Confusion about the description of the uncertainty principle

Question

I have heard two descriptions of the uncertainty principle, and I am quite confused about the uncertainty principle now.

The first one is dependent on the wave function of the particle, and it says that if you have something like the sine wave, you are very sure about the momentum, but not the position, because according to the sine wave, it could be almost anywhere, and if you have something that just rises up and goes down once, you are very sure about the position, but not the momentum, since you only have one wavelength to measure.

The second description I heard of is that in order to observe a particle, you must shine light at it. If you shine a large amount of light, you will be very sure about the position, but then your confidence in momentum goes down because the energy transferred from the photons to the particle you're observing, and if you shine a low amount of light, you are able to observe the momentum very well, but not the position.

Which one is the correct one, or are both correct?

Answer 1

The first one is correct, the second is not.

The second definition¹ is actually describing the observer effect. Explanations written by non-experts often mix the two up. But one key difference is that the observer effect only applies to situations where some external "probe" (like a particle) is interacting with the system. The uncertainty principle, on the other hand, applies even to a system which is isolated and not interacting with anything external.

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¹A couple other people have pointed out that these are not really definitions of anything, but I'll use that word for consistency with your question.

Answer 2

While other answers say the first one is correct, there is something that should be pointed out. The issue is with the beginning of your statement:

The first one is dependent on the wave function of the particle...

The Heisenberg Uncertainty Principle is very useful because it actually doesn't depend on the particular wave function. In other words, $\Delta x\Delta p\geq\hbar/2$ is true for all wave functions, not just sine waves.

There are more general uncertainty principles that do depend on the wave function, but those aren't as famous.

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Another thing to keep in mind is that neither of your two statements define the uncertainty principle. Your first statement is the closest to being correct, but even then it's more of an application of it, not a definition.

Also, the uncertainty principle isn't a statement of how "sure" or "confident" we are about the position and momentum of a particle, which seems to be a common idea in both of your statements.

Answer 3

Before answering the question, I would first look at HUP from more technical standpoint:

The uncertainty principle is given by noncommutativity of measurement. When you have wave function $|\psi\rangle$ the measurement changes it to another wave function $|\alpha\rangle$ - this is the famous collapse of the wave function - and produces number $a$, for example component of momentum of the particle. The measurement can be represented as operator: $$\hat{O}_a: |\psi\rangle\rightarrow |\alpha\rangle,$$ where $|\alpha\rangle$ is now the state of the particle with definite value $a$. Before that, the particle could have been in superposition of states with several possible values of measured quantity, but once you measured it, you collapsed the wave function to that particular state. Because, now the particle is in the state of definite value $a$, succesive measurement will produce the same number $a$.

Now what if you decided to immediately after this measurement measure different quantity? Again you measure the value of $b$ and collapse the wave function to wave function of this particular state: $$\hat{O}_b: |\alpha\rangle\rightarrow |\beta\rangle.$$

The uncertainty principle follows from the fact, that measuring the quantity $a$ first and then the $b$ is not equivalent to doing it the other way around. That is: $$\hat{O}_b\hat{O}_a \neq \hat{O}_a\hat{O}_b $$

To show this would take some time, but intuitively this makes sense. If the particle could have definite value of quantity $a$ and $b$ at the same time, then measuring it should produce those two values. But since the values are already given, then it should not matter which you measure first. We know, however, it does and therefore the particle cannot be in state with definite value of $a$ and $b$ at the same time. These two values are simply incompatible. If the particle is in the state of definite value of $a$, then it must not be in state of definite value of $b$. The most notorious example of such quantities is position and momentum you wrote about.

This is however not really property of the wave function of the particle as such. It is property of the operators $\hat{O}_b$ and $\hat{O}_a$, i.e. property of the measurement itself. Every such operator/measurement has some wave functions associated with it, which are wave functions of definite values of the measured quantity. And this wave functions associated to the operators/measurement are simply incompatible.

Now to answer the question:

The first "definition" is taken from the point of view of wave function. It says that when you have wave function with definite value of position, then it is not function of definite value of momentum and vice versa.

The second "definition" is taken from the point of view of operators. It tells you that measurement of position changes the wave function in such a way, that it is now in the state of superposition of many momenta and there is no answer to which of the these momenta particle has and vice versa.

They are therefore equivalent. But note neither of your "definitions" is really a definition. They are more like different interpretations of HUP.

Answer 4

They are in fact both correct, but the relationship is not obvious. Heisenberg explained his original argument for the uncertainty principle using a thought experiment called Heisenberg's microscope, which was essentially the second argument expressed a bit more thoroughly. Heisenberg took the argument further, showing that it leads to a reductio ad absurdum, meaning that position and momentum are not precise quantities, as assumed in classical mechanics, and can only be defined in terms of probability. His argument shows that this is a matter of principle, not one of technological limitation.

An observer seeks to measure the position and momentum of an electron of known momentum by bouncing a photon off it. If the photon has low energy, so that it will not disturb the momentum of the electron, then it has long wavelength and position cannot be discovered to any accuracy. If the photon has short wavelength, and correspondingly high energy, position can be measured precisely, but the photon scatters randomly, and an unknown momentum is transferred to the electron.

Heisenberg then noted that any attempt to measure the momentum transferred to the microscope necessarily results in a loss of knowledge of position of the microscope. Likewise, better measurement of the position of the microscope leads to loss of knowledge of its momentum. Attempts at more precise measurement of one property necessarily result in less precise determination of the other. This applies to the apparatus, and to any further apparatus used to measure the apparatus and so on ad infinitum.

Consequently, in the general case, the position and the momentum of an electron can only be stated in terms of probability distributions, where the more precise the probability of the one property is, the less precise is the other.

This argument can be used (as can others) to motivate the probabilistic structure of quantum mechanics, in which is based on the (mathematical) principle that probabilities can be expressed in terms of wave functions obeying the Born Rule. It is then possible to derive the relationship which is now called Heisenberg's Uncertainty principle:

• uncertainty in position multiplied by uncertainty in momentum is greater than Planck’s constant divided by 4$\pi$

The derivation was not given by Heisenberg, but by Earle Hesse Kennard, in 1927.

Answer 5

Your first interpretation is correct as the second one merely implies that the observational capabilities are the only things which seem to effect our measurements on the momentum and position of particles. This is not the case. H.U.P is not limited by our technological abilities but is integral to the entirety of quantum mechanics i.e. it is just as good as a law. I've often found that the second one is used to explain the principle to those who initially deny a probabilistic universe and believe in absolute measurements (like me ;D)

The first interpretation uses the Fourier series (for a detailed and fabulous description-look here) and showcases a more advanced understanding in how eventually on combining (or rather-deconstructing for Fourier) waves can show us an uncertainty in position and velocity. It is more appropriate as it gives you insights in the mathematics and helps us to understand that HUP is the explanation to a phenomenon.

As a prof. once told me: A universe with H.U.P may lead to many fabulous things but a universe with the laws of quantum mechanics must begin from H.U.P.

Edit: Look into the observer effect as pointed out in David Z's answer. Missed that out here.

Answer 6

The basic premise of quantum mechanics is that every sub-atomic particle is (or is associated with) a wave. To be more precise a wave packet of finite size. If the wave is very long, the frequency and wavelength (but not the position) can be determined with high precision . If the packet is short, Fourier analysis says it can be described as a superposition of many long waves with a spread of frequencies (and energies).