We are covering the uncertainty principle in my modern physics class, and I have a few questions regarding it.
I think my professor presented it in a rather cumbersome way. Using Fourier analysis, he showed how if you try and make a more narrow probability function for momentum, you get a broad peak for position. While this mathematical treatment is all well and good, I feel like we skipped the lecture on the underlying reasoning of it.
So, after a brief web search I found this article. It presents the historical development of the uncertainty principle, focusing on views of Heisenburg and Bohr. From what I can tell, Heisenburg’s argument is that at the quantum level, you cannot measure both position and momentum (or energy and time) with arbitrary precision because by taking a measurement you input energy into a system. This, and his argument of observing an electron in a microscope I can follow perfectly well.
However, Bohr’s ideas, outlined in the same article, I can’t really make heads or tails out of. For example:
Maybe I missed something, but the article doesn’t seem to explain why they cannot be simultaneously defined. Or is that a meaningless question? If the answer to it is “just because”, I’m suppose I’m willing to accept that, but it seems like I’m missing out on a deeper understanding.
So, in summary:
How is the uncertainty principle a consequence of wave-particle duality?
And how does Bohr’s view of it differ from Heisenburg’s?
I’m not sure if this helps, or even if I’m right (great, eh?), but, energy inputs and measurement technologies notwithstanding, I see it like this:
In order to absolutely measure the position of an object, you’d have to narrow down the period of observation to a single, dimensionless point of time (if that were possible), but within a zero-duration point in time, movement cannot be observed.
It’s not just a case of measurement. It is incorrect to talk of a quantum entity having an exact postition or momentum. I think the Bohr approach was that all you should consider is the result of your experiment or calculation, trying to mentally model what is “really going on” is looking for trouble.
I’m as confused as you are. The “duality” is something you impose on (say) an electron when you measure something about it. It’s not sitting there in the electron making it uncertain about where it is
As usual I step back and wait to be shot down by a real physicist.
Said real physicist will probably just say “Shut up and do the maths”. Quantum theory makes NO SENSE so don’t waste time thinking about it.
Heisenberg was wrong about one thing – the uncertainty principle is unrelated to measurement. Measurement is a way of trying to explain a tough concept that many people can understand. However, it holds whether you measure it or not.
As for why – there is no why. One of the frustrating things about quantum mechanics is that many of the phenomona are completely counter-intuitive, and there doesn’t seem to be a clear mutually agreed upon way to describe the phenomona outside of mathematics. The Uncertainty Principle is like that – once you get outside the math, the why’s don’t really make sense.
I think the issue that Bohr is concerned about is the duality of electrons. IIRC, and I very well could be wrong, electrons someimes behave like a wave and sometime behave like a particle. I know that light does this.
The difficulty arise sin trying to pinpoint an exact position for a wave.
That’s wrong, the uncertianty principle is certainly about measuremnt; on one side of the inequality is the product of the root-mean-square deviations of two observables for a set of repeated measurements. Infact quantum mechanics only tells us what happens when we make a measuremnt, it doesn’t tell us what goes on between measuremnets (though not everyone would agree with that).
To answer the OP, you have to rmeber that when Heisenberg first derived the HUP all he did was to prove that the unceratinty between postion and momentum was approximately equal to h (in the precise form it is greater than or equal to h-bar/2) and it’s certainly possible to derive this approximate relationship by considering optics and the wave-particle duality of photons (this as you’ve probaly guessed is related to Fourier transforms) there are many simplified versions of Heisenberg’s orginal argument on the web. Unfortunatley I really wouldn’t know which way to go to derive the precise form, except by considering the fundamnetal postulates of QM and Schwarz’s inequality
Because if you try and make a more narrow probability function for momentum, you get a broad peak for position.
Really, the functional analysis is the best way of doing it, since it makes no mention whatsoever of what experimental technique is under consideration.
This is a fundamental divide between those who interpret the Copenhagen Interpretation ontologically and those who interpret it epistemologically. That’s a Great Debate, not a General Question.
I don’t see where optics enter into it. The Fourier analysis enters into it because the Fourier transform is a change-of-basis in the Hilbert space of states. Expand in eigenstates of position (the operator f -> x*f), interpret components as probabilities and calculate the standard deviation of a position observation. Alternately, expand in eigenstates of momentum (f -> idf/dx) and calculate the standard deviation of a momentum observation.
Where does wave/particle enter into it? Simple: particles are position eigenstates (it’s here), while plane waves are momentum eigenstates (it’s moving that way).
If you consider your particle as a wavefunction, then you can precisely denote its wavelength (which is proportional to momentum), but then you have no idea of the particle location. A sine wave has no unique location in space. If you try to pin down location by describing the particle as a superposition of waves, you can localize it better and better the more terms you add. But each term is a different wavelength, so in the intermediate case you have a “wave packet” that is made up of a mix of wavelengths (and therefore a spread of momentum), and is narrowedc down to a certain region in space, so neither position nor wavelength/momentum are exactly defined. The extreme case is when you have precisely located a position in space (a delta function in position). But in order to construct such a wavefunction you need an infinite range of wavelengths. So if you know exactly where the particle is, you have no idea of its wavelength/momentum.
And all of this is independent of any measurement. You simply can’t even construct a theoreetrical waveform which simultaneously has a well-defdined wavelength and position.
The quantuj formalism itaself only speaks about the results of mearuemnts it says nothing about what ‘physically’ happens inbetween measurements (of course it does tell us about the time evoltuion of the wavefunction inbetween measuremnts, but the quantum formalism delibrately and sensibly does not attatch any physcial interpretation to the wavefunction), though some would certainlycontend that it is not impossible to say what happens inbetween measurements.
Optics certainly did entire into it for the original (but imprecise) derivation which was based on physical arguments specifically from optics (see Hesienberg’s The Physical Principles of the Quantum Theory 1930, which includes the orginal physical arguments from his 1927 formulation of the HUP).
Actually, the Fourier analysis method is the most elegant way to explain it. Since position space maps to momentum space thru a Fourier transform, the HUP drops out of that like a lead brick. What’s the FT of a delta function? QED.
I’m not sure it makes sense to talk about “what really happens” as opposed to what the equations predict. If we knew “what really happens” we’d change the equations to reflect that. Our mathematical represtentation is the best approximation we have of “what really happens”.
While using position and momentum to try to understand complementary may help to visualize it, there is always a danger that you come to reduce complements to just this one example. You shouldn’t because it’s much more basic than that. There are an infinite number of “cannonically conjugate” qualities that the uncertainly principle represents.
Go back to Schrödinger’s orignal statement on the subject as quotes in this article on Quantum Entanglement and Information from the Stanford Encyclopedia of Philosophy":
Forgive me. I see now that the Fourier transform is a rather elegant way of showing the relationship between two canonically conjugate quantities. I was just stuck with Heisenburg’s original assumption, that the uncertainty principle is a consequence of measurement. I guess there was comfort in the fact that I could visualize it.
Anyway, I’m certain now, QM is one big elaborate Zen koan.
