Timeline for Why ket and bra notation?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 29, 2016 at 18:05 | comment | added | Eric Towers | @leftaroundabout : The Question is in the context of quantum mechanics, not in the context of general inner product spaces. Since you seem to want to talk about some other Question, perhaps you should find one that fits and go talk there. | |
| Sep 29, 2016 at 17:14 | comment | added | leftaroundabout | No you don't. The inner product takes two primal vectors and gives you a number. In fact, that's the main thing that's interesting about an inner product, and the only reason you can simply switch between space and dual space: the mapping $\ast : \mathcal{H}\to \mathcal{H}^\ast$ is defined by $\psi \mapsto \psi^\ast := \langle \psi,\cdot\rangle$. The inverse of that mapping is even less trivial: it only exists in Hilbert spaces, not in general inner-product spaces; this is given by the Riesz representation theorem. | |
| Sep 29, 2016 at 13:43 | comment | added | Eric Towers | @leftaroundabout : Yes and no. The inner product requires that you feed it a primal and a dual. You still have to feed it an actual dual, which means you have to cast a primal to a dual when you use a primal in a dual slot. | |
| Sep 28, 2016 at 12:53 | comment | added | leftaroundabout | No, exactly that is not required! The inner product has complex conjugation built in. | |
| Sep 28, 2016 at 12:51 | comment | added | Eric Towers | @leftaroundabout : ... but that notation requires an adjoint notation ... $\langle \psi_{n+1}^*, H \psi_{n+1} \rangle$, which, contrary to your prior comment, marks up the dual vectors. | |
| Sep 27, 2016 at 16:06 | comment | added | leftaroundabout | Well, that is mainly inefficient to write because you can't use $n+1$ as an identifier without wrapping in in a bra/ket. However, if you wrote it $\langle \psi_{n+1} | H | \psi_{n+1}\rangle$ then this could perfectly well be translated to an ordinary inner-product expression $\langle \psi_{n+1}, H\, \psi_{n+1}\rangle$. No bras or kets here. What does get a bit annoying is when you actually want to talk about dual vectors as such, without executing an inner product – e.g. $\sum_i |v_i\rangle\langle v_i|$ becomes $\sum_i v_i \langle v_i,\cdot\rangle$, which may not be very clear. | |
| Sep 27, 2016 at 15:59 | comment | added | Eric Towers | @leftaroundabout : The idea of having no specific notation for primal versus dual vectors is ineffective in the quantum mechanical setting where the same symbols are used for an eigenstate and its dual. Consider $\langle n+1 \mid H \mid n+1 \rangle$. | |
| Sep 26, 2016 at 22:09 | comment | added | leftaroundabout | (Nevertheless: I rather prefer the maths convention of not using any special markup for vectors or dual vector at all; it should simply be declared in what space the quantity lives that some symbol refers to.) | |
| Sep 26, 2016 at 22:07 | comment | added | leftaroundabout | IMO the whole column vector vs colums vector issue is just an artifact of the over-reliance on matrices. I find nothing wrong with writing vectors $v \in \mathbb{R}^2 \equiv \mathbb{R}\times\mathbb{R}$ as tuples, i.e. $v = (v_0, v_1)$. If this leads to inconsistencies when it comes to matrix multiplication then that's an issue of the matrix notation, not the notation you use to represent vectors in some give space. And of course the component writings only work in the finite-dimensional case. Bra-ket notation avoids all these issues. | |
| Sep 25, 2016 at 20:11 | history | answered | Eric Towers | CC BY-SA 3.0 |