Stack Exchange
log in

Manas Dogra

 The h Bar

General chat for Physics SE (physics.stackexchange.com). For M...
4
Manas Dogra
Jul 4 10:21
I am a rising 2nd year student in PhD theoretical physics. I love physics. But that's also the problem: I love physics too much. This is hampering my research... I find peace in deriving either each microstep or get lost in relating ideas and finding alternate/beautiful solutions to already solved problems. I like reading papers, explaining it to others but that's it. This is like addiction to me. Don't think I am bragging but sometimes I am skipping meals and sleep for this. At the same time, somewhere deep inside I want to "be a physicist" which nowadays means publish a lot. Well, everyti…
Manas Dogra
Jul 4 10:21
This might not be the right place to ask the following (maybe I should go to therapy or something): but I have been feeling very low right now and I guess only fellow physicists can understand me. And hbar never disappointed me. Sorry if this is a wall of text.
Manas Dogra
Jun 18, 2024 08:04
Don't read the first three words in my above question: "If change under"
Manas Dogra
Jun 18, 2024 08:02
I know diffeomorphism invariance is a much discussed topic, but I don't know if this particular question has been raised previously and answered before
Manas Dogra
Jun 18, 2024 08:01
If change under general coordinate transformations is a global symmetry for theories in which metric is not the dynamical variable, why does it not imply that all theories are Lorentz invariant?
Manas Dogra
Jun 8, 2024 14:56
Idk. I am just trying to see why Schrodinger's equation is non-trivial at all...
Manas Dogra
Jun 8, 2024 14:55
Then maybe, it is something else. Schrodinger's equation is not a tautological postulate because we can't relate that operator to "quantization" of the classical Hamiltonian, and that the self adjoint operator in Stone's theorem is the quantization of a classical Hamiltonian, is the statement in Schrodinger's equation?
Manas Dogra
Jun 8, 2024 14:54
Even if we use the Stone's theorem, it implies existence of a self adjoint operator. Do we "define" that operator to be the Hamiltonian of our system? If so, then Schrodinger equation seems really tautological...
Manas Dogra
Jun 8, 2024 14:43
@ACuriousMind I thought that "quantization" of the classical Hamiltonian is a better definition just because, this definition makes it Schrodinger equation so obvious and I thought that maybe it isn't.
Manas Dogra
Jun 8, 2024 07:47
I mean if we do things the generator way (without doing canonical quantization), then why do books (even Sakurai) treat momentum and energy differently. Alright, we are doing non-relativistic QM as of now, but still it just hurts to see that something is "definition" and something other is "postulate"
Manas Dogra
Jun 8, 2024 07:45
So it might seem that canonical quantization is the go-to thing. But many books in QM, define momenta as the generator of spatial translations---that's not a postulate. But then they don't do this for temporal translations for some reason and declare Schrodinger equation as some postulate
Manas Dogra
Jun 8, 2024 07:43
@naturallyInconsistent But what if we avoid that by defining momentum as the generator of spatial translations? I mean, obviously then you would say. Hey then you go and define Hamiltonian as the generator of temporal translations, similarly... and my point in this text would be vague
Manas Dogra
Jun 8, 2024 07:25
Okay thanks everybody
Manas Dogra
Jun 8, 2024 07:21
@Sanjana Exactly. Do you have an answer, can you find a flaw in my chain of thought? Remember its just a motivation
Manas Dogra
Jun 8, 2024 07:06
oh ok ok. I was thinking there's some other way
Manas Dogra
Jun 8, 2024 07:00
@Slereah You mean, I have to use EOM or the action principle... in some way, right? I mean I tried to relate Hamiltonian to generator of time translations without EOM, but there seems to be no way
Manas Dogra
Jun 8, 2024 06:57
@RyderRude Hm
Manas Dogra
Jun 8, 2024 06:57
@Slereah I am looking at the naive quantization thing right now. But I feel that there's still some flaw in the above argument. E.g. I can derive that $\frac{\partial }{\partial t}$ is the generator of time translations, but is there any way of proving in classical mech. that this is generated by something called the Hamiltonian which describes the system?
Manas Dogra
Jun 8, 2024 06:50
@Slereah I didn't mean time evolution (=$e^{-iHt}$)in the literal sense, I meant the LHS of $i \frac{d}{dt} | \psi \rangle = H | \psi \rangle$.
Manas Dogra
Jun 8, 2024 06:47
Yeah. That occured to me and I said "assuming" that the same holds in QM... which is fishy enough... But still apart from that anything else???
Manas Dogra
Jun 8, 2024 06:46
@Slereah But that goes within the other postulates of QM, right?
Manas Dogra
Jun 8, 2024 06:45
Do we need to use Hamilton's equations for proving that Hamiltonian generates time translations? If that is so, then the above logic doesn't make sense cz we would need EOM for the logic to work, and we are looking for the EOM itself in QM in the above line of reasoning
Manas Dogra
Jun 8, 2024 06:44
Why isn't Schrodinger equation just obvious? It just says Hamiltonian generates time translations, which is known from classical mech. So "assuming" that the same holds in QM, we can postulate SE? I feel that this line of reasoning is fallacious in some way. Can somebody point out to me, how?
Manas Dogra
May 23, 2024 05:19
@naturallyInconsistent Thanks. It is much clear now.
Manas Dogra
May 22, 2024 14:07
@lucabtz Oh I was mistakenly taking the differential instead of the difference, so the $n$ was stayng there. THANKS
Manas Dogra
May 22, 2024 14:00
I just wanted a justification of that rule. I understand why the limit is to be taken...I just don't understand how to "derive" that rule
Manas Dogra
May 22, 2024 14:00
Now the book says that you have to replace the sum with an integral by the rule you just described.
Manas Dogra
May 22, 2024 13:59
@lucabtz I am just considering free particle in a box of size $L$ in a calculation where at the end of calculation I am to take $L \to \infty$. I have taken the b.c.s such that the putative energy/momentum eigenfunctions are $u_k(x+L)=u_k$ where $u_k(x)=\frac{e^{-i kx}}{\sqrt{L^{3/2}}$ that gives me $k_i =\frac{2\pi n_i}{L}$ for integers $n_i$.
Manas Dogra
May 22, 2024 13:51
Oh that's okay. I want to know where the $n_i$ goes
Manas Dogra
May 22, 2024 13:51
@lucabtz I understand the physical motivation, but not the math part
Manas Dogra
May 22, 2024 13:50
@lucabtz Where does the $n_i$ go? I don't see this mathematically
Manas Dogra
May 22, 2024 13:48
I know $k_i =\frac{2 \pi n_i}{L}$. Why does the measure look like that in the limit $L \to \infty$?
Manas Dogra
May 22, 2024 13:47
Why is $d^3k'=(\frac{2 \pi}{L})^3$ in case of box normalization of free particles in QM?
Manas Dogra
May 11, 2024 21:30
sorry for screaming :p
Manas Dogra
May 11, 2024 21:30
THANK YOU
Manas Dogra
May 11, 2024 21:30
OH MY...
Manas Dogra
May 11, 2024 21:30
THAT WAS IT?
Manas Dogra
May 11, 2024 21:30
AND THAT'S BECAUSE SU(2) HAS RANK ONE?
Manas Dogra
May 11, 2024 21:29
@ACuriousMind That was not my definition. The "hash" symbol # denoted "number of" as in cardinality/counting.
Manas Dogra
May 11, 2024 21:28
@ACuriousMind Do these quantum numbers have any relation with the definition of quantum numbers I am familiar with or is it just plain wrong?
Manas Dogra
May 11, 2024 21:27
I wrote the above before reading your latest reply which I am going to read rn, but anyway my last "long" text is my latest attempt to express my question clearly
Manas Dogra
May 11, 2024 21:26
I am under the impression that "conserved quantum numbers" are eigenvalues of Noether charges.
So to me #Noether charges=#conserved quantum numbers=dimension of algebra. But according to Pais, #conserved quantum numbers=dimension of CSA. Is my definition wrong? Is my reasoning wrong? What's going wrong: where's the discrepancy?
Manas Dogra
May 11, 2024 21:24
user image
Manas Dogra
May 11, 2024 21:22
@ACuriousMind Why do we associate the dimension of the CSA and not the algebra with the number of additively conserved quantum numbers?
Manas Dogra
May 11, 2024 21:21
Okay fine, let me give one last try
Manas Dogra
May 11, 2024 21:21
@ACuriousMind Ok...now what should I do, speak using set theory or something :p ?
Manas Dogra
May 11, 2024 21:20
I edited the question above
Manas Dogra
May 11, 2024 21:19
@ACuriousMind Why are there 2 quantum numbers when there are 8 charges in case of $su(3)$ color symmetry (global part) in standard model?
Manas Dogra
May 11, 2024 21:19
@ACuriousMind So what's the conclusion: there are 8 conserved charges but 2 quantum numbers?
Manas Dogra
May 11, 2024 21:18
(I know I am misunderstanding)