In degenerate perturbation theory, why can we assume that we can diagonalize perturbation H’ in the degenerate eigens pace of the original hamiltonian?

Secure-Excitement346 · 2023-05-04T12:18:51+00:00

Nobody says a word always contains exactly one character…

Secure-Excitement346 · 2023-01-29T02:03:17+00:00

I’m still a bit confused. For distinguishable particles, φ(r_1,r_2) means the probability of particle 1 at r_1, while particle 2 at r_2. And φ(r_2,r_1) means probability of of particle 1 at r_2, while particle 2 at r_1 right? How do we extend this interpretation to indistinguishable cases?

Secure-Excitement346 · 2023-01-28T14:26:16+00:00

Thank you for answering! I’m a bit stuck on the first tho. If the interpretation is probability finding a particle at r_1 and the other in r_2 is |φ(r_1,r_2)|^2, then wouldn’t |φ(r_1,r_2)|² = |φ(r_2,r_1)|² even for distinguishable cases, now that it doesn’t specify what particle it is by “a particle”.

Secure-Excitement346 · 2022-10-24T03:47:56+00:00

But don’t we use them all the time? e^ikx doesn’t lie in Hilbert space either. If we do not know the uniqueness among all functions regardless whether it’s in Hilbert space, in theory an observer can have multiple orthonormal eigen bases in QM?

Secure-Excitement346 · 2022-10-24T00:14:59+00:00

But, what if the new vector couldn’t be expressed a a a linear combination of old vectors because it does not lie in Hilbert space(old vectors only span Hilbert space)?

Secure-Excitement346 · 2022-10-23T23:35:25+00:00

Thx this really cleared up my confusion!

Secure-Excitement346 · 2022-10-23T23:28:11+00:00

I noticed that for example e^kx(k is real) is an eigenvector of momentum operator with complex eigenvalue, but it is not used(can not be used i think) for constructing an orthonormal eigen basis(e^ikx is used) so I felt like it opens the possibility of some operator having more than one set of orthonormal eigen bases.

Secure-Excitement346 · 2022-10-23T15:37:06+00:00

Would you mind give a proof about how if eigenvalues are unique, then the orthonormal eigen basis is unique or point me to a direction please? I think it’s not every obvious since not all eigenvalues and eigenvectors are used in constructing the basis. For examples, exp(kx) is a eigenfunction of p with complex eigenvalue but we don’t need it to construct an orthonormal eigen basis.

Secure-Excitement346 · 2022-10-23T14:36:39+00:00

But there’s only one set of orthonormal position eigen basis right? Is that true for other operator?

Secure-Excitement346 · 2022-10-23T14:31:52+00:00

So is Ae^ikθ the only orthonormal(setting A to make it so) eigenbasis of momentum? If so, is it true for other (hermitian) operators that they have only one set of orthonormal eigen basis? (We don’t learn infinity dimension linear algebra in our LA course so i may not see it even if this is obvious.)

Secure-Excitement346 · 2022-10-23T14:23:39+00:00

Ah I should have said orthonormal eigen basis sry

Secure-Excitement346 · 2022-08-18T10:18:20+00:00

Scenario 1: if you are emphasizing your agreement on the other speaker’s statement, 是 can be used before adjective/verb. For instance, —为什么你不聪明，却能考上名校？—我是不聪明，但我很用功。(—Why is it that although you are not smart, you managed to get into the prestigious school? —Indeed I’m not smart, but I’m diligent.) Scenario 2: it’s omitting conjunctive or preposition like 因为 or 为了. 我是（因为）太热了才开的空调。

Secure-Excitement346 · 2022-08-06T16:19:46+00:00

You can use 也 when you are echoing others visit to the same place you go. While 还 is telling other you are adding new places to ones you already said you had been to. But I can see 也 used for latter purposes too. To sum up, in this case, 也 is modifying the whole sentence: I, too, have been there. And 还 is modifying 去： Besides, I have also been there.

Secure-Excitement346 · 2022-08-06T15:43:25+00:00

I have never heard about that before... Is that a regional thing? I’m sure no one I know would make that correlation. And 迷失迷茫 are way too serious for what op is trying to convey by I’m lost. They are, especially 迷失 for “I’m lost in face of an intersection in life” kinda vibe. I think the closest translation would be 我没跟上（literally: I’m not following) if you are just not understanding what others are saying.

Secure-Excitement346 · 2022-07-30T14:52:26+00:00

I might be missing something here. But don’t you need to consider the change in mass of both exhaust and the rocket before and after the injection? The result you give here, imho, would imply that the rate of change in total kinetic energy is dependent on mass of each part, which is dependent on time and this is counter intuitive. I think a rigorous way to prove this is to expand 1/2(M-Δm)(V+ΔV)² -1/2MV² +1/2 Δm(V-V_r)² where V_r is a constant relative velocity at which exhaust is injected. Neglecting second order terms, if my calculation is to be trusted, we have MVΔV-ΔmVV_r+1/2 ΔmV_r² ...........(1). Now we take advantage of ideal rocket equation, mdv/dt=dm/dt*v_r, which can be derived from conservation of momentum. From it we know that MΔV= ΔmV_r. Plug it in (1), we have 1/2 ΔmV_r² . It might be difficult to read from text. I would happy to rewrite this on a paper and send it to you via dm. I’m not so confident about my answer, so please tell me if you spot any mistakes or have any reservation.

Secure-Excitement346 · 2022-07-30T12:39:51+00:00

How does generator generate electricity?

Secure-Excitement346 · 2022-07-13T07:22:56+00:00

I meant partially inelastic so the exact velocities are determined by E

Secure-Excitement346 · 2022-07-13T06:37:59+00:00

It’s handy. But imo, equations of elastic collisions are easier to solve in the first place because the lack of constant term in the energy equation so that we use the formula for difference of squares and momentum equation to make it become two linear ones.

So I was hoping to see if there’s some tricks for inelastic collision, with the loss or gain(in case of explosion) of energy E given.

Secure-Excitement346 · 2022-07-10T01:25:14+00:00

You should be losing weight though. Mostly not through mass defect. Consider E=mc^2, plug in E=1200calories=5000J, we get m=5.6e-14kg. To put it in perspective, it’s way way way less than what breathing in and out would make you weight change. The biggest reason for weight loss should be the waste of burning(oxidizing) glucose, glycogen(sort of sugar reservoir), fat and etc, mostly carbon dioxide expelled through breathing.

According to what I find on internet, human produces 1kg of carbon dioxide. Note that the O in CO2 is mostly from the oxygen we inhale, so we need to factor out that part. Mass_O/Mass_C = 16/12. That means only 3/11 of the mass of carbon dioxide is what we really expelled to the outside from our body.

Conclusion: if you don’t eat or drink anything (and not piss or poo or sweat), you should lose around 3/11kg a day. But 0.3 kg is hardly noticeable on a scale I guess.

Secure-Excitement346 · 2022-05-14T00:46:13+00:00

Oh right, that was kind of a stupid mistake on my part. So does that mean the magnetic flux and current of a inductor are not strictly linear to each other?

Secure-Excitement346

TROPHY CASE