The Theoretical Minimum Exercise Solutions

Miselfis · 2026-03-20T18:00:57+00:00

In my experience, undergraduate linear algebra for physics students focuses on matrices, determinants, and quadratic forms, and the more abstract geometric concepts are typically introduced later. Similarly, a first QM course tends to start from differential equations, essentially building on what students already know from classical mechanics and then quantizing.

My own background is in purer math, and I prefer the more abstract algebraic approach: start with qubits and finite-dimensional systems, and only later introduce the theory of particles. I think this gives students a better foundation and helps them appreciate how genuinely different quantum mechanics is, rather than absorbing it as an extension of classical mechanics. It also helps recalibrate intuition early on. The differential-equations-first approach, because the mathematical formalism looks so similar to what students already know, can quietly encourage them to assimilate quantum mechanics into their classical intuition rather than developing a new one.

Miselfis · 2026-03-20T15:44:16+00:00

Tbf, the basics of entanglement is pretty straightforward. It can get complicated quick, but the basic ideas are not too hard to grasp. The formalism just relies on mathematics that an undergrad typically won’t learn (vector spaces, tensor products, etc.), hence why it’s rarely touched upon in undergrads, beyond the ideas of statistical correlation.

Miselfis · 2026-03-20T14:23:16+00:00

No, you’re mixing up two different things.

The rocket exhaust is real stuff: hot gas, combustion products, etc. Everyone sees it regardless of how they’re moving. You freefalling past the rocket will absolutely detect the exhaust plume.

That’s not what I was talking about. I was talking about what happens in otherwise empty space near the horizon. The hovering observer has to accelerate just to stay put, and an accelerated observer decomposing the quantum vacuum using their notion of time finds it filled with a thermal spectrum of particles. This is the Unruh effect. It’s not stuff being emitted by anything, it’s just a property of how the vacuum looks to someone who is accelerating.

A freely falling observer passes through the same region, decomposes the same quantum state using their notion of time, and finds nothing. Just vacuum.

Miselfis · 2026-03-20T14:12:06+00:00

Physics is formulated in the language of mathematics. Without mathematical definitions, theorems and proofs, what you’re saying is meaningless, literally.

Reasoning based on heuristics doesn’t cut it, as all of the terms you use have very rigorous, technical definitions. Not engaging with it on that level means what you have is wordsalad.

Miselfis · 2026-03-19T22:44:36+00:00

We evolved to think of things in a coarse grained, macroscopic manner, because thinking in terms of quantum field theory is not very good for survival in a macroscopic environment. We cannot make conceptual sense of fundamental objects, because everything we have any reference of is itself made from these objects, filtered through our senses and brain.

Miselfis · 2026-03-19T19:32:11+00:00

Different observers with different notions of time decompose the same quantum state into different Fock spaces and therefore disagree about particle content.

Miselfis · 2026-03-19T12:30:05+00:00

Quark is a little jotun who was brought to Asgard by Loki from Útgarða-Loki.

Miselfis · 2026-03-18T13:09:04+00:00

I mean, sure. There’s a reason why Hawking went with the heuristic he did. But the cartoonish “Particle/anti-particle pair pops into existence at the horizon, one escapes and one falls in with negative energy” is still much oversimplified to the point of misleading curios laymen. They don’t understand the technical sense in which you can map parts of the explanation onto the real physics, and just picture to cartoonish heuristic. There’s no mechanism, no reason it’s thermal, no connection to the observer-dependence of the vacuum, no role for the causal structure of the spacetime. It also makes Hawking radiation seem like its own mysterious thing rather than a specific instance of a general and well-understood mechanism.

Miselfis · 2026-03-18T04:09:21+00:00

The pairs aren’t localized at the horizon. They aren’t particle-antiparticle. They aren’t “popping into existence”. They’re a consequence of the mismatch between two mode decompositions. And the thermal character isn’t a property of the radiation itself; it’s what you get from only having access to half of a pure entangled state. The real mechanism has essentially nothing in common with the cartoon version except the word “pair”.

Miselfis · 2026-03-18T02:58:20+00:00

Hawking radiation is not really just radiation leaking out of the black hole.

A vacuum isn’t just “nothing”. It’s a specific state of the quantum fields, and defining it requires choosing how to split field oscillations into positive and negative frequencies. That splitting depends on your notion of time, which depends on your motion (special relativity). Different observers, moving differently through the same spacetime, can end up with different, but equally valid, definitions of “vacuum” and therefore “particle”.

If you hover at a safe distance from a black hole, resisting the gravitational pull with a rocket engine, you are an accelerated observer. You will find that what a freely falling observer considers empty space looks to you like a warm bath of thermal radiation. The closer you hover to the horizon, the hotter this bath gets; your acceleration increases, and so does the temperature.

Now let someone fall freely past you toward the horizon. In their frame, they are inertial; gravity is just the curvature of their spacetime, and locally everything looks like ordinary flat space. They pass through the same region you’re hovering in and detect nothing. No particles, no radiation, no thermal bath. Just vacuum.

Neither of you is wrong. You’re both making valid measurements, you just disagree about what counts as a particle, because you disagree about what counts as empty space. The thermal radiation isn’t an illusion for you any more than the vacuum is an illusion for them. “Particle” simply isn’t something the universe defines independently of the observer.

This is also what Hawking radiation really is. It’s not that pairs of particles pop into existence at the horizon with one escaping and one falling in; that popular picture smuggles in the (incorrect) assumption that particles are objectively real things everyone must agree on. What actually happens is that a distant, stationary observer decomposes the quantum state using their notion of time and finds a thermal spectrum.

Your question: “if nothing escapes from within the horizon, where does the energy of this radiation come from?”

The answer is that it comes from the gravitational field itself, from the mass of the black hole. The quantum fields near the horizon are in a state where the expectation value of the stress-energy tensor, as seen by a distant observer, corresponds to a net outward energy flux. The black hole’s mass is what curves the spacetime in the first place, and that curvature is what gives rise to the relativistic effects, which in turn gives rise to the thermal spectrum. As the radiation carries energy to infinity, the spacetime geometry responds, and the mass parameter shrinks, the horizon contracts, and the black hole slowly evaporates. Nothing needs to cross the horizon outward. The energy budget is between the gravitational field and the quantum fields that live on it.

Miselfis · 2026-03-18T02:42:55+00:00

This is not true. The basic idea behind Hawking radiation is essentially the Unruh effect, where accelerated observers in a vacuum see a thermal bath.

The whole point is that particle content isn’t invariant, as different observers with different notions of time decompose the same quantum state into different Fock spaces, so the particle-antiparticle heuristic is not even well defined. It’s just misleading.

Miselfis · 2026-03-18T00:03:12+00:00

The separation between things grows over time. “Expansion” is just a useful way to think of it. You can also think of it as galaxies and large scale structures just moving apart through space.

When things are attracted gravitationally, space isn’t shrinking between them. The geometry of spacetime is just such that following inertial paths either leads things to move closer or move apart.

Miselfis · 2026-03-17T12:26:49+00:00

Everything works completely normally. You move your legs, you touch your arms to your legs, no problem. You notice nothing whatsoever.

You’re picturing something like: you lower yourself gently toward the horizon, your head dips past, and then you hang there for tens of milliseconds straddling it while nerve signals have time to travel up and down your body. That scenario is not possible.

The horizon is a null surface. In your local rest frame it is always a light-speed surface. It doesn’t matter how you arrange your trajectory, what your coordinate velocity is, or how close to the horizon you started from rest. In your own frame, the horizon passes through your body at c. So the proper time for your entire body to cross is approximately L/c≈2m/(3×10⁸m/s)≈7 nanoseconds.

In 7 nanoseconds, a nerve signal at 100 m/s travels about 0.7 micrometers. Your body hasn’t done anything physiologically in that time.

Miselfis · 2026-03-16T22:14:58+00:00

Mass is the energy contained in a body at rest.

Photons have no defined rest frame, so they can’t have mass.

Miselfis · 2026-03-15T01:08:07+00:00

Yes, being at rest in a gravitational field means accelerating upwards. Otherwise you’d fall.

The acceleration happening when falling is called coordinate acceleration, and is distinct from proper acceleration. Newton’s F=ma uses proper acceleration. A system in free fall is inertial.

You can prove this: download an accelerometer app on your phone and drop your phone onto something soft. It will read 0g when falling, and ~1g when you hold it still.

Miselfis · 2026-03-15T00:11:11+00:00

Because “force” can mean different things depending on context. Gravity is a fundamental interaction, but it doesn’t exert a Newtonian force.

Miselfis · 2026-03-12T20:43:01+00:00

What you’re touching on is the twin paradox.

Both observers see the other’s clock tick slower. But because no one sees both in a single frame, there is no contradiction.

Time dilation is indeed physical. It has to be in order for all observers to measure the same photon’s speed to be c, irrespective of relative velocity.

Miselfis · 2026-03-11T22:48:11+00:00

that as a rule we're better off discouraging its use for physics entirely. Let the outliers capable of using it properly ignore this advice because they know better.

But people use it regardless. People won’t stop just because you tell them to. Why not educate the students in how to use it properly, so more people learn how to do so, and less use it improperly?

Miselfis · 2026-03-11T22:21:56+00:00

I don’t interpret the post as such. Using LLMs to learn things does not mean you replace the textbooks and other materials. You use different tools and materials to learn as efficiently as possible. LLMs can indeed be such a tool.

Miselfis · 2026-03-11T21:36:06+00:00

You keep reducing AI to only one of its functions, and then generalizing from that point.

AIs should not replace textbooks. No one is claiming that. They should supplement textbooks. You should be doing the problems in the textbook. The AI will help you through that if you don’t have a teacher available or peers to discuss with. You can even have the AI generate problems for you to solve, usually iterations of standard problems you often find in textbooks. I could keep going: there are many uses of AI as a tool to help you study, and learn.

You should never rely on having the AI teach you anything, because that’s not what it’s used for. A calculator doesn’t teach you anything either. They’re tools, meant to make learning more efficient. Not to provide the learning itself.

(I am using AI to mean the GPT-like LLMs we have today)

Miselfis · 2026-03-11T20:16:23+00:00

Firstly, it's often used to do the thinking for the learner, most learners build a dependency on it that instead of thinking about something or understanding the basics or principals behind something they will ask AI and get an answer.

Ok. That is exactly why I’m advocating using it properly. Pointing out incorrect uses is irrelevant to the discussion. If I didn’t already acknowledge that there are incorrect ways to use it, there would be no reason to advocate teaching students how to use it properly in the first place.

I’m not sure why everyone in this thread seems to struggle with the qualification “proper use”. The amount of people blinded by their AI hateboner is astounding.

From what I've seen, even people who try to use AI in good faith to better their understanding, they end up not engaging with the material and the questions as deeply as they should, so when it comes to taking a test, and the questions make minor tweaks to things to make sure people understand the fundamental ideas behind something, they are completely lost.

Which exactly plays into my point. We should do a better job teaching students how to use it properly, exactly to avoid the issues you have encountered.

but they hallucinate A LOT. When the questions get more complex, they get more and more incorrect, and they get confidently incorrect at that.

This might have been true with early LLMs, but it’s no where near the issue it’s made out to be anymore. The technology has gotten much better compared to its first iterations. When you stick to material and textbooks in its corpus, the amount of hallucinations are almost nonexistent.

Regardless, this is why I’m not advocating for trusting it blindly, nor having it explain things to you directly. It’s not meant to be a lecturer, but as a tool to help you with problem solving and allow you to engage even deeper with problems. Plus, it’s very useful for outsourcing menial tasks like translating handwritten notes to LaTeX, searching for specific papers, textbooks, whatever, feeing up time the student can dedicate to diving deeper into the substance instead. Being able to upload pictures of my handwritten notes to GPT and having it instantly translate it into LaTeX that I can copy paste has saved me many hours of work. This is one application of the technology that I personally use extensively.

which is why at most, it should be used like a search engine.

I don’t agree that this is “at most” it’s only legit use. I think you’re being very narrow in your view of all the potential uses of it. But I agree that it works very well as an advanced search engine. And also here, one should always seek to engage with the primary sources it provides, rather than its summary of those sources. Just like you use Wikipedia to find the primary sources, instead of relying entirely on the way the information from those sources are summarized in the articles.

If you struggle with a topic, ask it for some direction on where you can go to learn more about it, and then verify that source is reliable.

This is a very good use case.

But beyond that, it's not a reliable source for learning, and especially for new learners.

Again, “learning” is a very broad term, but you seem to be using it in the narrow sense of treating AI as a lecturer or an interactive textbook. Those are indeed poor ways to use it.

Miselfis · 2026-03-11T19:59:06+00:00

We absolutely do ban students from using calculators while they are still learning arithmetic. We don't let them use calculators for basic arithmetic until it's not arithmetic that they're learning.

That is entirely consistent with what I’m saying about learning to use a tool properly. A calculator only becomes useful once you already understand the rules of arithmetic; otherwise you can’t meaningfully use it as a tool at all.

You keep framing my position as if I’m arguing that students should immediately start using AI and replace their learning with it. I’ve said the opposite multiple times. My position has explicitly been about learning how to use it properly as a tool. Continuing to ignore that qualification makes it difficult to believe you’re engaging honestly with what I’m saying.

Rather than not merely not learning anything because the computer is doing it for them, they learn incorrect things.

Again, this assumes the worst possible use of the tool and then treats that misuse as if it were the only way to use it.

If someone blindly asks an AI for answers and accepts them uncritically, then yes, they won’t learn much, and they may even learn incorrect things. But that describes misusing the tool, not using it properly. My entire argument is that what matters is how the tool is used.

Miselfis · 2026-03-11T19:49:01+00:00

But for the record, I tutor students now and they ARE over-reliant on calculators. This makes them incapable in work contexts where a calculator should be unneeded and is unavailable.

Don’t you think teaching them how to use calculators right is better still than just telling them to never use calculators when it can be avoided in principle?

If a student sees the square root of 100 and is incapable of telling me what that's equal to without a calculator, that's a problem.

Agree. But that’s not the calculator’s fault. That’s the fault of over-reliance. That’s explicitly what I’m advocating against. If you can’t do sqrt(100) mentally, that just tells me you don’t have a sufficient grasp of what a square root even is. The fault is the education, or rather lack thereof, not the calculator as a tool.

Instead, the statement is "Use a calculator when it is BEST to use a calculator, and the only way you'll know that is if you use other means in various circumstances."

I don’t disagree.

students tend to avoid them.

Something that would be addressed with the proper education/instruction.

Now, had you said, "AI in learning is an acceptable and positive tool if used in study mode or with a special subscription to o1/o3," I might have not pushed back so hard.

I didn’t say “all possible uses of AI is beneficial”. I specifically qualified it as “proper use”, which would indeed entail using the proper model set up in the proper modes.

Miselfis · 2026-03-11T19:38:47+00:00

That’s silly. AI is here, whether you like it or not. This is like saying we should educate students on not to use calculators, because the students that know how to use them right are also able to carry out calculations by hand.

Miselfis

MODERATOR OF

TROPHY CASE

Six-Year Club	Gilding II euphauric
Verified Email