sec2 - II. The Machinery of Quantum Mechanics Based on the...

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Unformatted text preview: II. The Machinery of Quantum Mechanics Based on the results of the experiments described in the previous section, we recognize that real experiments do not behave quite as we expect. This section presents a mathematical framework that reproduces all of the above experimental observations. I am not going to go into detail about how this framework was developed. Historically, the mathematical development of QM was somewhat awkward; it was only years after the initial work that a truly rigorous (but also truly esoteric) foundation was put forth by Von Neumann. At this point, we will take the mathematical rules of QM as a hypothesis that is consistent with all the experimental results we have encountered. Now, there is no physics or chemistry in what we are about to discuss; the physics always arises from the experiments. However, just as Shakespeare had to learn proper spelling and grammar before he could write Hamlet, so we must understand the mathematics of QM before we can really start using it to make interesting predictions. This is both the beauty and the burden of physical chemistry; the beauty because once you understand these tools you can answer any experimental question without having to ask a more experienced colleague; the burden because the questions are very hard to answer. A. Measurements Happen in Hilbert Space All the math of QM takes place in an abstract space that called Hilbert Space. The important point to realize is that Hilbert Space has no connection with the ordinary three dimensional space that we live in. For example, a Hilbert Space can (and usually does) have an infinite number of dimensions . These dimensions do not correspond in any way to the length, width and height we are used to. However, QM gives us a set of rules that connect operations in Hilbert Space to measurements in real space. Given a particular experiment, one constructs the appropriate Hilbert Space, and then uses the rules of QM within that space to make predictions. 1. Hilbert Space Operators Correspond to Observables The first rule of QM is: all observables are associated with operators in Hilbert Space . We have already encountered this rule, we just didnt know the operators lived in Hilbert space. Now, for most intents and purposes, Hilbert Space operators behave like variables: you can add them, subtract them, multiply them, etc. and many of the familiar rules of algebra hold, for example ( Z Y X , , are arbitrary operators): Addition Commutes: X Y Y X + = + Addition is Associative: ( 29 ( 29 Z Y X Z Y X + + = + + Multiplication is Associative: ( 29 ( 29 Z Y X Z Y X = However, the multiplication of operators does not commute : Multiplication does not commute: X Y Y X We already knew that this was true; in the case of the polarization operators we showed that x P and ' x P do not commute: y x x y P P P P ' ' Thus, the association of observables with operators allows us to...
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