There is a popular idea that Heisenberg's principle is about metrology.
It is a fact that Heisenberg's principle was developed as a problem of measurement of the location of an electron around the nucleus. Then, Heisenberg used a thought experiment with a hypothetical high-energy gamma rays microscope in order to provide a "picture" of the principle; you can find the analysis of this hypothesis in this site: http://www.aip.org/history/heisenberg/p08b.htm
Actually, the principle of Heisenberg is not "portrayed" adequately by the microscope’s experiment, when the uncertainty of the measured results is seen as a problem of accuracy in measurement. The issue is not about a problem of methodology of measurement. As it is noted in the above link: “In fact Heisenberg's microscope, although it was a big help in developing and teaching the quantum theory, is not itself part of current understanding”
Regardless of method of measurement (and even under theoretical study), it is not possible to value certain pairs of variables of physical terms, like the pair "position/ momentum" of electron, with the same certainty of precision at the same time for values of both variables.
The principle of Heisenberg resolves this problem by providing another point of view: since we can not value both variables of the pair at once with precision, we value one of them with certain precision and then we use the proper relation between the variables in order to construct a single physical “picture” of both of them, that takes in account the other valued variable too, which is not valued with certainty. This is something like a compensation of our ignorance about the inaccurate value of the variable of the one member of the pair by the definition of a ratio between itself in comparison with the more accurate value of the other variable of the pair. This ratio is not a static ratio, but it is a ratio of probability distribution of the certainty of the values of each variable of the pair over a period of time.
This means that when we study an electron we can be certain where its position might be now, but at the same time we can not be certain where its position will be afterwards or where it was before. Or, we can be certain where an electron's position will be after a period of time or where it was before a period of time, but at the same time we can not be certain where it is now.
This paradox is due to a paradoxical intrinsic way that the physical variables of position and of momentum, of the electron, are related among themselves. The uncertainty for the value of the one variable of the pair compared to the certainty of the value of the other variable is not a problem of methodology of accuracy of measurement of the values of these variables and it is not uncertainty in value introduced by influence of the methodology of the measurement over the measured variables of the measured system.
The relation among these variables of the physical pair is expressed by a probability distribution. That is, the uncertainty of the value of one variable of the pair multiplied by the uncertainty of the value of the other variable of the pair is always bigger than a certain numeric value expressed by the formula of Heisenberg's principle.
And now, let’s face the physical meaning of Heisenberg's principle. The meaning of Heisenberg's principle is that we can not observe, or theoretically study, the same electron for both values, of its physical variables “position” and “momentum”, at the same time with the same certainty of precision for both of them! In order to achieve certain precise measurements for both position and for momentum, we need to perform different distinct measurement, one measurement for the position and another measurement for the momentum. These two measurements/calculations though, can not be simultaneous; they have to be successive.
Once we have observed, or have calculated, the current position of an electron with certainty we can not share the same degree of certainty about its previous position or of its next position. And once we have observed or calculated the previous position or the next position of an electron we can not share the same degree of certainty about its current position. ( current, next and previous refer to time ).
This means that the trajectory of an electron is meaningless unless we are constantly observing its position with precision, but at the same time we can not value the momentum of the electron in each position of the trajectory with the same certainty, with precise values. In an analogy it is like having precise certainty about the trajectory of a moving tennis ball and at the same time being unable to share the same certainty about the momentum of the tennis ball in each position of ball’s trajectory. This is paradox.
Same thing happens when we increase the certainty of the momentum, as the certainty of the position is decreased. In an analogy it is like having precise certainty of the value of the momentum of a moving tennis ball and at the same time being unable to share the same certainty of its position as a well defined trajectory of the ball.
But we know that this is NOT happening in the case of a tennis ball in real life dimensions, why is it happening in the case of an electron in subatomic dimensions?
Well, it can happen analogically in real life too, but we have to change the analogy: In real life we are used to observe the physical phenomena in a well defined 3D space. If we introduce in real life dimensions a 3D space that is not well defined, the position of the tennis ball and the momentum of the tennis ball will also become a pair of physical variables related with a mutually uncertain relation. For example, if we perceive the wind in a stormy day as a 3D space distortion rather than a force, then the position of a tennis ball and the momentum of a tennis ball, moving in the turbulence of wind, have similar mutually uncertain relation with the relation of electron’s momentum and position.
I do NOT say that in the above example, of a tennis ball moving in a wind, is explaining what happens in the case of the electron. In electron’s case there is no external cause which produces the uncertainty relation of position and of momentum; this relation is intrinsically set in the electron. But, the 3D space distortion in real life, by a wind, can be used as an analogy of how two physical variables of real life can become non-commuting, in being related by a relation similar to the relation of uncertainty of values of variables of observable quantities of quantum particles (although this has to be used only as an analogy, and it can not explain the quantum paradox of Heisenberg's principle).
Leandros
