Uncertainty Principle & Measuring Momentum: Is There a Confusion?

[spidey]

we have the relation p= E*C then why do we say that we cannot measure momentum accurately? if we know E then we can find p, and then we can find position at ease,whats more difficult here?
are we confusing with uncertainty principle? or is Uncertainty principle confusing us?
Or am i missing anything?

 

[tiny-tim]

… we can find position at ease …

Hi spidey!

(that equation is for photons, of course …)

Nooo … finding position is actually quite difficult … the more you know about momentum, the less you can know about position.

You can get energy, and therefore momentum, as accurately as you like … but then you won't be sure about time or position!

 

[ZapperZ]

we have the relation p= E*C then why do we say that we cannot measure momentum accurately? if we know E then we can find p, and then we can find position at ease,whats more difficult here?
are we confusing with uncertainty principle? or is Uncertainty principle confusing us?
Or am i missing anything?

As has been pointed out, that relation only works for photons because it is for a massless particle.

But a more general question would be the relationship between "p" and "E". You will note that p and E (or to be accurate, H, the Hamiltonian) need not necessarily commute, except for a free particle. If they don't commute with each other, then measuring p will not tell you E with the same certainty. You'll have the same issue as p and x.

There's also a more common misunderstanding of the HUP here that I'm seeing very often in this forum. The HUP doesn't talk about the uncertainty in a single measurement. You can measure p and x as accurately as you want for a single particle. These accuracies depend on the instrumentation accuracy, i.e. how small is that spot made by that particle on the CCD. That isn't the HUP. The HUP comes in in 2 different ways:

1. If I have made a determination of x with an uncertainty of $\Delta x$, then how accurately can I predict $p_x$?

2. I make many repeated measurement of x and many measurement of $p_x$, and look at the spread in values for those observables.

There's nothing in the above to prevent you from measuring x and $p_x$ as accurately as you want from each of the single measurement.

Zz.