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.

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    $\begingroup$ (1 of 2) I don't think the sort of expanation/analogy you are looking for is going to help them. Just tell them it is shorthand notation and how it works. $\endgroup$
    – guest
    May 26, 2018 at 1:14
  • $\begingroup$ (2 of 2) Getting into the sort of analogies you want will confuse them further. And remember these guys are pretty new to the whole thing. Better to get familiarity BEFORE going back for deeper understanding. For that matter, even in a regular QM course (let alone where you are, that should be easier), I prefer the approach to learn time independent Shroedinger equation first. It is a bad pedagogic idea to introduce new notation along with new ideas. Keep it diffy Qs for now. So you are hurting rather than helping by getting in to the "handy" notation at this level. $\endgroup$
    – guest
    May 26, 2018 at 1:15
  • $\begingroup$ I think this is a hard problem. Possibly consider this video: youtu.be/LyGKycYT2v0?t=11m18s (and more generally, the whole "essence of linear algebra" series, which could serve as an adjunct for those who had a shallow look at the subject). $\endgroup$ May 26, 2018 at 3:27
  • $\begingroup$ As you must sense, the best solution is to have linear algebra as a prerequisite. $\endgroup$
    – user52817
    May 26, 2018 at 4:17
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    $\begingroup$ You don't need everything to be a prereq for everything. An extreme amount of QM can be learned (consider the start point of knowing none) for a student in this setting by just using diffy Qs. Even the bra ket can be learned and taught and used just fine without a course in LA (I learned it in grad QM chemistry that way and did not find the gap). Also, much of LA course is not used in "freshman physics QM (really more of a waves course in 3rd semester and not even junior year physics QM). So doing a whole semester of LA to have a better rationale for one operator is bad cost benefit. $\endgroup$
    – guest
    May 26, 2018 at 19:21

1 Answer 1


The symmetry between vectors and co-vectors in your example (i.e. the possibility to identify a vector with a co-vector) exists only since you have a scalar/hermitian product on the vector space $V.$ The general notion of duality in linear algebra doesn't have that symmetry, and I would rather break the symmetry if I wanted them to understand this more general duality.

If my aim were for them to "only" understand the difference between $|v\rangle$ and $\langle v|$, I'd tell them that the hermitian product on $V$ induces two different maps of type $V \to (V \to \mathbb{C})$. The first one is $v\mapsto |v \rangle$ and the second one is $v\mapsto \langle v|$. I'd then tell them a little bit about the funny notation and why it is useful.

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    $\begingroup$ I'd tell them that the hermitian product on $V$ induces two different maps of type $V \to (V \to \mathbb{C})$. The first one is $v\mapsto |v \rangle$ and the second one is $v\mapsto \langle v|$. Thanks for your thoughts, but these are sophomore engineering majors. My question asks for down-to-earth analogies or simple, concrete examples. In the question, I said: 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. $\endgroup$
    – user507
    May 30, 2018 at 15:49
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    $\begingroup$ You know better what you can tell your students, but notice that my formulation is not talking about linear operators on ket spaces (that would require that you first define what a ket space is and what a linear operator is.) It "only" requires that students know what a set and a mapping is. $\endgroup$ May 30, 2018 at 16:22
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    $\begingroup$ @BenCrowell It puzzles me that your students should feel comfortable describing quantum mechanical particles with complex valued wave functions and use integrals like $\int \Psi^*\Psi dx$, while on the other hand the notion of a set and a map is considered way to abstract for them. $\endgroup$ May 30, 2018 at 21:16

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