QM Questions: Intro & Two Conceptual Issues

[hotcommodity]

I'm preparing for an upcoming course in QM by working out of a textbook, and I've encountered a handful of conceptual questions that I'm having trouble answering. I figured it best to ask my questions in a single thread, so as not to clutter the subforum. Also, I chose the general physics subforum because it's unlikely that I'll be asking anything that may be considered advanced physics (but feel free to move the thread if need be).

To start with:

1. The wavelength of a matter wave is, of course, given by $$\lambda = \frac{h}{p}$$. If an object has zero momentum, does its wavelength get arbitrarily large? And if so, why doesn't its wave nature dominate?

2. Let's say that a beam of electrons are moving through a single slit (with a width comparable to the wavelength of the electron). The electrons would then show up all over the screen/detector that lay behind the single slit. Could we say that the electrons show up at all places on the screen at a constant rate? Or would we only be able to speak in terms of the average rate of electron detection for any given region on the screen?

Any help is welcome and appreciated.

 

[PiratePhysicist]

1) This is a bit of a trick question, what you should be asking yourself, is: Is it possible for something to have something with zero velocity? (think in the quantum realm, not classical)

2) For this one, remember that quantum mechanics is all about probability.

 

[HallsofIvy]

Another way of saying the same thing PiratePhysicist said, IF the momentum were 0, what would the uncertainty principle say about its postition?

 

[hotcommodity]

Thank you both for the replies.

Another way of saying the same thing PiratePhysicist said, IF the momentum were 0, what would the uncertainty principle say about its postition?

If the momentum were zero, there would be zero uncertainty in the object's momentum, which would make the uncertainty in the object's position infinitely large. But I'm not sure what physical concept that reveals... Does it mean that the object is equally likely to be found anywhere?

 

[PiratePhysicist]

Yeah, so basically, it's impossible to have something with zero velocity.

 

[hotcommodity]

That makes sense, thank you.

I have another question in regard to the uncertainty principle.

3. My book states "Because the width of a diffraction pattern, related to $$\Delta p_x$$, is inversely proportional to the slit width, related to $$\Delta x$$, the uncertainties are inversely proportional. (using the example of a single slit pattern).

So my question is, when we're speaking of uncertainties in position and momentum, are we speaking of (a) the uncertainty in the position & momentum of the particle while it's in the slit? (b) the uncertainty in the position & momentum of the particle just after it leaves the slit? (c) the uncertainty in the position & momentum of the particle once it registers on the screen? Or (d) (As I believe the quote above implies) the uncertainty in the position of the particle while it's in the slit, and the uncertainty of the momentum of the particle as it registers on the screen?

Again, any help is appreciated.