Detector Watch S6 V1

I thought I'd try to explain the Heisenberg thing better, as most peoples' eyes glaze over when you mention it. It's not really that hard to follow, as it comes down to a matter of counting.

Suppose you have a repetitive wave passing by you. It wiggles up and down as the crests ( high points ) and troughs ( low points ) reach your position. Imagine you don't know where it came from or where it is going, but you have been asked to measure it's frequency. That is, how many times ( a number ) does a whole wave pass by in some given time interval ( some number of seconds for instance ). We define a whole wavelength to be the ( shortest ) 'piece' of the wave between a specific phase value. The phase is the measure of what bit of a wave are you examining - a crest, a trough or anywhere in between. We suppose that the wave shape is the same during our monitoring of it. I say 'shortest' above as you could count two full wave cycles between two crests say, but not mention the one crest in between. So I'm after a single cycle from one crest, say, through to the very next one.

So there we are with some clock to know how long things take, and we count a full wave as it passes by. The trouble is that nothing is perfect, we can't measure in as fine increments as we like. This is an essence of quantum mechanics which differs from the preceding classical program. So both the time interval and our assessment of phase are fallible. So I don't quite exactly know when a single full wave has passed precisely by.

But all is not lost. I can do something like the following : count not a single up and down cycle but many wave cycles passing by, then divide the number of waves by the time interval on my clock which refers to those cycles. If my phase accuracy is within say, one hundredth of a cycle - this will be true regardless of how many cycles I measure. For one cycle this is a 1% error, but for 1000 cycles this will be a 0.01% error. So my error in calculating frequency will be 1000 times less over that longer time interval. You can treat any error in my clock's function in a similar vein as it will lead to a corresponding inaccuracy in the exact arrival time of a crest say. But still any absolute timing error is relatively diminished over a longer total time.

So this leads to the following general rule : if you want a better measurement of frequency then you have to do that measurement over a longer time interval.

It's not often mentioned but there is a thing called wave number, it's analogous to frequency. It's the number of full cycles in a certain distance. So while frequency is cycles per second ( Hertz ), the wave number is quoted as cycles per meter. One can do a similiar analysis for wavenumber. I'm still counting wave cycles but for this I'm comparing against a length standard not a time one. So if I could freeze the wave movement in time I could move back and forth in space and count cycles. And I will have inaccuracy in defining the phase exactly as before, also the distance too.

Hence another general rule : if you want a better measurement of wavenumber then you have to do that measurement over a longer space interval.

Thus we have variables that vary inversely :

frequency_error varies like (1 / time)

wavenumber_error varies like (1 / distance)

Because of this relation frequency and time are called conjugates of each other. And wavenumber and distance are also each others' conjugate.

Now quantum mechanics associates frequency with particle energy ( Einstein ), and wavenumber with particle momentum ( De Broglie ). So we now have :

energy_error varies like (1 / time)

momentum_error varies like (1 / distance)

OR re-arranging and introducing a number to define the exact proportions of things ( h = Planck's constant )

energy-interval x time-interval = h

momentum-interval x distance-interval = h

h is one of those numbers which gives us the scale of things in the universe. In these instances it gives a definite answer as to how much uncertainty in one variable leads to how much uncertainty in it's conjugate. But of course there may be other factors that reduce our success in measurement quite apart from counting waves and defining time and length intervals. So

energy-interval x time-interval > h

momentum-interval x distance-interval > h

meaning that :

energy-interval > h / time-interval

momentum-interval > h / distance-interval

So that the energy and momentum estimates could actually be worse ( > h ) than the case of equality ( = h ). The case of equality is called the quantum limit meaning that when we have removed all other sources of inaccuracy in measurement we are left with basic way the universe behaves. If I wanted to measure energy/frequency exactly then I'd have to study the wave over an infinite length of time. If I wanted to measure momentum/wavenumber exactly then I'd have to examine the wave over an infinite distance. My time/distance inaccuracies may be small but they are not zero - or equivalently Planck's constant is small but it is not zero. We simply can't do better than that. Our Universe is just built that way ....

Cheers, Mike.

( edit ) I ought point out that these wave measurement issues are not confined to quantum mechanics ( QM ). Any other phenomena involving waves will have the same concerns. Radar operators may be familiar with relations that limit accurate determinations of a target's position & movement because of wave character. We don't have to invoke QM for this ( though you could! ) and you definitely don't have the luxury of infinities to make matters more precise. The QM breakthrough was in assigning wave behaviours to what were previously thought to be ( point ) particles.

Also in the scenario of measuring rotations of things around some pivot, the conjugate variables are angular position and angular momentum. Basically if you want to do better at finding out how quickly something is rotating around a point then you have to follow it for a larger arc. You can derive a similiar uncertainty relation from the ones given above. Alas I can't remember it, but you'd probably be throwing in PI as well. :-)

( edit ) Thanks to Chipper, he'd reminded me of the earlier post of mine which I was looking for. It has some pictures to help too ... :-)

( edit ) Depending upon who derives it there may also be a factor of 2 * PI dividing the h in the above relations. It's not important for the overall ideas, but expresses differences in how one might define the 'width' of a wave packet. As you can't have a single exact frequency/energy then 'real' particles are modelled as being a superposition of a group of pure sine waves with close frequency values. PI is involved 'cos it's trigonometry ..... :-) :-)

( edit ) Whoops, can't stop elaborating. Please note that I have not mentioned the interaction of the observer with the thing being observed. What the modern explanation of QM uncertainty refers to is that objects just do not possess exact values for energy and momentum. Heisenberg & Bohr originally phrased matters in terms of probes disturbing the system under measurement. Like a photon being used to find where an electron is, but with the photon inevitably disturbing the electron because we used it at all. It has energy. Here measurement = interaction. This is a true problem of course and will involve h and whatnot, yielding similiar relations. But that's a distinct issue. Well, opinions do differ ... :-)