In the last couple of weeks, I’ve been using Hermite Polynomials in my work. I won’t go into what they are (look them up here if you like) suffice to say that they are one of many contributions to mathematics from Charles Hermite (1822-1901), who was himself one of many french mathematicians whose work has laid a foundation for much of modern theoretical physics. A physicist would generally know these polynomials (when modulated by a gaussian function ) as the solutions to the 1-dimensional quantum harmonic oscillator, although that’s not why I’m using them.
The 1-dimensional quantum harmonic oscillator problem is a textbook problem that gets inflicted on generations of students. I remembering suffering the algebra that went with it. At the University of Waikato, we save our second year students the algebra by just talking about the solutions, but then spring it on them in third year. For those who like that kind of thing, it’s an interesting analysis, but for those that don’t, it really is quite horrible.
Perhaps that is what motivated Paul Dirac to come up with (in my opinion) a really elegant complementary approach to solving the 1-dimensional quantum harmonic oscillator problem. While his approach is easily found in text-books, what I haven’t been able to track down is a description of how he came up with it. The same seems to be true of many of the analyses that get wheeled out to students. While they look clean and tidy when presented now, I’m left with the question “How did they come up with this?”. That tends to be overlooked in favour of the end product. Did Dirac spend weeks pondering over this, thinking “there must be a better approach – the symmetry between p and x in this equation should surely be exploitable somehow…”, was it a sudden revelation, did he try twenty different approaches till something worked, or what? My text books don’t say.
What Dirac did was to reformulate the problem in terms of ‘raising’ and ‘lowering’ operators. He realized the problem as a ladder of energy-levels, and showed rather elegantly that these energy levels were equally spaced. Moreover, some rather neat operators, that he defined, could move a quantum state ‘up’ or ‘down’ the ladder. That’s a very creative way of looking at the problem, and has been taken much further since then. For example, when analyzing problems with many electrons (which generally means just about anything electronic) we can now formulate the problem in terms of operators that create and destroy electrons. Whether electrons really are being created and destroyed is a moot point, but the formulation is a neat one that helps us to analyze what is going on. Theoretical physicists consider it a really useful ‘tool’ of the trade, even though the history behind its construction tends to be overlooked when we teach it.
So what is the point of me telling you this? Well, it’s about teaching. Just how do you teach creativity, especially in something that is, on the face of it, as tedious as physics. Physics isn’t actually tedious (if it were I wouldn’t be sitting here writing this) but we do tend to make it unnecessarily so at times. I wonder whether that’s because that’s the easiest path to take for undergraduate teaching. At PhD level and beyond, there’s some really creative research going on, but do our undergraduates really see this? Likewise, from what I’ve seen at school science fairs, there’s some great creativity at primary and intermediate school level, but that then vanishes late in secondary school in favour of ‘content’. Somehow, we tend to smother out creativity and elegance in favour of ‘something-that-gets-the-job-done.’ But truly great physicists, Dirac included, have never ‘just-got-the-job-done’.
Open-ended projects are a way to go (and we manage to some extent to do this with our engineering students), but, as many readers know, we run into trouble with time, the need to prepare students for exams, fitting in with timetabling requirements, and so forth. The problem may go much deeper than we think – indeed, does the whole secondary and tertiary education structure smother-out creativity from students (at least in physics)?
And with that, have a creative Christmas, and Happy New Year to you all! I’ll be heading southward next week to the Canterbury hills – a part of the country I haven’t been to before.