This might be a slightly naive question, and if so I apologize, but I am currently a little confused as to why the Heisenberg Uncertainty principle should apply to particles, i.e. our system (say an electron) after we observe it and collapse it’s wave function.
From what I understand, the Heisenberg Uncertainty principle just comes from the fact that momentum is the Fourier transform of position (wave number technically I think, but all the same since momentum is related to wavelength which is related to wave number). The more localized one is, the less localized the other will be because ‘localized’ things require a larger distribution of frequencies to localize them.
Nonetheless, it seems as those this should only hold, if our object is treated as a wave, but if we treat it like a particle, it feels like this should just go away. Even if you represent a particle like a wave by using something like the Dirac delta function or whatnot, you would get essentially an infinite number of corresponding wave numbers, in other words total uncertainty on the momentum which seem strange if we think of things like particles classically. It just feels like in order for Heisenberg to hold, things always need to be ‘wave-like’ in some sense. I apologize for the long winded question, but any help would be appreciated.
Edit: Thank you all for your responses. I think my confusion has been cleared up.
Best Answer
I understand your confusion. It is due to an old-fashioned way of introducing Uncertainty relations based on wave formalism, dating back to Heisenberg, but probably quite misleading.
Quantum mechanics (QM) does not say that particles are waves. That was de Broglie's original point of view, but today is untenable. Particle dynamics may be described using waves. But this is not the same as saying particles are waves. There are many reasons for that. I mention a couple of them: quantum wavefunctions for more than one particle are not functions of a single space point; in measurements, nobody ever measured a fraction of charge, spin, or any other property of the particle like it would happen if the physical properties would have been spread over an extended field.
QM is a probabilistic theory from which we can extract consequences on the statistical behavior of many measurements on equally prepared systems. However, in most cases, the outcome of an individual measurement is a random variable. Moreover, QM can be formulated differently, and wavefunctions in a Hilbert space are just one of the possibilities. The real issue is the calculation of probabilities.
The actual content of the Heisenberg relations is captured by the Robertson-Schrödinger theorem: $\Delta x \Delta p_x \geq \frac{\hbar}{2}$ is a statement about the variances of the random variables corresponding to independently measured position and momentum in an ensemble of equally prepared particles. As such, it is neither a statement about the measure of both momentum and position of a single particle nor an effect of the interaction with a measurement device.
Limits for combined measurements on the same system exist, but it is a different story, and there are strong indications that such limits differ from the usual Robertson-Schrödinger result.