The folks at Noncommutative geometry have given me a useful hint to embed mathematical formulas directly in a blog or web page: the use of TeXify. Here is an illustration with the collapse of the wave function written using the formalism of the theory of incomplete measurements: .
There are just three issues with that tool:
- The text they give you to copy for web pages is, strictly speaking, not very good: no quotes, no self-closed img tag.
- The resulting image is not anti-aliased, so it does not look very good in the middle of anti-aliased text on any modern display.
- In that blog, it insists on putting some blue line around the images. I assume it has something to do with the style sheet, I need to look. But it’s annoying.
Anyway, it’s better than nothing.
Uh… the collapse of the wave function? Are you sure?
Maybe I should explain why the above equation is the collapse of the wave function. This is probably not obvious to anybody who did not read the TIM article. In the above equation, is the measurement being considered, and is the system the measurement applies to.
What the equation translates is the choice we make for our measurement instruments to give repeatable or stable results. If I connect a voltmeter to a battery and read “9V”, then without changing anything, read again, I expect to still read about 9V, not -32V. The output of the voltmeter should be stable when measuring a system that does not change. It would not be very useful as a measurement instrument without that property.
What the equation expresses is that what is learned by performing the measurement () is the same thing that what is learned by applying again to the result (). It’s really very simple.
Why does it lead to the collapse of the wave-function? The more formal explanation is in the article, but the intuitive explanation is quite simple. We deduce that the state after the first measurement is an eigenvector of the measurement observable from the fact that another measurement will always give the original eigenvalue. In other words, if a measurement instrument was not giving a stable result, we would be unable to deduce that the wave function after the first measurement is an eigenvector at all. There would be no collapse of the wave function.