Timeline for Mathematical physics without partial derivatives
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13 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Mar 7, 2020 at 17:46 | comment | added | Federico Poloni | So how do you actually compute stuff in this model? You mentioned diagonalization, so I guess one of the "primitives" that one needs to compute is the eigenvalues and eigenvectors of a given Hermitian matrix. Are there other ones? Do you need to take matrix sums? Products? | |
| Mar 7, 2020 at 17:37 | comment | added | user21349 | [...] Sure, I can "compute" wavefunctions in this model. It's a finite-dimensional vector space. Here's a wavefunction that I computed in some basis: $\left(\begin{matrix}1 \\ 0\end{matrix}\right)$. But I don't think this is what you had in mind as computation. | |
| Mar 7, 2020 at 17:19 | comment | added | user21349 | @FedericoPoloni: No, that is a more general statement that is independent of applications or bases: if you allow matrix algebra among the things that you are allowed to do, then you can use it to compute the derivative of any function that you can compute. I think you're misunderstanding. In a model like the interacting boson model, we do not know how to compute any wavefunctions. All we know is the matrix elements of the Hamiltonian. As a concrete example, here is a matrix: $\left(\begin{matrix}1 & 2 \\ 3 & 4\end{matrix}\right)$. Please compute some derivatives for me. | |
| Mar 7, 2020 at 17:05 | comment | added | Michael Engelhardt | @AidanRocke - I don't know how to say it any more clearly than I already did: If my physics problem doesn't require differentiation, then I won't need your nifty biological system to do any differentiations for me, will I? | |
| Mar 7, 2020 at 16:46 | comment | added | Aidan Rocke | @MichaelEngelhardt Why would it be self-defeating? I consider automatic differentiation in biological systems to be the most likely scenario, given what we know, but as a scientist I think it is important to carefully consider the alternative possibility. | |
| Mar 7, 2020 at 15:08 | comment | added | Michael Engelhardt | @FedericoPoloni - The irony is that the clever idea of automatic differentiation, and its putative realization through a biological system, become irrelevant if we formulate our physics problem such that it does not require any differentiation anymore, as the OP is asking us to do. That's where the OP lost me - the line of questioning seems completely self-defeating. (You probably had a similar reaction, going by some of your comments to the OP). | |
| Mar 7, 2020 at 14:56 | comment | added | Federico Poloni | No, that is a more general statement that is independent of applications or bases: if you allow matrix algebra among the things that you are allowed to do, then you can use it to compute the derivative of any function that you can compute. | |
| Mar 7, 2020 at 14:18 | comment | added | user21349 | @FedericoPoloni: If I'm understanding you correctly, then you're assuming that the function has been expressed in the position basis. The point of my answer is that you can work with these models without ever even knowing any wavefunctions in the position basis. In the interacting boson model, nobody knows what the wavefunctions would be in the position basis. | |
| Mar 7, 2020 at 9:57 | comment | added | Federico Poloni | Arguably, linear algebra allows one to compute derivatives: given a rational function (or an analytic function as the limit of its Taylor series), you can evaluate it in the matrix argument $\begin{bmatrix}\lambda & 1 \\ 0 & \lambda\end{bmatrix}$, and the result you obtain is precisely $f(\begin{bmatrix}\lambda & 1 \\ 0 & \lambda\end{bmatrix}) = \begin{bmatrix}f(\lambda) & f'(\lambda) \\ 0 & f(\lambda)\end{bmatrix}$. This is, essentially, automatic differentiation recast as linear algebra. So matrix algebra is, essentially, equivalent to derivatives. | |
| Mar 6, 2020 at 16:16 | vote | accept | Aidan Rocke | ||
| Mar 5, 2020 at 23:21 | history | edited | user21349 | CC BY-SA 4.0 |
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| Mar 5, 2020 at 22:18 | history | answered | user21349 | CC BY-SA 4.0 |