Tangibly Explaining HUP to General Chemistry Students

[Zeeboozoo]

I understand this topic has been brought up before, however there appears to be "truth" or "lies" and very little in between when it comes to explaining the Heisenberg Uncertainty Principle (HUP) to students of freshman chemistry.

I'm teaching university general chemistry to students who've had no physics. Some have vague memories of trigonometry.

I need them to have an understanding of HUP. I've gone the route of explanation using compression of sin waves before with no luck - they're simply left confused. Treating an electron like a particle and speaking of uncertainty in momentum introduced via conservation of energy principles upon measurement works, however is argued to be "heresy." Some students can "see" the wave function is collapsed once position has been determined (via a picture), however they don't understand it nor can they envision it.

If anyone has come up with a good educational model for students who have not yet taken physics, I'd be very grateful to hear from you.

Thanks!

 

[vanhees71]

The uncertainty relation (at least the standard one, called Heisenberg-Robertson uncertainty relation, HRUR) has nothing to do with being able or not able to measure quantities accurately. That's the first important point in teaching QT to begin with. The uncertainty relation is a restriction of how accurate you can determine simultaneously two incompatible observables for the given system. The general uncertainty relation says: For any (pure or mixed) state ##\hat{\rho}## and any two observables ##A## and ##B## represented by the corresponding self-adjoint operators ##\hat{A}## and ##\hat{B}## one has
$$\Delta A \Delta B \geq \frac{1}{2} |\mathrm{tr} (\hat \rho \mathrm{i} [\hat{A},\hat{B}])|.$$
Here ##\Delta A## and ##\Delta B## have the meaning of being the standard deviations of the observables for the system prepared in the state represented by the Statistical operator ##\hat{\rho}##, i.e.,
##\Delta A^2 = \mathrm{tr} \hat{\rho} \hat{A}^2 - (\mathrm{tr} \hat{\rho} \hat{A})^2.##

I don't know what you mean by "compression of sine waves", but it's clear that you cannot use a "sine wave" to exemplify the HRUR. A sine wave is a superposition of two plane waves, which cannot represent a particle state, because they are not square integrable. They are generalized eigenvectors of the momentum operator, i.e., distributions. They live in the dual of the domain of the momentum operator, which is larger than the Hilbert space of square integrable functions (the domain of the momentum operator is of course smaller, but it's a dense subspace).

Although you'll not be able to explain the freshmen students these mathematical subtleties, you do them a big favor, by not telling them the usual hand-waving ideas you find even in some (otherwise maybe even good) textbooks! At best the students don't need quantum theory later on, and then forget the imprecise hand-waving without any loss, but if they later on need quantum theory they'll have a hard time to unlearn the unsharp concepts again, and it's harder to unlearn wrong concepts than to learn the right ones from the very start, although it may take a bit more time and effort first. That's at least my own experience with the hard path learning quantum theory, and the math isn't the hard part but it's the physics concepts!

 

[zonde]

Do you need quantitative or qualitative explanation?
I can propose simple qualitative explanation: particles in a beam don't take the same trajectory i.e. you can't squeeze beam of parallelly moving particles over some limit. This seems to describe the things one would observe in experiments.

 

[sandy stone]

Have you tried something like this? https://www.google.com/search?site=...mg..0.9.327.kb45TzMhxz4#imgrc=ruFRSw_zyNc5jM:

 

[Zeeboozoo]

First, I'd like to thank everyone for their responses. The students are general chemistry students (age 18), meaning they have no calculus, no physics, and essentially no point of reference upon which to base tangible comparison. What is desired is mostly qualitative with a little bit of quantitative to make it reasonably "not a lie," hence the line of reasoning proposed by Zonde is most closely related to what my students need. Every tangible explanation I've seen is based upon the "observer effect."

Vanhees, I do have a question as this appears to be an area wherein you are quite knowledgeable. How do we say to 18 year old students, with them completely understanding us, that HUP has nothing to do with being able to measure quantities accurately... but has to do with how accurately you can determine simultaneously two incompatible observables for a given system? In other words, what is the difference amongst the words "measure," "determine" and "observables" in the sense that you've employed them. Unambiguous definitions and word associations quite often prove requisite in teaching. Thank you.

 

[vela]

You could do a qualitative overview of Fourier analysis. You don't have to go through the gory mathematical details, but you can show illustrations of the same signal represented in the time domain and in the frequency domain. Point out that this is what the spectral analyzer on their stereo is doing. Then show how if you narrow one representation, the other one spreads out. Then note that quantum mechanics revealed to us that position and momentum have the exact same relationship. This tends to avoid the whole confusion that arises from bringing up measurements. The HUP is just a requirement that the two representations of the electron's state have to satisfy.

Then you can talk about how this requirement manifests physically, like zonde's example or how the HUP limits how small an atom can be.

 

[zonde]

You could do a qualitative overview of Fourier analysis. You don't have to go through the gory mathematical details, but you can show illustrations of the same signal represented in the time domain and in the frequency domain. Point out that this is what the spectral analyzer on their stereo is doing. Then show how if you narrow one representation, the other one spreads out.

Speaking about spectral analyzer on stereo and Fourier transformation another example is the difference between wav (amplitude encoding) and mp3 (frequency encoding) audio formats. So maybe it is possible to make some arguments about compressing signal in wav format and mp3 format and how you can't pick a signal that makes both files smaller. ... just an idea.

 

[vanhees71]

First, I'd like to thank everyone for their responses. The students are general chemistry students (age 18), meaning they have no calculus, no physics, and essentially no point of reference upon which to base tangible comparison. What is desired is mostly qualitative with a little bit of quantitative to make it reasonably "not a lie," hence the line of reasoning proposed by Zonde is most closely related to what my students need. Every tangible explanation I've seen is based upon the "observer effect."

Vanhees, I do have a question as this appears to be an area wherein you are quite knowledgeable. How do we say to 18 year old students, with them completely understanding us, that HUP has nothing to do with being able to measure quantities accurately... but has to do with how accurately you can determine simultaneously two incompatible observables for a given system? In other words, what is the difference amongst the words "measure," "determine" and "observables" in the sense that you've employed them. Unambiguous definitions and word associations quite often prove requisite in teaching. Thank you.

This is difficult, if not impossible. I don't think that you can explain quantum theory properly without the underlying math. To explain the uncertainty relation qualitatively, you have to first define what an observable and a state is and what quantum theory tells us about these concepts. The content of the Heisenberg uncertainty relation is that for two incompatible observables there is no state, where both observables have a determined value. Now you have to explain that quantum theory tells us that the state only provides statistical information about the observables, i.e., probabilities for the outcome of meausurements of observables. The uncertainty relation tells us that for two incompatible observable if one observable is pretty well defined, i.e., if its standard deviation given by the statistics entailed in the state is small, the other observable is quite undefined, i.e., it has a large standard deviation.

So to explain the HRUR properly, you have to explain a minimum about probability theory and that a state contains only probabilistic information about the outcome of measurements. The less math you can use, the more difficult it is to explain and also to understand by the students.

 

[Zeeboozoo]

Everyone, thank you for your responses. They've all made me think. Upon searching for video illustrations for background and tangibility.





This, in combination with some embellishment, will probably get across to 50% of the population (pardon the pun please).

Happy Holidays!