I was watching a video lecture on quantum mechanics given by Leonard Susskind, wherein he was explaining the uncertainty principle. He pointed out by saying that the momentum of a photon is inversely proportional to its wavelength. For a wavelength on the order of magnitude as an electron, there will be enough momentum to knock it away such that you can’t say anything about it’s current position. He didn’t generalize it, however, such that I could get my head around what happens if the particle is really heavy, say 99% of the Planck Mass. Is it the case there that you know the position of the particle only to within one Planck Length and that if the particle is lighter, your measurement is even more imprecise? Also, once we hit the particle, can we say with arbitrary position where it was? Finally, won’t the photon that we struck it with be reflected and checking the wavelength of the reflected photon, can’t we work out the momentum we imparted to the electron?
[del]I’m pretty sure that anything below a Planck ____ (length, mass, length of time) is meaningless in the quantum model. Therefore, the question cannot be answered by that model.
But I am a theoretical physics dilettante. I welcome correction.[/del]
From Wikipedia: Unlike all other Planck base units and most Planck derived units, the Planck mass has a scale more or less conceivable to humans. It is traditionally said to be about the mass of a flea, but more accurately it is about the mass of a flea egg.
I also thought of another question. According to the U.P., you cannot know both the position and momentum of a particle to arbitrary precision at the same time. IIRC, there are other pairs of properties which are similar in nature. If that is so, what are they?
Time and energy are the major other complementary pair. This is what makes virtual particles possible. If the energy exists for a short enough period of time it can create particles that immediately wink out. Virtual particles have all sorts of implications for the way things work, but I’ll let a physicist discuss that.
My memory (always a bad cite) tells me there are many other pairs. Ah, here’s the right Wikipedia page: conjugate variables.
Yes. Without going too far into the math, there are a couple of more general uncertainty relations. One relates time and energy, and is similar to the position-momentum one (but requires a bit more care to derive and use). In addition, there’s a general uncertainty principle for operators that don’t commute (modulo some details I’m not omitting here). For motivation in the simplest case, taking a measurement in quantum mechanics corresponds to forcing the wave-function into an eigenstate of a (for simplicity) finite-dimensional operators. Unless operators commute, eigenstates of one aren’t necessarily eigenstates of the other.
What Blue Blistering Barnacle said might be usable as a rule of thumb, but it leaves out a lot. For instance, two different components of angular momentum don’t commute with each other (and hence have an uncertainty relationship), but any single component of angular momentum commutes with the total angular momentum.
OK, I think I have enough to go on to learn about conjugate variables. But what about the other questions, namely:
As the particle’s mass approaches Planck Mass, does precision of position and momentum measurement uncertainty approach Planck’s constant (or the reduced one, I can’t remember)?
Can we say with arbitrary precision where a particle was as opposed to is right now?
Can we work out from the reflected photon the new momentum of the particle with arbitrary precision? How about any momentum we added to the particle?
First, you should not attach too much weight to the heuristic ‘Heisenberg’s microscope’-style derivation of the uncertainty relation. (In fact, it’s been a point of contention recently if this ‘measurement-disturbance relation’, as it’s usually called, holds as such, though without going into the details, I think the side arguing that it indeed does hold has some better arguments.)
What’s really important is that in the case of canonically conjugate variables, which are ‘translated’ to noncommuting operators in the quantum theory, the mathematics forces an uncertainty relationship on us, the so-called Robertson-Schrödinger relation. That this holds is beyond contention (well, to the extent quantum theory itself is, of course); unfortunately, it doesn’t have a nice visualizable picture behind it.
But the picture in the Heisenberg’s microscope thought experiment is actually misleading: it makes it seem as if the particle’s position/momentum is merely influenced by the act of measurement, and that thus, we simply can’t know it exactly, but if we could enact some form of measurement without this disturbance, then we could, in principle, know both a particle’s momentum and position exactly. But this kind of thinking actually leads to contradictions with quantum mechanical predictions, and experiments supporting them; this is encapsulated in the Kochen-Specker theorem, which roughly says that not all observable quantities about a system can simultaneously have a definite value. Additionally, Heisenberg’s microscope can’t explain all the uncertainty relationships between other quantities, such as the components of angular momentum, for instance.
So basically, to try and answer your questions, regardless of the mass of the particle, the uncertainty relation holds always, and the product of the uncertainties in our simultaneous knowledge about a particle’s position and momentum will always be bounded by half the reduced Planck’s constant. We can know either the particle’s position or its momentum (at the time the measurement took place) with (in principle) arbitrary exactness, at the cost of being totally ignorant of the value of the conjugate quantity. There’s no way to ‘cheat’ the uncertainty principle: according to the Kochen-Specker theorem (which, as I said, has experimental consequences), simultaneous values for conjugate quantities simply cannot exist, or if they do, their value must depend on the kind of measurement you’re performing.
I would go further than that, even: The various uncertainty relations aren’t just a limit on what we can know; they’re a limit on what actual, true values the properties can have. A particle with a very well-defined momentum simply does not have a well-defined position, regardless of whether or not anyone is attempting to learn that position. If you want to pin down the position, you can, but the cost is that the particle no longer has any particular momentum.
Yes, this is to me an even greater mind-fuck. And many a thread, and many a comment by the respected physicists here, has been to correct/widen the understanding of the UP as “merely” a measurement limit.
You know, but aren’t they out there … ? Just, you know, being somewhere at some time? … No? … :: mind fucked ::
ETA: Wait–serious question: with that sentence using “they,” you fcan get an answer “Yes,” right? Statistical and all that. But it’s the “it” that prompts a real “No”–right?
Conversation I
**Me: **“You know, but [isn’t it] out there … ? Just, you know, being somewhere at some time?” …
**You: **No." Me: :: mind fucked ::
Conversation II
**Me: **“You know, but [aren’t they] out there … ? Just, you know, being somewhere at some time?” …
**You: **“Yes, but only as an aggregate, as a statistical measurement approaching certainty but never ever so.” Me: :: mind fucked ::
