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I need some examples of the Heisenberg uncertainty principle on a basic level, or if possible in daily life. Or maybe a simple explanation for validity of the principle in easier words. I cannot get any example except for the measurement of position and momentum of electrons with the help of photons so I decided to ask it here. Are there any simple examples?

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    $\begingroup$ If simple examples of the Heisenberg uncertainty principle occurred in everyday life, then it would have been discovered before 1927. $\endgroup$ – tparker Jul 27 at 15:45
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    $\begingroup$ "Daily life" narratives about the uncertainty principle are inherently going to be wrong, because they don't (and can't) take into account that quantum mechanics describes a deterministic evolution of probability models of possible observed states, not a probabilistic evolution of discrete but unknowable states. $\endgroup$ – R.. Jul 28 at 5:26
  • $\begingroup$ It might be a good idea to watch this youtube video about the bandwidth theorem that I found on a SE Physics question. Not exactly Heisenberg, but same equation and real life radar examples. $\endgroup$ – axsvl77 Jul 28 at 13:23
  • $\begingroup$ Perhaps this might help? physics.stackexchange.com/questions/114133/… $\endgroup$ – Saturnix Jul 28 at 13:58
  • $\begingroup$ imgc.artprintimages.com/img/print/… $\endgroup$ – Hot Licks Jul 29 at 3:12
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If by "daily life" you mean things we experience on a day to day basis at the macroscopic level (as opposed to the microscopic level), while the principle applies it does so insignificantly.

The principle essentially states that you can never simultaneously know the exact position and speed (momentum) of an object because all objects behave like both a particle and a wave at the same time. If you know the exact position, there will be some error in determining the momentum, and vice versa. At the macroscopic levels of every day life the behavior of a ordinary object is overwhelmingly particulate in nature.

Are there any simple examples?

Consider a baseball of mass $0.145\ \mathrm{kg}$ moving at a velocity of about $40\ \mathrm{m/s}$ ($90\ \mathrm{mph}$). The De Broglie wavelength of the baseball is on the order of $10^{-34}\ \mathrm m$. The diameter of an ordinary atom is on the order of $10^{-10}\ \mathrm m$. Consequently, the wavelike behavior of the baseball is too small to observe, and therefore the error in determining the simultaneous position and momentum of the baseball is infinitesimally small.

So what is the relation of wavelength and diameter with the uncertainty in measuring position and momentum. I didn't completely get it.

OK, let me put it another way. First, the uncertainty principle is

$$\Delta x\Delta p \geq \frac{\hbar}{2}$$

or

$$\Delta x \geq \frac{\hbar}{2\Delta (mv)}$$

Where $\hbar$ is the reduced Plank's constant of $1.0546 \times 10^{-34}\ \mathrm{J\cdot s}$

Now our baseball has a mass of $0.145\ \mathrm{kg}$ and speed of $40\ \mathrm{m/s}$ as measured by a radar gun. Assume the radar gun has an accuracy of $1\ \%$. Therefore the uncertainty in our speed (and momentum given constant mass) is also $1\ \%$ or $0.4\ \mathrm{m/s}$. Given this uncertainty, the uncertainty in the position of our baseball, $\Delta x$, is about $9 \times 10^{-34}\ \mathrm m$, which would be many orders of magnitude smaller than the diameter of an ordinary atom. In other words, the uncertainty in the position of ordinary objects is essentially zero.

Compare this to the uncertainty in the position of an electron of mass $9.1 \times 10^{-31}\ \mathrm{kg}$ moving at the same speed with the same uncertainty, which would be about $1.4 \times 10^{-4}\ \mathrm m$. That is 6 orders of magnitude greater than the diameter of an ordinary atom.

A more practical example for an electron is determining the uncertainty in its speed when moving around the nucleus of an atom given that its position is confined to the diameter of the atom. For example, the diameter of a hydrogen atom is about $1 \times 10^{-10}\ \mathrm m$. That would make the uncertainty in the speed of the electron confined to the atom of about $0.6 \times 10^6\ \mathrm{m/s}$.

Bottom line: Although the uncertainty principle applies to all objects, its application is not relevant to the objects we encounter in daily life.

Hope this helps.

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    $\begingroup$ @Aaron Stevens. Thanks for the edit. But isn't it uncertainty in the momentum or position? Of course, the mass is presumed constant for the baseball. $\endgroup$ – Bob D Jul 27 at 16:56
  • $\begingroup$ Well, is the uncertainity in measuring the momentum same as the momentum?ie is del.p=p=mv. Didn't you just put the value of del.p as m×v. Or i didn't get it? $\endgroup$ – Akil Jul 28 at 0:18
  • $\begingroup$ @Akil, No, there are not the same. $\endgroup$ – Shishir Maharana Jul 28 at 4:02
  • $\begingroup$ @AaronStevens Yes Aaron, but I am trying to keep things in terms of momentum rather than velocity because when you apply this to photons you can only refer to momentum as they have no rest mass like the electron and baseball in my example. $\endgroup$ – Bob D Jul 28 at 13:00
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    $\begingroup$ @AaronStevens Now I understand what you are getting at. I erroneously used the actual speed (momentum) of the baseball in my example and not the uncertainty in the speed (momentum). I will revise my answer (the $\Delta x $ uncertainty of 9 x $10^{-34}$ m was based on a 1% uncertainty (radar gun accuracy) in an early draft. I will also revise my electron-hydrogen atom example to calculate uncertainty in speed given confinement of position within the hydrogen atom. Do you think that will do it? Thanks. $\endgroup$ – Bob D Jul 28 at 16:44
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While I agree with tparker saying "If simple examples of the Heisenberg uncertainty principle occurred in everyday life, then it would have been discovered before 1927.", your question reminded me of a problem from Sakurai's Modern Quantum Mechanics:

1.22 Estimate the rough order of magnitude of the length of time that an ice pick can be balanced on its point if the only limitation is that set by the Heisenberg uncertainty principle. Assume that the point is sharp and that the point and the surface on which it rests are hard. You may make approximations that do not alter the general order of magnitude of the result. Assume reasonable values for the dimensions and weight of the ice pick. Obtain an approximate numerical result and express it in seconds.

I really liked this problem because I learned two things: first, that there is a Uncertainty relations between angular momentum and angle, a fact systematically overlooked in teaching of Quantum Mechanics at undergrad level. Second thing is that, while HUP effects in this example is practically unobservable, the answer to the problem was very surprising low: ~100 seconds!

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Maybe I missed it, but somehow no-one seems to have mentioned the Bandwidth theorem. The Heisenberg Uncertainty Principle is simply one manifestation of it. Basically if you have a function $\psi\left(x\right)$, and its Fourier transform $\tilde{\psi}\left(\nu\right)$, then you can have $\psi\left(x\right)$ being zero everywhere except a small region of $x$-axis (precise position), or you can have $\tilde{\psi}\left(\nu\right)$ being zero everywhere except a small region of $\nu$-axis (precise frequency or momentum). BUT you cannot have both. The proof of this is entirely classical.

Manifestations of this are:

(1) cross-talk on radio stations. If you transmit at a specific carrier frequency (momentum) you can only modulate this freqency so fast in time (position) before the bandwidth needed for your signal starts spilling over into nearby channels.

(2) how easy it is to get echo with a sound from a clap. the reason being is that clap is short in time, thus broad in frequency, whilst echo usually occurs at very precise frequency, so if you do not know which frequency the echo is at, a short (broadband) clap will ensure that you will hit that frequency (and others)

(3) Diffraction limit in optics. If you use a specific wavelength (frequency, momentum) for your imaging, then there is a practical limit on how well you can resolve small objects (position) due to tendency of light to diffract. Note that this is classical - no photons needed

There are many more examples. My message is this.

Do not put the Uncertainty Principle on the pedestal. It is a specific manifestation of a VERY common phenomenon.

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    $\begingroup$ Yes it is manefestation of a common phenomenon, but it is the physical implications of it in the context of QM that gives it the status it has. You could say the same thing about many things in QM. For example, a vector being expressed as a linear combination of basis vectors. It's very common, even classically. But the physical implications of it in QM are very interesting when compared to classical understanding. $\endgroup$ – Aaron Stevens Jul 28 at 12:43
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    $\begingroup$ IMHO There are no physical implications of the uncertainty principle. There are implications of Schrodinger equation, or its equivalent in the Heisenberg picture, yes. But where have you seen someone rigorously using Heisenberg principle? Usually it is mentioned in a hand-wavey part of the introduction, and is no-where to be found when the calculations begin (unless you make additional effort to stick it in). $\endgroup$ – Cryo Jul 28 at 12:50
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    $\begingroup$ To me drawing attention to this principle is simply confusing for students. The principle is there, but it is a consequence, not a cause. And the only reason there is confusion is that the students have first been taught to think of electrons as partciles and then told to forget this, and now thing of waves, and then told to forget both and think of quantum fields. $\endgroup$ – Cryo Jul 28 at 12:50
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    $\begingroup$ @Cryo I do agree with your assessment that it is a consequence. I also agree that it does get grossly misinterpreted, confused, and applied incorrectly. However, I don't agree with your reasoning that because of these things and because there are classical manifestations of it that it isn't important. $\endgroup$ – Aaron Stevens Jul 28 at 13:02
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    $\begingroup$ This is wrong. The uncertainty principle for position and momentum looks like the manifestation of the Bandwidth principle, but there is a much more general Robertson-Schrödinger uncertainty principle for arbitrary operators in QM which you cannot explain with such classical "wave mechanics". Fourier analysis cannot explain the uncertainty principle, it is an accident that it looks like it can for the specific case of position and momentum. $\endgroup$ – ACuriousMind Jul 28 at 13:24
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Though it was mentioned, I will reiterate diffraction of light, esp. through a slit. Specifically, light impinging normally on a single slit.

And infinite plane wave as zero transverse momentum; but once it goes through a slit, it's spatial position is confined to the width of the slit, there by leading to an uncertainty in the transverse momentum, which, in the photon model matches the HUP.

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  • $\begingroup$ What happens if there are infinitely many slits? $\endgroup$ – Shishir Maharana Aug 1 at 9:54
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    $\begingroup$ @ShishirMaharana it's just a linear superposition. So 2 slits with no separation is exactly equal to one double-wide slit. And so on to infinity. Meanwhile the double slit experiment is a super wide slit minus a big block in the middle--so then who cares "which" slit the particle goes though? There's only one, with a left and right side. $\endgroup$ – JEB Aug 3 at 2:18
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See, the Heisenberg uncertainty principle is about quantum mechanics, which happens on a microscopic scale. However, we are living a macroscopic life, too big that we cannot see quarks, even atoms, with our eyes. So, you wouldn't be having examples of the heisenberg uncertainty principle unless you're a particle scientist whose "daily life" is about operating hadron accelerators.

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A very practical example-- My wife and I can only determine either the position or speed of the constituent parts the other's body, but not both at the same time. Therefore, it is quote impossible for us to dance together.

The best we can do is to individually stimulate movement in rhythm to some music (which hopefully exists) and maintain a safe distance for independent motion. I've heard that some couples have somehow managed to consistently dance in synchronicity. If physicists ever learn how this is done, I expect there would be tremendous forward progress in quantum theory, computation, and practical engineering applications. As well as much awkward dancing.

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Some lottery or Keno games in the everyday world have their winning numbers selected by a physical (as opposed to algorithmic) random number generator that uses quantum events such as nuclear decay, shot noise or photon reflection / transmission by a semi-transparent mirror. More about this in a Wikipedia article.

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    $\begingroup$ And how is this connected to the uncertainty principle specifically rather than the probabilistic nature of quantum mechanics in general? $\endgroup$ – ACuriousMind Jul 28 at 14:05

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