Analogies or explanations for duality, at the college sophomore level mathematics electricity equations physics

This semester I taught the third semester of my college’s freshman physics sequence. Nearly all the students are engineering majors. Compared to previous semesters when I’ve taught this course, I went into a little more detail on quantum mechanics, including introducing the very handy bra-ket notation. My students were first exposed to the description of a wavefunction $\Psi$ as a complex-valued function of position, including expressions such as $\int \Psi^*\Psi dx$ for probabilities. All of them had been exposed to complex numbers before. Later we did wavefunctions that were not expressed in position space, and they had to do things like taking the ket $i|\uparrow\rangle$ for an electron in a spin-up state and constructing its bra version $-i\langle\uparrow|$. In mathematicians’ language, these notations describe a vector and its dual.

At this stage I felt that there was some awkwardness, as expressed by student questions such as "What’s the difference between $|\uparrow\rangle$ and $\langle\uparrow|$?" Most but not all of them had had a semester of linear algebra. With those students, I tried making the analogy to row and column vectors, and that sort of worked. However, it didn’t seem like they had been exposed to the idea of taking the complex conjugate when doing a Hermitian adjoint. I don’t know how much of their lin alg course was done over the reals only, but I get the impression that most of that course consists of mind-numbing concrete computations, with page after page of homework problems involving matrices with integer elements. It’s also a little awkward to say that bras and kets are row and column vectors, since many of the vector spaces we deal with are infinite-dimensional.

I could have just told them that a bra is a linear operator on ket space, but I think that would have been much too abstract for their level of mathematical development. And really it seems unaesthetic to me to break the symmetry between the two mutually dual vector spaces by describing one as more fundamental and the other as derived from it. The way they occur in quantum mechanics, and the way in which they relate to experimental observables, is completely symmetric. (In this respect they are different from the covariant and contravariant vectors of general relativity, where the convention is that infinitesimal displacements, i.e., elements of the tangent space, are represented by contravariant vectors, and we do have quantities like frequency that are then more naturally represented as covariant vectors.)

With students who hadn’t had any linear algebra at all, I seemed to have some luck when I tried explaining the general idea of duality. The best example I could think of using math they would know was that you could have a function and its derivative, with each one basically expressing the same information, two different representations of the same thing. But this isn’t a completely accurate example of duality, because we have constants of integration, so a round-trip transformation loses information.

Can anyone suggest other analogies or examples to use for the idea of duality, ones that would be accessible to people whose most advanced training is a year of calculus? Maybe it would be helpful to introduce complex conjugation itself as an example of duality. Post navigation