(no subject)

[lhexa]

Last night was good. I completed some eight easy Quantum Mechanics problems, mostly stuff relating to 1D problems. But I had forgotten how strange such work makes my dreams... I can't relate the details of last night's first dream, partly because much of it has slipped away and partly because it was halfway mathematical in content. It somehow involved attempting to establish a means of communication with another world. The dream eventually forced me awake as it became more abstract, even though I quickly lost any ability to articulate the (mathematical) communication going on. I awoke with angular Mayan patterns (half-felt, half-visualized) tracing their way down my legs and patches of heat on my skin that moved from place to place. Eventually my thoughts and sensations became normal enough to bring a minor realization: an ideal measurement creates an infinite probability current. (Said current uses a delta function in time.) I puzzled over that fact for a time, eventually writing it down on a certain list in case it ever turns out to be significant. Then, because this is the way such nights work, I toyed mentally with the probability current for a couple of hours more before being able to fall asleep again.

I still exist. Last semester was good, this semester is very bad. More details after it's over.

I needed to be humbled, anyway.

Comments

[otter3.livejournal.com]

Eventually my thoughts and sensations became normal enough to bring a minor realization: an ideal measurement creates an infinite probability current. (Said current uses a delta function in time.)

This statement obviously intrigues me greatly.

[lhexa.livejournal.com]

Normally, there's a result called the local conservation of probability which requires probability to move smoothly from place to place. But in an ideal measurement the wave function jumps immediately from whatever it was before to an eigenfunction of the operator corresponding to the observable (since only an eigenfunction has a single value of that observable). The probability has a discontinuity in time, requiring the probability current (http://en.wikipedia.org/wiki/Probability_current) to have a delta function (http://en.wikipedia.org/wiki/Dirac_delta_function) (a special kind of infinity) in it: when a delta function is integrated over an infinitesimal interval it gives a finite difference.

The moral, I guess, is simply that there are no ideal measurements, but it looks like even a real (but good) measurement would induce a very high probability current. That's neat.

[khyle.livejournal.com]

I am glad you still exist! :) *hugs* Interesting stuff that what you're dreaming though... Let us know how things go.

[lhexa.livejournal.com]

I will.

[guardlion.livejournal.com]

That's some rather neat imagery there with the patterns and heat spots. :-) Sensations from my dreams rarely cross over into the waking world. The most I ever get is a violent body spasm when I hit the ground in a dream after falling some distance. And that usually wakes me up suddenly. If you ever decide to take up doodling and can remember what the patterns looked like, consider sketching them out on a generic 2D body form. I'd be curious to see.

Regarding your realization, I was able to figure out what probability current is, but I'm lost about what an "ideal measurement" is and how it relates to this. Would you care to educate me?

And don't plex too hard about the semester. Everyone has a bad one now and then. An what's done is done. Do what damage control you can and keep on truckin' with the rest of things. I have faith in you. :-)

Oh, BTW, if you want an interesting surprise, do a google search for "angular Mayan patterns". :-)

[lhexa.livejournal.com]

*smiles* My body spasms too, dog-like, but it tends to happen as I'm falling asleep.

An ideal measurement gives you an exact value for some observable quantity, but in doing so it alters the wave function of whatever's being observed. It forces the wave function into an "eigenfunction" of the observable, a state in which the quantity is definite. Since the wave function jumps discontinuously to its previous state to the eigenstate, its rate of change (and thus probability current) is for an instant infinite. (It's a delta function, which is a special kind of infinity.) For other details look at my comment above.

Thanks for the supportive words. :) And did you mean to indicate the fact that the search string points back to this journal?