More on debunking the mysteries of quantum mechanics

As promised (or threatened ) whatever way you want to take it, Here is another post on trying to debunk the alleged mysterious nature of quantum mechanics. This post will try and set the framework for more detailed analysis of the alleged paradoxes. To do this we have to look at the basic mathematical structure of quantum mechanics. I can do no better than quote from Richard Feynman who in the first chapter of his book on the subject gives the following summary of the essential features these are

(1) The probability of an event in an ideal experiment is given by the absolute value of a complex number a which is called the probability amplitude

                                    P = probability
                                    a = probability amplitude
                                    P = |a|^2

(2) When an event can occur in several alternative ways the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference

                                           a = a1 + a2
    
                                    P = |a1 + a2 |^2  note this is not the same as |a1|^2 + |a2|^2

This is crucial, all the stuff about interference etc which bedevils much of the agonising about say the two slit experiment boils down to the fact that in order to get the total probability for an event to occur we must add the sum of the pobability amplitudes together and then take the square of the total sum rather than simply adding the sums together.

(3) On the other hand if it is possible to determine whether or not one event has actually occurred then the probability of the event is the sum of the probabilities for each alternative.

The crucial point to recognise is that we are simply dealing with probability amplitudes when I take the modulus squared of the probability amplitude taking into account all possibilities this gives me the probability distribution associated with the situation of interest. This approach was first taken  by Dirac in his founding book 'The principles of quantum mechanics'. He showed that the mathematical structure of quantum mechanics was that of a linear vector space over the field of complex numbers. This was extended by Von Neumann in his book The mathematical structure of quantum mechanics which tidied up some of the loose ends when extending the dimensions of the vector space to infinite dimensions. Note these dimensions are essentially the dimensions of the probability state space associated with a given system they are not real fields in an infinite dimensional space. For those who want more detail on the underlying mathematical structure of quantum mechanics I refer to the freely available notes by David Mermin

http://people.ccmr.cornell.edu/~mermin/qcomp/CS483.html

In this set of notes he spells out how the formalism of quantum mechanics can be seen as a generalisation of probability applied to classical systems by making real probablilty amplitudes complex numbers. It's interesting to note that Dirac's approach is quite different to the standard approach based on solving Schrodinger's wave equation. The point is that the solutions to the wave equation i.e Schrodinger's wave function are only one way (albeit a convenient one) of representing the function space associated with quantum systems. Because Schrodinger's wave equation is usually solved in coordinate space the impression is given to many people that the wave function is some how a real wave in a multi-dimensional space time. This is unfortunate and misleading I would contend that the abstract formalism initiated by Dirac gives a much clearer picture as it shows that we are not tied to one particular representation. Indeed for complicated problems one can abstract the essential features by listing the possibilities. It's interesting a lot of the current problems in quantum mechanics to do with the relatiionship between information theory and the possibility of quantum computiation generally speaking do not need to solve Schrodinger's equation at all. It may well be that a new generation of physicists bought up on the Dirac approach might be more readily inclined to accept the abstract nature of quantum mechanics than an obssession with whether or Schrodinger's wave function  is a real wave or as I would argue simply the complex square root of a probability distribution function. It is my contention that all the paradoxes etc beloved of the popular literature stem from seeing Schrodinger's wave function as a real field in a multi dimensional space time. I would urge people to have a look at the Mermin lectures.