the particle is emitted or not, and both states may be equally probable. That is described as a superposition of states, but according to the paradox, a consequence is the cat is also in a superposition of states, neither dead nor alive.
 
The problem involves the conclusion that the square of the amplitude of a wave function gives the probability of making an observation. If the objective is to compute the probability of something happening, and you consider the states to represent probabilities, such states have to be superimposed. The probability of something that exists, such as the cat, must be 1. In this case, the cat is either dead or alive, and the probabilities have to be summed. The same thing happens with coin tossing: the coin will be either heads or tails, and there is equal probability, hence two states must be considered to make a prediction.
 
Herein lies a logic issue: the fallacy of the consequent. Thus while we have, "if we have a wave function, from the square of its amplitude we can obtain a probability" it does not follow that if we can ascribe a probability, then we have a wave function. What is the wave function for a stationary coin, or a stationary cat? It most certainly does not follow that if we have two probabilities, then the state of the object is defined by two superimposed wave functions. A wave function may characterize a state, but a state does not have to have a wave function.
 
A wave function must undulate, and to be a solution of the Schrodinger equation, it must have a phase defined by exp(2pi.i.S/h), where i is the complex number, and S is the action, defined as the time integral of the Lagrange function, which can be considered as the difference between the kinetic and potential energies. (The requirement for the difference arises from the requirement that motion should comply with Newton's second law.) As can be seen, a complete period occurs whenever S=h, which is extremely small. For the spinning coin, any wave function period returns to its initial value so quickly it is irrelevant, and classical mechanics applies. In this light, the Schrodinger cat paradox as presented says nothing about cats because there is no quantum issue relating to a cat.
 
What about the particle? When Aristotle invented logic, he allowed for such situations. To paraphrase, suppose one morning there is a ship in harbour fully laden. The question is, where will it be tomorrow? It could be somewhere out to sea, or it could be where it is now. What it won't be is in some indeterminate state until someone observes it because it is impossible for something of that size to "localize" without some additional physical effect. Aristotle permitted three outcomes: here, not here, and cannot tell, in this case because the captain has yet to make up his mind whether to sail. Surely, "cannot tell" is the reality in some of these quantum paradoxes?
 
As a final comment, suppose you consider that with multiple discrete probabilities there have to be multiple wave functions? The probabilities are additive and must total 1. The wave amplitudes are additive, and here we reach a problem, because the sum of the squares does not equal the square of the sum. Mathematically, we can renormalize, but what does that mean physically? In terms of Aristotle's third option, it is simply a procedure to compute probabilities. However, it is often asserted in quantum mechanics that all this is real and something physically happens when the wave function "collapses".
 
Now, you may well say this is unnecessary speculation, and who cares? Quantum mechanics always gives the correct answer to any given problem. Perhaps, but can you now see why there can be exceptions to the Woodward Hoffmann rules, and where they will be? The key clue is above.
Posted by Ian Miller on Aug 23, 2011 11:53 PM Europe/London

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