I have been daydreaming about the role of time in quantum mechanics, clearly without bothering to look in the literature. I find it difficult to believe that a "time operator", that is a conjugate of the Hamiltonian, cannot be defined. The usual argument is that since time is unbounded, so would be the energies and therefore you end up with an unphysical system. However, I am doubtful since, for instance, you cannot define a suitable conjugate operator to position when the position is bounded (open boundary conditions I mean, if you have periodic bc momentum is well defined but it has discrete eigenvalues). In light of this, and the similar to x and p relation between time and energy, I don't fully understand what is the problem. Anyway, I found this article where it is argued that the confusion arises originally from mixing x, the operator position, and q, the eigenvalues of this operator. In classical mechanics the difference accounts to x being the coordinates of points in space, and q the coordinates of a point particle in space. With this in mind, some funny time operators are constructed, which applied to some clock-systems have time as eigenvalues. It works pretty well from what can I see. The energy spectrum of these clocks can be unbounded of course (which wouldn't pose a problem if they are isolated if I understood correctly), or bounded if we choose a periodic time (a round clock of course :). 

All very nice, I will have to think about this for a while to see if it makes full sense...

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Posted by fercook to Science at 5/11/2006 02:10:14 PM

Labels: reviews, science