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Thank you! I was not worried; I just curious as to how fundamental this empirical fact is to our theoretical understanding of particles.

Ok, this is making much more sense. I think the last follow-up question is this: I can see that if we somehow discovered a particle that was a mix of spin-0 and spin-1/2, this would force us to fundamentally reconsider our understanding of quantum mechanics as a whole. If we discovered a...

Right, I do know this. My question here was why that is the case, specifically if it was a theoretical necessity or an empirical observation. What I have gathered so far is that the superposition of integer-spin and half-integer spin is forbidden by theoretical necessity. The superposition of...

Right, I am talking about the term symbols -- i.e., for carbon ##^3D## is forbidden, since #L# is even and #S# is odd. I don't know how that works. Perhaps a mentor can move this post there, if it is appropriate.

Thank you for the answer. It leaves me with this question though: wouldn't the same logic prohibit, say, the state ##\frac{1}{\sqrt{2}}(|210\rangle + |220\rangle)## (expressing states as ##|nlm\rangle##? My understanding is that a state like that would be allowed. Additionally, I checked out...

Consider electrons in atom, and let's mostly ignore interactions between the electrons for now. What I mean by that is that the lowest energy level is the doubly degenerate 1s, then the doubly degenerate 2s, then the 6-fold degenerate 2p, etc. Textbooks like Griffiths use term symbols...

Hi all, it has been quite a while since I've been on here. I am curious, as the TLDR summary says, if it theoretically possible to have a particle in a superposition of states with different ##S^2## eigenvalues. For example, a particle being in the state...

Right, but I'm trying to figure out this with matrices/rotating coordinates to diagonalize ##A##. My understanding of what you wrote is that if you have ##\vec{y}=P\vec{x}##, this means (assuming ##P## is a constant nxn matrix) ##\partial_\vec{x}=P\partial_\vec{y}##. Is that correct? And if so...

I've been trying to get change of variables in PDEs down (I don't particularly like my textbook or professor's approach to it), and I want to ask here if I am getting this right. Let ##\vec{x}=(x_1,x_2,...,x_n)^T## and ##\partial_\vec{x}=(\partial_{x_1},\partial_{x_2},...,\partial_{x_n})^T##. I...

Force varies with distance from the center of Earth. Where does this come from? I may just be missing something obvious here.

Where does one find the solution? Wolfram Alpha didn't come up with anything. I could try to reverse engineer it, or in any case check a few things with the solution that I've wanted to look at.

That's pretty much it. If there is a very basic strategy that I am forgetting from ODEs, please let me know, though I don't recall any strategies for nonlinear second order equations. I've tried looking up "motion of a free falling object" with various specifications to try to get the solution...

And yes. I see though that it would all be meaningless if they were invertible matrices, as then the span of A would be all of ##\mathbb{R}^n##, and a vector in ##\mathbb{R}^n##'s projection onto ##\mathbb{R}^n## is just itself.

Ah, that's what I was missing. Ok, thank you.

Aren't they the same thing? If so, why would textbooks write the former? Ex: https://textbooks.math.gatech.edu/ila/projections.html or http://www.math.lsa.umich.edu/~speyer/417/OrthoProj.pdf or https://en.wikipedia.org/wiki/Projection_(linear_algebra)#Orthogonal_projections Thank you!

Thank you! I just coded a python program using my recursion relation (which I convinced myself was correct), and put it into OEIS for various values of N. I see now that ##f_N(d)## is asking how many partitions of d are there with no term greater than N. Thank you! The only remaining question...

This is an interesting combinatorics problem that I thought of. Oddly, I think I know of an application of said problem to physics, but I could not find any problem like it online (the closest I got was the Knapsack problem, which is just about optimization). My initial instinct was to look for...

It definitely checks out in the high temperature limit (##T=\frac{\epsilon q}{Nk}##) with two identical solids. I can't seem to figure out it it works with solids of different sizes. EDIT: I figured out that in the high temperature limit, it agrees with Newton’s law of cooling for any sized...

When I learned about Einstein solids in thermal physics, we assumed the fundamental assumption of statistical mechanics. For two interacting Einstein solids, I completely understand why this is valid after a considerable amount of time has passed. But, how can we model these solids as they get...

Here is an example code in python, describing a class of fruits: class Fruits(object): def __init__(self, color, taste): self.color = color self.taste = taste If, in general, we ALWAYS do this: class ExampleClass(object): def __init__(self, property1, property2)...

"abc".upper only produces an error message. I still don't understand why the code is not upper("i want this in all caps!"). Is there a good resource where I can learn about methods? I just don't get the concept, and what the difference between acting on an object (or having the object in the...

What do you mean by "functions belonging to an object"? I just don't see why the string would not be the argument of the function upper or lower, i.e. the script being upper("i want this in all caps!").

I am still on the basics of python, so the odds are if you use a lot of fancy terms I will not understand the answer yet. It is just really confusing me. Thank you!

Hi, I want to make sure my understanding of calculating surface integrals of vector fields is accurate. It was never presented this way in a textbook, but I put this together from pieces of knowledge. To my understanding, surface integrals can be calculated in four different ways (depending on...

Though I know it's highly unlikely that you have found pieces of wood or animal fossils...it sure does look like that's what it is. This could be like cloud watching for astronomers ("look, that rock looks like a gorilla!")

I have always thought of the Fourier Transform the way you are thinking of it. There is no link to your source that says it is not-- and I would likely be equally puzzled by that assertion. One thought I have, without seeing the source, is that there may be some mathematical semantics at play...

Hi! As I mentioned in the post in the other thread, that sim is my favorite on the PhET website-- but they have a bunch of other simulations for QM. If you look on https://phet.colorado.edu/en/simulations/filter?subjects=quantum-phenomena&sort=alpha&view=grid you will see all of the quantum...

Pretty much, but with a caveat. Technically, it's not just saying that the variance is zero, but the state of the particle has changed to the state where the variance is zero (there's still a caveat). Although I think this does tread a little bit into interpretations, as some interpretations...

That book is written for an introductory thermodynamics/thermal physics course. I also disagree that a problem should be self-contained and independent of the context of the course. Introductory physics textbooks don’t include the phrase “in the nonrelativistic approximation” in every...

How do you figure that?

I think that’s far too deep for an introductory thermodynamics course. At the level the OP is asking, the system is approximately adiabatic and quazi-static.

The first thing is about the book you cited. It's popular science-- not true, rigorous physics. Lots of what is in that book is misleading, or worse. I don't think you've fully grasped the concepts you're talking about here. There is no entanglement going on in the double slit experiment. The...

Yes-- say it is a very heavy piston, with the container on its side. The process would be very slow, but close to quazi-static. I think that the problem should have specified that it's a quazi-static process, but you could probably assume that given that it is likely all you have talked about in...

Yes. However, thermodynamics becomes far, far more difficult for non-quazi-static processes. You are correct that the equation will not give a 100% correct answer when applied to the real world...unfortunately, no other equation in classical physics does either. Physics is full of...

Although the specific patch does not exert a force on itself. It’s the other charges on the other sides of the conductor that create the pressure.

Something seems a little off with those pictures, so I figured I would add a picture with my work for this. Note that the only reason I am including equations in the picture is so that they are color-coded, so that you can match the terms to the vector on the diagram: Notice that for the sheet...

I think the wave-particle duality is still useful to think about, if you think about it in terms of what quantum objects behave like, as opposed to what they are. You are right that I should clarify: when I say localized, I really mean approximately localized. It is of course impossible to...

As Griffiths states, the electric field directly above or below the surface is caused by two things: the charged patch itself (##E_{patch}=\frac{\sigma}{2\epsilon_0}##) and any other electric field not related to the patch (##E_{other}##). That is ##E_{near surface}=E_{other}+ E_{patch}##, where...

The key to this is the identity $$\delta \left(f(x)\right)=\frac{\delta(x-x_0)}{\left| f'(x_0)\right|}$$ where ##x_0## is a zero of f(x). In this case, ##f(x)=x^2 - x_0^2##, ##x=p_0##, and ##x_0=\sqrt{\vec{p}^2+m^2}=E_{\vec{p}}##. This gives me the correct answer.

That's a major misunderstanding about quantum physics. The wave-particle duality is NOT about something being both a wave and a particle. That, as you have noted, would allow one to separate the electron wave and the electron particle, or the light wave and the light particle. This is real the...

Note: This question involves both classical and quantum physics, so I didn't know where to put it. I'll start with the coin flip: People often compare electron spins to a coin flip, citing that coin flips are random. I am wondering if that is a true analogy, or just another faulty analogy...

In mathematical terms, I would think one can describe a field F as a function ##F:\mathbb{R}^{4}\rightarrow X## (I'm not positive, but for relativistic physics I think it would be ##F:\mathbb{R}^{3,1}\rightarrow X##). What X is would then determine the "type" of field (i.e. scalar, vector, etc)...

If you need some resources to help you understand relativity better, I'm sure some people here can help you out with that. I can also let you know some pointers I found helpful when I first learned it (I did do a fairly intensive study of relativity, I just had to get a refresher in this thread...

When you say you are increasing the size of the solid, are you adding oscillators/atoms (increasing N) or increasing the distance between oscillators/atoms?

Well, good luck on your test! Just a quick thought-- you can easily check your work by noting that ##A=PDP^{-1}## where $$D= \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}$$ and $$P= \begin{pmatrix} 1 & 0\\ 1 & 1 \end{pmatrix}.$$ It is easy to compute ##e^{Dt}##, and then you can just use...

It won't let me edit the post as I made a few mistakes, so see the above quote for my revised post.

This is your problem. The statement ##e^{X+Y}= e^{X}e^{Y}## holds if XY=YX. Otherwise, it is not necessarily true. As you have seen, it is not true in your case. Notice that in the case where $$ B=\begin{pmatrix} 2 & 1\\ 0 & 2 \end{pmatrix}=N+D= \begin{pmatrix} 2 & 0\\ 0 & 2 \end{pmatrix}=...

Assuming throughout their lives they have only gone at nonrelativistic velocities, then there appear to be two cases: 1. The birth frame is at rest wrt. Bob and Alice, in which case Alex is older. 2. The birth frame is at rest wrt. Alex, in which case Alice is older. Is it even possible to...

The reason I phrased it as I did is that if Alex took a picture of Bob's watch, accounting for the time it took light to get there, he would conclude that Bob's watch hit the "?" time simultaneously with Alex crossing Alice. Bob would disagree with this, but that's okay, as spacelike events need...

I drew out the Minkowski diagrams for this situation-- I just wan't to make sure I did this right (note I have color coded the three people on the diagrams) First, I do know that d0>d', and tA=tB>t0 (right?). My question is on the time I labeled with a ? on both diagrams. If I am correct, that...