Recent content by haisydinh

Hi, a friend of mine gave me a math problem which I've spent hours trying to find different methods to solve. But none of them work and I'm now out of ideas. The problem goes like this: So for example, ##17^2 = 1^2 + 2(12^2) ⇒ 17 = 3^2 + 2(2^2) ## or ## 3^2 = 1^2 + 2(2^2) ⇒ 3 = 1^2 + 2(1^2)##...

Thanks for your answer, PWiz! However, even after reading through the article, my trouble still remains. I understand that energy can be conserved if we consider gravitational field to be negative; but how does that actually work in a universe that is accelerating due to the dark energy? The...

Can you please elaborate a bit? At least from what I currently understand, the quote above of Sean Carroll basically says that it's ok to claim that the law of energy conservation holds in general relativity, and thus the conservation law also holds for the case of dark energy. But I don't...

I know there has been a lot of threads on the forum about this topic, but my question is slightly different from the others. I have recently read this article by Sean Carroll (http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/); now, I'm completely aware (& convinced)...

Well, I think you got it. You can re-write the quote by just replacing the word 'variation' with the word 'derivative' (because they mean basically the same thing as far as differentiation is concerned). In other words: "the first-order [derivative] in S (action) has to be zero for S to be at a...

For simplicity, let's just deal with the 2-dimensional case. It's true that at the minimum, whichever way you go, you will always 'go up the hill' (so to speak). Hence, it's obvious that f(x-h) >f(x); and f(x+h) >f(x) (for any small variation h). So visually, the graph decreases as it goes...

thanks very much for your answers. However, I'm still a bit sceptical about this. I mean I have tried traveling in a roller coaster before, and I know that it's a super terrible feeling to fall down freely. But if free-fall is equivalence to floating in zero-gravity, then does that mean that I...

Here are the 2 situations: Situation A: a person floating in space far away from any mass (i.e. the person is in zero-gravity environment) Situation B: a person undergoes a free-fall from the top of a roller coaster (assuming that there's no air resistance) My question is, would the person in...

Yes, you are absolutely right! How silly of me! Thanks for pointing that out! Well, it's clear that my proof is wrong here. But the problem still remains though. Maybe it's more relevant to physics rather than to mathematics. Maybe in quantum mechanics, they are talking about a different type...

Maybe I should be more clear with my question. Let's say that ##M = \begin{pmatrix} a & c+id \\ c-id & b \end{pmatrix}## is our Hermitian matrix, then the polynomial is: ##det(M - I\lambda) = det\begin{pmatrix} a-\lambda & c+id \\ c-id & b-\lambda \end{pmatrix}## ##...

Hi, thanks for replying. However, the theorem says that "the polynomial associated with a Hermitian matrix always has ##n## distinct roots". In other words, this is a special property of a Hermitian matrix. I can easily prove this with a ##2×2##- Hermitian matrix or a ##3×3##-Hermitian matrix...

Hello, I am currently trying to study the mathematics of quantum mechanics. Today I cam across the theorem that says that a Hermitian matrix of dimensionality ##n## will always have ##n## independent eigenvectors/eigenvalues. And my goal is to prove this. I haven't taken any linear algebra...

Hi, I am a high-school student who recently finished the chapter on QM. I thought I completely understood it, but when I started to look back at what I’ve learnt, then suddenly, nothing really makes sense. And one of the things I find really really hard is the nature of electrons. My...

Hi, I am reading about the quantum model of the atomic structure, and recently encountered the Schrödinger’s model. However I am a bit confused about the nature of the electrons. Can we think of an electron a wave or as a particle when it is inside an atom? I know that in Schrödinger’s...

Yes, this was really my source of confusion. It seems I forgot that the definition of potential energy created by a conservative force needs to be changed as well (i.e. a negative sign must be added); because otherwise, the resulting potential energy would be positive, which could not be right...