Heisenberg's Uncertainty Principle question

[guardian]

Heisenberg's uncertainty principle states that "The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa"

 

My question is this, why do we say momentum is less/more precise when p=mv ? To elaborate a little, isn't the mass precisely known based on what particle is being measured and so it is just the velocity component that is less/more precise? Or even more to the point, isn't either distance travelled OR the time travelled less/more precise?

 

I may be misled, but I was under the impression that the mass in the momentum equation is the rest mass and is invariant, so with that, why cloud the imprecision?

[swansont]

Yes, you can pull the mass term out of it, but even masses can have uncertainty in their measurements. The relationship is [math]\Delta x \Delta p > \hbar[/math] Since different particles have different masses, a given uncertainty in velocity gives a different uncertainty in position, so momentum makes it a more useful reltionship (position and momentum are the conjugate variables involved) Also it may not apply - for a photon, p = E/c

[guardian]

Sorry, I had replied, and changed it since I had more time to think (told wife I'll be in bed in a mo).

 

Ok, I suppose I can live with mass being uncertain although would this be the precision to which we have deduced the masses of particles to be ie. in the St. Model ? - these more or less (well most of) came from collider/accelerator experiments, from what I've sourced. Am I on the right track with this or am I still missing something?

 

With regard to the photon momentum, with a photon having a certain speed c, that leaves E. This would be E=hf for a photon (or momentum p=h/lambda), which, in this instance would still leave us with uncertainty in either time (for frequency) or distance (for wavelength).

 

I'm not trying to dispute the principle, just trying to understand why it is in terms of momentum. For mass particles if I've got the second paragraph above right then that explains it, but for the massless what then is the uncertain part?

 

So what I am really asking is what are the fundamentally uncertain parts of the principle.

[swansont]

Position and momentum are the relevant conjugate variables.

[Severian]

Ok' date=' I suppose I can live with mass being uncertain although would this be the precision to which we have deduced the masses of particles to be ie. in the St. Model ? - these more or less (well most of) came from collider/accelerator experiments, from what I've sourced. Am I on the right track with this or am I still missing something?

[/quote']

 

Mass insn't uncertain in the sence of the uncertainty principle. Presumably every fundamental particle has a well defined mass which you can specify to as many significant figures as you can measure. (Actually mass changes with energy scale, but you can define something like m(m) to define it precisely so this is beside the point.)

 

With regard to the photon momentum, with a photon having a certain speed c, that leaves E. This would be E=hf for a photon (or momentum p=h/lambda), which, in this instance would still leave us with uncertainty in either time (for frequency) or distance (for wavelength).

 

Energy is conjugate to time and momentum is conjugate to distance. The important thing to realise is that a state with a definite position is a superposition of lots of states of different momentum. This is simply a Fourier transform:

 

[math]f(x) = \frac{1}{(2 \pi)^4} \int F(p) e^{ipx} d^4p [/math]

 

f(x) is the distribution of the object in position, while F(p) is the distribution in momentum. Now if F(p) is a delta function, ie. the particle has a definite momentum, then f(x) will be spread over all space - it won't have a definite value for the momentum. Similarly (via the inverse transform) if the particle is at one point, its momentum distribution will be spread out.

 

So basically, the HUP is just a statement about Fourier transformations....

[guardian]

That clears it up. Till next time...thanks.

 

Severian, if you have time, I have posed a question that you may want to have a look at in the Astronomy - 'Where does space end....' thread. It is sort of in reply to one of your posts...I hope you may be able to help.