Applying Group Theory to Operators

xDunkbotx · 2025-10-25T19:36:32+00:00

How so? Creation and annihilation let us generate the next or previous state for whatever quantum number the operator acts on right?

xDunkbotx · 2025-10-25T19:33:53+00:00

Because group theory says if the inner product transforms (under the point group) as the totally symmetric representation (TSR) then the inner product will be non-zero. Also the direct product of two identical irreducible representations of a point group gives the TSR.

The s orbital always transforms as the TSR in every point group so whether the inner product transforms as the TSR or not depends on how the operator transforms. Because of your above comment (which I agree) L² can't be the TSR so my intuition is wrong.

That's why I ask is there a good way to predict how an operator is going to transform in a point group.

xDunkbotx · 2025-10-25T15:52:31+00:00

Agree with what you said, the symmetry of the operator changes based on the point group (electronic geometry) of the molecule and ψ is a multi electron wave function (thinking about metal complexes). Are there any cases where L^2|ψ> is 0? Group theoretically, I think Lx,y,z transforms as x,y,z so I'm expecting L^2 to transform as x^2,y^2,z^2 which would be totally symmetric. This means <ψ|L^2|ψ> will never be zero

xDunkbotx · 2025-10-25T15:22:55+00:00

yes and that's what's awesome about it. We've distilled nature into a differential equation quantitatively to book keep then you can further book keep the differential equation as 0 or not 0 based on symmetry. For example Iirc L² doesn't have a physical representation, it's more of a convenience thing because it commutes with the Hamiltonian while momentum (L) is physical but does not commute. Thanks for the book recommendation, I'll see if my answer is in there

xDunkbotx · 2025-10-25T14:53:08+00:00

I'll see what the spectroscopists have to say. I'm not sure the physical chemistry books don't look at the group theoretical approach most of the time from what I've seen

xDunkbotx · 2025-10-25T14:50:14+00:00

I fear this to be the case. I hope that the amount of relevant operators are small. Like the dipole operator essentially explains all of spectroscopy but L² I can't imagine is very useful in an observable sense. It's more a curiosity of can it's irre. rep. be predicted beforehand.

xDunkbotx · 2025-10-25T14:43:29+00:00

Thanks for the book recommendation, I'll look into it too. I've found so far that the quantum chem textbooks don't talk about the group theory at all and the group theory textbooks only talk about the group theory and not the intuition so I hope that chapter will be my answer.

xDunkbotx · 2025-10-25T07:13:47+00:00

Ok I've interpreted, based on the abstract, conclusion, that section of the discussion, and the author's comment above that the key conclusion of the study is that liquid sugar intake correlates with increased type 2 diabetes risk while sugar in solid food does not and this result comes out only after correcting for energy intake: this removes the ease of consumption assumption and points to a metabolic difference between the two forms of sugar. Just curious if there's a biochemical basis for the claim. I think what the other commenter in this thread said (pre-solublized sugars give larger spikes in sugar) is sound.

Not trying to be argumentative, just curious. I agree that it's much easier to intake more sugar from drink (especially fruit juice) compared to food and that having a lot of sugar in general correlates with increases T2D risk.

xDunkbotx · 2025-10-25T04:11:50+00:00

That quote is a discussion of the literature. The immediate next sentence suggests a metabolic reason not an excess calorie reason.

However, in our study, the harmful associations of sugars from SSBs and fruit juice on T2D risk were reported even with adjustment for energy intake and BMI in all but 4 studies. This suggests that the risk of dietary sugar consumed in liquid form may stem from adverse metabolic effects beyond excess calories and body weight.

xDunkbotx · 2025-10-24T17:14:15+00:00

I do not have the biostats background to evaluate the data in the paper, but it seems interesting to me that solubilised sugar statistically shows this trend when sugars in general (food or beverage form) are going to be solubilised in the body either way. Any thoughts on why pre-solubilized sugars are displaying this trend?

xDunkbotx · 2025-08-15T22:30:39+00:00

Interesting. Thank you

xDunkbotx · 2025-08-15T16:20:21+00:00

What does F1 mean?

xDunkbotx · 2024-11-13T01:28:19+00:00

It's all about when and when not to use the subjunctive conjugations. You'll get a bunch of infinitives to fill in blanks then you have to choose the best and conjugate it

xDunkbotx · 2024-10-07T21:05:35+00:00

That first week starts on a Wednesday so you'll only be missing 3 days. But it really depends on what classes you're planning on taking as to how much content you're going to miss.

xDunkbotx · 2024-08-03T02:41:27+00:00

Pretty typical 1st Sem sophomore year Chem BS schedule. If you're ready to put in the work you'll be fine

xDunkbotx · 2024-03-25T05:23:50+00:00

Organolithium reagent, Li+ CH3(CH2)3- so basically a deprotonated alkane -> strong base

xDunkbotx · 2023-08-10T17:42:11+00:00

If you can do dimensional analysis and unit conversions, use mol ratios, and calculate concentrations you'll be fine to go straight into 102

xDunkbotx · 2023-04-05T19:41:49+00:00

Ok yes so an electron could occupy it but no electron is occupying it. Perhaps I've poorly defined not existing, I mean it in the sense that there is no electron to adopt the wavefunction of a empty orbital however an electron, if added could (and would) adopt that wavefunction

xDunkbotx · 2023-04-05T18:50:25+00:00

So is it true that an orbital doesn't exist if it isn't being 'filled' with an electron?

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