Teacher needs help: Bra–ket notation for parabolas?

[whitsona]

I am about to do a unit on the wonders of vector decomposition for year 2 in a 3 years science curriculum where the kids follow along with the historical developments of science.

It's the science class I wish I'd had before becoming an engineer at MIT.

In year 3, we are with Einstein and Feynman. We do Markovski diagrams OK. But, the kids eventually get a shell-shocked look about them as the math turns into squiggles before their eyes and they stop believing me when I tell them there is stuff they can grok in that soup as we march through to Schrodinger and Bell's Theorem.

But, in year 2, the kids really grab hold of a lot of concepts with the parabola unit. And, the vector decomposition is a big win. I have students every year who are really worried that their ping pong balls won't trace one out as we film them because they built their catapults incorrectly. And, then they have a moment of pure physics joy when those simple. little invisible vectors work together an make that beautiful parabola happen.

This has me wondering if I could express a parabola in Bra-ket notation for them now-- so that next year they will have something familiar to grab onto when we get this far.

But, I never covered this part in my formal education (just an engineer, not a physicist) and I'm not sure how to get the notation right. Anyone inspired to help me out? Many thanks!

 

[mfb]

We do Markovski diagrams OK.

Minkowski diagrams? There are also Markov chains but they don't fit in thematically.

Bell's theorem a year after free fall sounds ambitious.

This has me wondering if I could express a parabola in Bra-ket notation for them now

Technically you can ( $|p \rangle$ ), but I don't see any useful way to do that.

You can predict interference patterns and compare them with the experiment in the classroom. If the students learned about waves before this should be quite intuitive.
While you can't do that experimentally, you can tell that the same experiment can be done with electrons, atoms and even small molecules, and you can calculate why you can't demonstrate that.

If it has to be something related to parabolas: We can let neutrons bounce above a surface, and measure their quantized "parabolas". Here is an article from 2002.

 

[whitsona]

Ha! Markovski was brain fog. Did Markov chains when I was a working engineer. This gig and this subject are not old hat for me yet. (Did not even know that Minkowski was one of Einstein's professors who refused to give him a reference until I started this teaching thing.)

I think I wasn't clear about the question I was asking. [restated at the end] I am up at 3 AM tonight taking apart this phenomenal pair of videos and turning them into a lecture on Bell's Theorem in a really, really simplified way when we start talking about superposition and quantum erasers. We are not doing the math. But, like they did in this video, I spell out relationships and we get a general sense of things.

Bell's Theorem: The Quantum Venn Diagram Paradox by Minute Physics.
Some light quantum mechanics (with MinutePhysics) by 1 Brown 3 Blue

What am I really teaching? A really advanced version of what they used to call Earth Science-- as a prep course for kids who will quickly go into AP specialization. We use the context of the historical development and the excitement that they have about doing real Quantum stuff to keep the broader for longer and get cross-disciplinary. It's a series put out by the NSTA and Smithsonian by Joy Hakim. And, it's amazing. Sounds terrible. But, it's really a fresh look at stuff like phase change, Snell's law, etc. Simple stuff with "things get Quantum" tie-ins.

What I am doing _right now_ in year 2 has a lot of implications for where things are falling apart or teetering in year 3 for the kids right now. In year 2, they see that what happens to Galileo's projectiles is a superposition of:

• the "a body in motion tends to stay in motion" in the x-direction resulting in even steps every second because there are no opposing forces
• and the velocity in y being reduced every second by gravity so that our upward velocity in y becomes a downward one

What I want to do is overkill. I want to use ket notation to talk about how the up-down is happening at the same time as the side-to-side and they are superimposed on each other-- when we simply launch a cannonball and make a parabola. But, I'm not familiar enough with the notation to be sure I am getting it right.

 

[whitsona]

Does it help to see what I want to do if I say-- don't think about the need for the numbers to be imaginary. The imaginary axis is just the y-axis. So, to determine the position of the parabola, I think I superimpose $y(t) | y \uparrow >$ and $x(t) | \rightarrow>$

So, would I write that as $y = v_{y initial} \cdot t + \frac{1}{2}at^2 |y \uparrow>$ and $x = v_{x initial} |x \rightarrow>$? Or some variation close to this? I want to get just a glancing familiarity with the symbols without butchering them too much. Dealing with fear of math symbols is 90% of my issues with them. Their understanding is phenomenal when they can see through the math anxiety enough to answer intuition questions.

 

[protonsarecool]

So, just to be clear, you just want to write the position vector $\vec{r}=\begin{pmatrix} v_{x, ini} t \\ v_{y, ini} t + \frac{1}{2} a t^2 \end{pmatrix}$ in braket notation, right? So you can just call the unit vector in x direction $\begin{pmatrix} 1 \\ 0 \end{pmatrix} = |\rightarrow>$ and the unit vector in y direction $\begin{pmatrix} 0 \\ 1 \end{pmatrix} = |\uparrow>$ and then write the whole thing as
$$|position> = v_{x, ini} t |\rightarrow> + (v_{y, ini} t + \frac{1}{2} a t^2) |\uparrow> . $$
I'm not sure if there is much value in this, as you cannot do a lot with the notation at this stage. It might just seem to the kids as an unnecesary complification of the notation. The real use of brakets is in the intuitive writing of linear algebra concepts like decomposition of operators and scalar products, stuff like
$$\hat{A} |b> = \left(\sum_{i} a_i |a_i><a_i|\right) \cdot |b> = \sum_{i} a_i <a_i|b> |a_i>. $$
But those things are obviously way over the head of these kids at this stage and completely unnecessary to describe simple parabolic trajectories. So overall i would probably rather stick with the standard column vector notation.

 

[mfb]

I don't think that helps, and I would expect it to lead to more confusion.

 

[bhobba]

If you want to introduce Bra-Ket notation simply teach some linear algebra that way - it's not the standard notation for that area, but its widely used in some areas like QM, distribution theory, Rigged Hilbert Spaces and White Noise Theory. Of those only distribution theory could really be taught at lower levels. But it is very elegant, even more elegant than the standard notation IMHO - I personally prefer elegance over practicality - but that's just me.

Thanks
Bill

 

[whitsona]

If you want to introduce Bra-Ket notation simply teach some linear algebra that way - it's not the standard notation for that area, but its widely used in some areas like QM, distribution theory, Rigged Hilbert Spaces and White Noise Theory. Of those only distribution theory could really be taught at lower levels. But it is very elegant, even more elegant than the standard notation IMHO - I personally prefer elegance over practicality - but that's just me.

Thanks
Bill

Bill,

Yes! You have it exactly. I am working through some intro Linear Algebra stuff right now on 3blue1brown's page now that will also be incorporated in this extra extension that has the parabola in it. I'm afraid that I was one of the people who went through linear algebra with more memorizing of computational formulas than understanding. So, I have to go slowly when taking what I do know and moving it into something I never used like the ket notation. But, I think I am getting the hang of it now.

Tha parabola is a "hands-on" part of the Unit. It's a great "get up, move around, do math" activity. Can you think of any other hands-on linear algebra activities that I could include? (kets or no)

 

[whitsona]

So, just to be clear, you just want to write the position vector $\vec{r}=\begin{pmatrix} v_{x, ini} t \\ v_{y, ini} t + \frac{1}{2} a t^2 \end{pmatrix}$ in braket notation, right? So you can just call the unit vector in x direction $\begin{pmatrix} 1 \\ 0 \end{pmatrix} = |\rightarrow>$ and the unit vector in y direction $\begin{pmatrix} 0 \\ 1 \end{pmatrix} = |\uparrow>$ and then write the whole thing as
$$|position> = v_{x, ini} t |\rightarrow> + (v_{y, ini} t + \frac{1}{2} a t^2) |\uparrow> . $$
I'm not sure if there is much value in this, as you cannot do a lot with the notation at this stage. It might just seem to the kids as an unnecesary complification of the notation. The real use of brakets is in the intuitive writing of linear algebra concepts like decomposition of operators and scalar products, stuff like
$$\hat{A} |b> = \left(\sum_{i} a_i |a_i><a_i|\right) \cdot |b> = \sum_{i} a_i <a_i|b> |a_i>. $$
But those things are obviously way over the head of these kids at this stage and completely unnecessary to describe simple parabolic trajectories. So overall i would probably rather stick with the standard column vector notation.

Thanks! This is perfect. I get your concern about adding more confusion. I am not going to have them do any operations with it. I just want them to see the symbols and the concept along with the activity so that the symbols and some of their meaning are familiar to them when they get to later classes. And, I wanted to make sure I didn't present the notation in an incorrect way that someone would later have to correct in their minds.

 

[protonsarecool]

Ok, i can understand this approach. Glad i could help. :-)

 

[bhobba]

Can you think of any other hands-on linear algebra activities that I could include? (kets or no)

Eigenvalues and eigenvectors in solving differential equations. Why Markov chains usually converge and what's going on when they 'oscillate', then its relation to imaginary eigenvalues.

Their are many books on applied linear algebra for beginners to give ideas eg
https://www.amazon.com/dp/0131473824/?tag=pfamazon01-20

Thanks
Bill