Timeline for mathematically explain why add operators with different physical unit is wrong?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Oct 3, 2018 at 16:52 | comment | added | md2perpe | @UniverseMaintainer. No, not really. I just wanted to mention that "incompatible" things can be added. But I haven't seen it in QM and hardly in physics in general. The closest is probably geometric algebra, related to Dirac's $\gamma$-matrices, where scalars, vectors, bivectors and so on can be added. | |
| Oct 3, 2018 at 16:35 | comment | added | Universe Maintainer | @md2perpe yes. I agree with you. Do you know any case in QM where we use this new “fruit”? | |
| Oct 3, 2018 at 16:16 | comment | added | md2perpe | @UniverseMaintainer. Even if we can't make a new fruit by adding some apples and some bananas, we can construct formal sums like $2 \text{ apples} + 3 \text{ bananas}.$ The set $\{ x \text{ apples} + y \text{ bananas} \mid x,y \in \mathbb{R} \}$ can easily be made into a linear space. In the same way we can construct formal sums of $J_z$ and $J^2,$ and the eigenvalues of such sums will be formal sums of eigenvalues of each operator. | |
| Oct 3, 2018 at 12:56 | comment | added | md2perpe | To be clear: I just said that we can do it. I didn't say that we should do it. | |
| Oct 3, 2018 at 12:04 | comment | added | J.G. | @md2perpe Fine, let's define "adding" $J_z$ to $J^2$ as forming an ordered pair, but "adding" $J_z^2$ to $J^2$ the normal way. I don't see the point, but OK... | |
| Oct 3, 2018 at 12:01 | comment | added | md2perpe | In mathematics physical dimensions normally don't exist. In mathematics we can also construct formal sums of very different linear spaces. So we can write $1\ \rm{m} + 1\ \rm{m}^2$ although it has no deeper meaning than constructing a pair $(1\ \rm{m}, 1\ \rm{m}^2)$ and making this an object in a linear space in a natural way. | |
| Oct 3, 2018 at 5:04 | comment | added | J.G. | @UniverseMaintainer What's $1\text{m}+1\text{m}^2$? What's $100\text{cm}+10000\text{cm}^2$? | |
| Oct 2, 2018 at 21:51 | comment | added | Universe Maintainer | @J.G. Do you suggest even in math, operators with different dimensions can not be added? | |
| Oct 2, 2018 at 21:31 | comment | added | md2perpe | @UniverseMaintainer. They act on the same Hilbert space, but they live in different operator spaces. Let $\mathcal{O}(\mathcal{X})$ denote the space of operators with values of physical dimension $\mathcal{X}$, and let $\mathcal{J}$ denote the physical dimension of angular momentum. Then $J_z \in \mathcal{O}(\mathcal{J}),$ but $J^2 \in \mathcal{O}(\mathcal{J}^2).$ | |
| Oct 2, 2018 at 21:21 | comment | added | J.G. | @UniverseMaintainer Technically each dimension for operators obtains a different HS because they map the original kets to different spaces, because expressing the image in the original basis requires dimensionful coefficients. | |
| Oct 2, 2018 at 21:15 | comment | added | Universe Maintainer | J^2 and Jz they do live in the same Hilbert space, I don't really get your point. | |
| Oct 2, 2018 at 20:36 | history | answered | J.G. | CC BY-SA 4.0 |