* This joke is for everyone who’s ever loved (or hated) the quantization of total angular momentum, J. The z-component of total angular momentum is denoted by Jz.
For a given j (the quantum number that characterizes the magnitude of total angular momentum as J=ℏ√j(j+1) ), Jz can have 2j+1 different values.
* In case you don’t really follow pop-culture, the joke refers to Jay-Z’s song “99 Problems,” which has the infamous chorus line: “I got 99 problems but a bi*ch ain’t one”.
The semester started last week. I am again teaching a junior/senior elective for majors and it looks like it might be a rough semester.
The course I am teaching follows a basic, required course in the major. I find the students are poorly prepared, more poorly than the class I had last semester. The students are quiet and look positively terrified. I know it takes a little while for people to warm up and start answering my in-class questions, so that will come with time. But I am being quite alarmed by all the things that they say they have never seen before, because, if that’s true, then I have to significantly rethink the class. Sure, I suppose they might be fibbing, but I do believe think most of them have simply never seen the material or, if they have, then it really didn’t stick at all and they genuinely don’t remember it.
Last week and this week, we are reviewing the material from the previous course, and it’s going very slowly. I may have to take more time simply to get them up to where they would actually need to be, which means cutting out some of the new stuff.
Also, the lack of facility with math always rears its ugly head, but at least that’s not particularly surprising. Any physical science field that requires a lot of physics has to be taught math really rigorously, and the math department does a very good job. The problem is that, owing to a recent
idiotic progressive change in the curriculum for our major, which is supposed to give the students more flexibility to freely choose easy courses outside of the major customize their program of study, some important formerly required math courses have now become electives (e.g. how can linear algebra and differential equations not be required just fuckin’ blows my mind) and now many students elect not to take them. Also, many of the required math courses are mismatched in timing with the relevant courses in the major.
Perhaps more importantly, on top of pure math, the next layer is often missing, and that is the layer where the students are taught physics while using the math tools. This is where they should simultaneously be taught how to build their intuition about the physics with the help of math (math is your friend, people!!!) and how to better understand the ability of math to capture the physical world. What we need are slower-paced calculus-based physics courses and less jam-packed syllabi in the lower-level courses for our major. The way physics for non-physicists courses are taught right now is woefully inefficient: there is too much material in each one of these courses, everything is only touched upon, and the kids retain absolutely nothing. It’s a complete waste of time. Considering that many students haven’t had physics as a standalone subject until college, maybe I shouldn’t lament but should be in awe that the kids have as much proficiency as they do.
People say that we discourage our physical science majors by throwing so much math and physics and chemistry at them when they join the university. That it’s boring and kills their natural creativity and that we should get them more chances to design right away and whatnot. First, if you are going to be a professional scientist or engineer, you need to know that stuff. There is no way around it. You cannot do/create/design anything new and have it work without being able to recognize whether or not it violates the basic laws of nature. So there is no doubt in my mind that a solid foundation in basic math, physics, and chemistry is the core of physical science education. I don’t know how we make it less boring and more appealing — I thought all of it was fascinating to begin with. There are freshman design courses sprouting all around the country, many with a humanistic component, where kids are taught to interface with the communities and solve actual existing problems. I think that is great and helps motivate a lot of kids, but we can’t forget that in order to be independent scientists and engineers we have to give them a lot of basic science tools — sure, it’s cool to make a product for someone in the community as a freshman, but don’t forget that there was an instructor there to catch the (often obvious) fallacies in the many iterations of the design. For most kids, we are not stifling their unique unadultarated genius with these basic courses; we are giving them the tools so they would be able to work independently to express their creative ideas once they have their diploma in hand.
But perhaps what would help more than anything is somehow magically undoing the years of programming in middle and high school that tell kids math is stupid and boring and useless, and that only hopeless nerds like science and engineering…
This is a cute movie on the famous double-slit experiment, featuring a caped and bespectacled superhero Dr Quantum. I show it in class to my undergrads as they first get exposed to quantum mechanics. Enjoy!
While it would be totally cool if we were to reveal that Erwin Schroedinger, one of the fathers of quantum mechanics, had been a proto-blogger of some kind, I am afraid no such luck today.
Schroedinger’s cat is a well-known thought experiment that emphasizes how critical the concept of measurement is for “knowing” things in quantum mechanics. The setup is the following: inside a closed box are a cat, a radioactive atom, and a bottle of poison. There is also a mechanism which, if the radioactive atom decays, opens the bottle of poison and the cat dies. As long as the atom hasn’t decayed, the cat is alive. The decay of the atom is a stochastic process, you can only talk about probabilities that the atom has or hasn’t decayed, and therefore about probabilities that the cat is alive or dead, unless you open the box. The part that is relevant to this post is that you don’t know for certain if the cat is dead or alive until you open the box.
If you have no interest in physics, you can skip this paragraph and the next two (start reading again after the cat pic) but I can’t help but geek out a little bit. There are a number of issues with this thought experiment, so I am just briefly going to sketch some that come to mind. People talk about the cat being in a superposition of dead and alive states; superposition is a term used to denote a linear combination of vectors. For instance, let’s look at the vectors in a plane, call it the xy plane. We will use x and y to denote the unit vectors along the corresponding axes. Any vector in the xy plane can be expressed as a linear combination of x and y. For example, vector v=2x+5y is a linear combination of x and y, with 2 and 5 being the coefficients in the linear combination; the coefficients tell us about the proportion of x and y that we have in the mix that is v, as well as about the total length of v. So imagine you can introduce a “unit vector” akin to x or y, but which corresponds to the cat being alive and denote that vector by |alive>. Similarly, introduce another vector that corresponds to the cat being dead and denote it by |dead>. Then the cat “state”, a vector telling us about cat viability, might be written as |cat viability>=coeff_dead|dead>+coeff_live|alive>. Now, the two coefficients have to do with the probability of finding the cat dead or alive: the magnitude squared of each coefficient is the corresponding probability, so the squares of the two coefficients add up to 1. These probabilities, and thus the coefficients, would really be obtained by having many identical boxes with many cats and atoms, and having them sit there closed for the same amount of time, and then opening them and counting all the dead cats. This would obviously be a serious PETA issue. Luckily, we know enough about the radioactive decay of atoms that a cat genocide can be avoided. Basically, if the atoms have a characteristic lifetime τ, then, after time t, exp(-t/τ) of atoms have decayed while 1-exp(-t/τ) have remained undecayed. So depending on how long the lifetime τ is and for how long you have decided to keep the poor cat in there, the probability of finding it alive versus dead is quite different. This is in contrast with people ubiquitously putting the two coefficients above to be 1/√2, which really does not hold in general.
Moreover, in quantum mechanics the act of measuring does in fact disrupt the system and causes the state of the observed system to “collapse” into one of the possibilities. In this thought experiment, you do open the box, but you don’t actually do anything to the cat — its liveliness is exactly the same at a certain point in time regardless of whether you open the box or not. An example that might better capture the spirit of what a measurement really does to the measured system would be that, instead of the atom, you have a mechanism where opening of the box rolls a die, and if the number on the die is 1, 2, or 3 the cat lives, whereas if it’s 4, 5, or 6 the mechanism somehow kills the cat. So the act of opening would now affect the irreversible “collapse” of the cat viability wave function (although, in all biological fairness, only the dead cat outcome is truly irreversible). Anyway, a lot has been written about Schroedinger’s cat, so I leave it to your powers of googling to look into it more if so inclined.
As great as Schroedinger’s contribution to physics has been, he will remain in history as one of the first who has realized a grand, culture-traversing truth: cats make everything both memorable and funny.
Anyway, quantum mechanics musings aside, it is true that you don’t know for sure if Schroedinger’s cat is alive or dead until you look.
While I am new to this space, I am not new to blogging, an activity with which I have long had a pretty torrid relationship. I really enjoy writing, and the experience of blogging has really made it clear for me. Often, there are many things on my mind that feel the need to write down. I can certainly open a Word document or a grab a legal pad and just jot it all down. I can write unabashedly, whatever I want and however I want. While the cathartic effect of writing alone is enough is some cases — especially when I am enraged to the levels of blinding, white-hot fury — usually it’s not enough. The downside is that nobody but me reads that stuff so writing just for myself lacks the sharing aspect and makes the effort feel somehow incomplete.
Often I think that writing a blog is inherently exhibitionist and therefore a bit skeevy. (But then again, I am not on Facebook or Twitter, and I blog pseudonymously, so perhaps I am just an Internet prude.) Does that mean that anyone who has ever written anything with the intention of others reading it online or in print should also be referred to as an exhibitionist to some degree? I can’t honestly decide if it’s okay to put one’s thoughts out there for others to read and perhaps think about, or if one should just keep one’s thoughts to oneself because the very act of wanting to share in a public forum is just plain crazy. So, to me, it’s the never-ending struggle of wanting to share versus not wanting to get exposed.
I am always torn between writing what and how I want, and writing things to be read. When I write for myself, the impetus to edit and clarify is diminished, and the sharing and commiseration is missing. When writing for others to read, I am more compelled to work on the language and flow, but I can feel uncomfortably exposed, even though it’s nice to be connected, to read about others who share in the same trials and tribulations.
When I have an essay in me, it’s as if the unwritten text is in a superposition of for-me and for-others states. The act of writing is like a measurement. Once a piece is done, there is no going back — once the essay is written with qualifiers, tempered down, and cleaned up for an audience, it is no longer the raw essay it used to be inside my head. More importantly, after having written it for an audience, the essay can de facto no longer go back to its raw, idea-only self. Once out, it takes on a form, but its form and substance are so tightly interwoven that I cannot go back and write another related piece one just for me: the process of writing with others in mind irreversibly changes how the essay looks in my head, its existing form erases how it might have looked if written just for me.
So I keep struggling with what I want the writing to be — the solitary, journal-like experience achievable in the absence of an internet connection, or a more public, shared experience through a blog, where writing for others improves the clarity and form, but dulls the edges, tempers, and qualifies. Once the writing starts to take form, the more I have one but not the other, akin to very slowly opening the box to reveal the life or death of Schroedinger’s cat.