This has probably been brought up before but most introductions into quantum physics state that the reason for the uncertainty principle is that we cannot measure the momentum and velocity of a particle with absolute precision because the measurements thmselves will affect one or other of these variables. However it is also said that a particle cannot have a precise position and momentum at the same time (inherently). To me these are completely different concepts. If the latter is true, can somebody explain in laymans terms why it is true (from what was known of quantum theory before Heisenbergs "discovery")? Even a link will do. Also, if it is true, why do textbooks and introductions completely bypass this important explanation and instead start talking about how measurements affect particles?
So we measure motion with a ruler and a stop watch. That is, we measure distance traveled and elapsed time of travel. Obviously, in order to have measured the time it has already elapsed. That means that we are looking at the results of a previous motion, it has already happened, the stop watch stopped and the results of the measurements of distance and time are complete. Say the object traveled 10 ft in 1 second. If the velocity was a constant, any time period you look at the math will add up. At the .5 second mark it had traveled 5 ft. At the .25 second mark it had traveled 2.5 ft, and so on. You can never pinpoint an exact location, just reduce the visibility of the motion, ie, 2.5 ft in .25 seconds is 10 ft/sec. You are only reducing the visibility of the motion in proportion to the reduction in distance and time, it is still 10 ft/sec. You are losing visibility of the motion in order to try and pinpoint an exact location the object was at any "point in time." There is the problem right there. A duration of time is not a point in time, and you will never get it to be one by reducing the distance and time of an object's previously measured motion in proportions, as it will always be 10 ft/sec, and the object was never at one point in space at any one point in time, as the object was in motion.
Which is exactly what I said we are taught at the start but then we are supposed to accept that a particle does not even have a precise momentum and position. That's the part I don't understand.
I am not surprised. You said I don't think his is quite correct. The uncertainty principle states that we cannot simultaneously know the position and momentum of a particle to arbitrary precision. Note that that momentum, velocity and energy are intimately related via the de Broglie postulate that \(E = \frac{p^2}{2m}\) for any particle. (The following was given to me by a drunken physicist late one night on the back of a beer mat. I have tried to fill out some details) To any particle \(p\), (edit: this is not the same \( p\) that appears as the numerator in the de Broglie equality above; the first is momentum, the second is a particle. Sloppy, sloppy!) one can associate a wave function \(\psi\) with the property that \(H\psi = \epsilon \psi\) as the allowable eigenvalues for the energy of \(p\), where \(H\) is the Hamilton operator. (It is an elementary fact from linear algebra that to any operator we can associate a spectrum of eigenvalues ) Then it seems that \(\psi^2\) represents the probability of finding \(p\) at any specific place. (I suspect this is probability density) This makes sense; clearly the solution to \(\psi\) is in the real interval \([-1,1]\), thus the solution to \(\psi^2 \in [0,1]\), which is a definite requirement for any sort of probability. So to the meat of the matter. a priori we have no information about our particle's energy, so we must allow all permissible \(\epsilon\) for which \(H\psi=\epsilon \psi\) holds. Fine. Now suppose we actually know where \(p\) is, say at \(x\). Then here, \(\psi^2 = 1\) and \( \psi^2 =0\) at all other locations. But the only way to generate such a \(\psi^2 = 1\) at \(x\) and \(\psi^2 =0 \) elsewhere is by allowing all its possible (first order) wave functions to interfere destructively except at one place \(x\). Hence we cannot make any assumptions about its exact energy. Conversely, if we know the exact energy of our particle, there is one and only one (first order) wave function that will describe this energy, and hence no interference can occur, so that to second order there is no non-probabilistic way to describe its position. But wait for a sober physicist to check it - if such creatures exist.........
Sober Professor of Physics here..Please Register or Log in to view the hidden image! Here's a try using plain language. One of the things I find interesting about Quantum Mechanics is that the mathematics is fairly straightforward, but the concepts are foreign to us -- primarily because we have educated our intuition by interacting with the universe at large distance scales where quantum phenomena are more difficult to discern. The quantum concept here is that any description of a particle's state has an average and a distribution centered around that average. This is true whether it is position, momentum, angular momentum, etc. So, as a result of our intuitive understanding of the universe, when we talk about a particle's position we intuitively think the particle is described as being at a particular position, and not at other positions. Same with momentum. But in general, this is not true. The quantum mechanical description of where the particle is would have a description of its average position and a description of all the positions it occupies at the same time (with some distribution that insures the particle has a probability of existing equal to one). The difficult part here is that the way in which you think about particles before you have studied quantum mechanics doesn't correspond to reality. The problem is in how you think about particles. You don't think about them as fuzzy objects, somewhat indeterminate in where they are or where they are going, but that is the quantum mechanical reality. (Here is where language usage also gets in the way - we use the word particle, which brings to mind a distinct infinitesimally small object, not a distribution -- that is until you study quantum mechanics.) Once you get your head around the fact that particles really are described by these fuzzy distributions, the next thing to realize is that some of these distributions are coupled. This is what the mathematical explanation above is demonstrating. By coupled, I mean that while a particle can have any average position and any average momentum, their respective distributions around those averages are inversely coupled. If one is small (a tight distribution around the average), then the other one is large (a very broad distribution around the average). Similar relationships occur for other variables like angular momentum and angles, as well as energy and time. The hard part of all of this is getting your intuitive response to be in line with quantum mechanical laws. Most of the difficulties, seeming paradoxes, etc. with quantum mechanics stem from deeply held thoughts about the way things are that are not in tune with quantum mechanical law. If you can get your mindset totally consistent with these, then you find there is no problem. The problem lies in the way we have come to think about the universe by only observing it at large distance scales. I hope that helps.
Excuse me please, that was a typo which is now corrected - I meant of course position. OK, first question - why do we have no information on the particles energy exactly? OK, so are you saying that the so-called "collapse of the wavefunction" is actually destructive interference of the wave at other values? Is there not only one wave describing the probability of, for example, position? If there is only one wave, how does interference occur?
The measurement piece I forgot to talk about the measurement piece. The basic problem with understanding measurement is that what we measure for any dynamical variable like position or momentum is not necessarily a statement about what state the particle was in before the measurement. When a measurement requires a result within a particular range of uncertainty, the wavefunction will collapse to a value within that range, but that doesn't mean it had that range of uncertainty before the measurement. The measurement itself interferes with the state. So both of the descriptions you originally mentioned are basically correct, they are just talking about different aspects of the quantum state.
Hi sciguy137, thanks for your contribution and welcome to Sciforums! Believe me, I have no problem at all with those quantum concepts and I understand roughly how the quantum world works as explained by popular science. However I have just never heard the proper explanation of the uncertainty principle explained in any textbooks and instead they ususally resort to the easier explanation that it s all to do with the measuring equipment etc. The paragraph that I have quoted above is what interests me most as this is answering my question. Presumably you are talking about the complementarity pairs which I have indeed heard of before. What's the reason that some of these wave functions are paired like this to each other? For example, why is it position with momentum and energy with time? Is it something to do with relativity?
Complementary dynamical variables There are many ways to answer your question, all of which are related. I often prefer to start with the simplest description using a minimum of mathematical sophistication and then increase the level depending on the background of the questioner. So, since we are trying to represent wave-functions, we could look at waves in space of the form sin(kx) or cos(kx) using 1-dimension for simplicity. Note that x is position and that (hk/2*pi) is momentum. If I want to create a function that is localized in space I need to add a lot of waves that are nearby in k value. To have the function constrained to a very narrow range in space you need to add together a semi-infinite set of waves with very different k values. Since k is directly related to momentum for each component wave you can see that if we are to create a wavelike disturbance in space then the width of the spatially localized distribution is inversely proportional to the distribution of k states, or momentum states, that make up this localized spatial distributrion. To go one level higher in mathematical description, the distribution in space and the distribution in k-values are Fourier Transform pairs. Each distribution determines the other. To go another level higher in mathematical description, the variables x and p (or hk/2*pi) do not commute -- i.e. xp(psi)-px(psi) does not equal zero (where psi is a wavefunction). So any two variables that are involved in a single description of a wave ( like k and x) must be related by an uncertainty principle. In the same way, a wave like sin(Et/(h/2*pi)) shows that E and t are relatedby an uncertainty relationship. I hope that helps.
