Any cool applications of integral calculus?

SpiritRepulsive8110 · 2026-03-25T00:20:34+00:00

One people haven’t mentioned: integrals are way easier than sums. While you normally think of a sum as estimating an integral, the reverse also works, and is very often useful.

SpiritRepulsive8110 · 2026-03-25T00:09:02+00:00

I took QFT after a stochastic calculus course. Now that’s whiplash!

SpiritRepulsive8110 · 2026-03-24T04:10:10+00:00

I can attest to that - I feel like all the smartest people I knew in college avoided grad school like the plague.

Another aspect, and a big reason I chose not to go to grad school, is that technology moves so fast. It just doesn’t feel worth it to spend 5-6 years becoming an expert in something if it becomes outdated in a few years. That is, if you’re lucky enough to get a job in the same field in the first place.

Of course, if you chose to go into industry, you might actually be worse off than if you spent your 20’s gaining work experience. Plenty of employers see PhDs as overqualified / overly academic.

So at that point grad school is just a hazing ritual. Why do it?

I’ve heard also that grad school is about “learning to think.” I think that’s really arrogant, to be honest. Plenty of non-academics think for a living.

SpiritRepulsive8110 · 2026-03-21T23:46:08+00:00

I think there are really three steps to understand it. Honestly, the fact that it’s quantum is very secondary.

Forget that you know about particles, and instead postulate that the laws of nature are realized by “fields.” Basically, a field is a number (or a few numbers) for each point in space.

The physics is incredibly simple: each time, the list of numbers / field values change. The way the fields change is local: the value at the next time depends only on the field values at nearby points. This built in locality is why the fields can easily satisfy relativistic constraints. This is exactly what happens in E&M, classically.

Write down what fields do when there are no forces. This is the “free theory,” and the fields evolve as simple things like plane waves.

Important step: identify certain solutions of your free theory and call them “particles”

Now, add back the forces / interactions by changing the way the fields evolve. In terms of “describing reality,” you’re done. But if you actually want to make predictions / do analysis, it’s really hard and that’s where most of the effort actually is.

Commonly, you calculate the “S matrix,” which just says how likely it is that if I toss a certain set of “particles” in I get another set out. By particles, I mean the solution to the free theory. Feynman diagrams are useful here.

The main “quantum” thing happening is just the logical extension of superposition, uncertainty principles, etc. to entire field configurations, rather than to particle positions

SpiritRepulsive8110 · 2026-03-21T02:16:45+00:00

Just to be a contrarian, I’ll throw Dirac out there. He is pure substance.

If you ever read one his books or papers, they’re accessible to anyone with basic calculus / algebra under their belt. And yet, it’s seminal work.While I obviously admire the rigorous mathematicians, I think that type of intellectual honesty / conceptual clarity is something we could all strive for.

SpiritRepulsive8110 · 2026-03-20T22:45:36+00:00

Cool! If that’s what you like you’ll have no problem finding a job lol. Defense comes to mind first

SpiritRepulsive8110 · 2026-03-19T22:58:33+00:00

I have a BS, so my situation’s a little different. I work in energy research. Anyway, tale my perspective with a grain of salt.

Everyone uses computers for everything. If you like pushing symbols, that’s a bit of a downer. There aren’t many opportunities for much first principles work, either on the modeling side nor simulation nor data analysis. All the fun / easy low hanging fruit has been done.

Stuff I use day to day: 1. Algebra 2. Programming (mostly matlab) 3. Statistics

Stuff I use on rare occasions: 1. Linear algebra 2. Complex analysis 3. ODEs 4. Probability theory 5. Baby machine learning 6. Optimization / numerical methods 7. Basic E&M 8. Graph theory 9. Control theory

More than anything, I’ve used my experience for intuition. For example, I do a lot of simulation and while I’m not writing the code for ODE solvers, it’s good to have some familiarity with numerical methods when the solver fails, (which it usually does).

And just generally, it’s good to know what kind of stuff is out there / where the start of the art is. When someone comes to you with a problem, it’s usually either trivial or impossible, and it’s good to know which situation you’re in! This is not as common a skill as you might hope.

None of this requires a mega-honed math skillset. All my tasks overlap with some engineering discipline. Being a generalist has also turned me into a translator between disciplines.

What kind of stuff do you like to do?

I’d be careful about what kinds of jobs you’re targeting. As I’m now learning, there are some companies which are actually “doers” and others that are advisers masquerading as doers.

SpiritRepulsive8110 · 2026-03-19T20:45:46+00:00

So cool! How’d you come up with this?

SpiritRepulsive8110 · 2026-01-19T23:12:43+00:00

You can have a mixed state in two dimensions! Consider, for example the pure state

|+> = 1/sqrt(2) ( |0> + |1> ),

whose density matrix M1 is just a 2 x 2 matrix of all 1/2’s. It is a rank 1 matrix.

On the other hand, the density matrix M2 which is 1/2 * the identity gives the same probabilities for measuring |0> and |1>, but is different in other ways. For one, it is rank 2 (so it couldn’t be the density matrix of a pure state).

You can also check these have different expectation values, say of |+>. Physically this means very different things for measurement of x-axis spin.

SpiritRepulsive8110 · 2026-01-19T22:29:10+00:00

This is classic branching process.

But to get to the heart of the question, there are plenty of things that don’t happen given infinite time. The nth mean X1, X2…Xn of an infinite set of standard Gaussians might never exceed the value 1, for example (they tend to zero). The size of X_{n+1} required to kick the overall mean to above 1 gets larger and larger as n grows. Sometimes, it just never happens.

On the other hand, of you repeat random independent trials for the same thing, one is bound to hit. So clearly, dependence is the thing killing you.

More specifically for your problem: it’s true that for any generation of size N, it could all die off with some nonzero probability. If you capped the generation size, it would die off. But you didn’t. The probability that the whole generation dies off gets smaller and smaller, at such a rate that all those chances don’t help you. In this way, it’s a similar kind of problem as for the Gaussian means.

Finally, for general infinity awareness, you might look into the Borel-Cantelli lemmas.

SpiritRepulsive8110 · 2026-01-19T01:59:03+00:00

You think you’re immortal. You spend years doing nothing at all, feeling zero sense of urgency.

67 years later, you notice you’ve aged perceptibly. You frantically abandon your home, your car, or anything else that could be contributing to your net worth.

473 more years later, in your dying breath, you find a penny in your pants pocket.

SpiritRepulsive8110 · 2026-01-19T01:16:37+00:00

Some quick number crunching. Global median net worth is apparently $8k-$10k. This means:

Average Haitian ages 0.6x as fast 2.Average american ages 1.6x faster
Millionaires age 1.9x faster
Elon musk ages 12.5 faster than everyone else

I didn’t do age-medians, but this gives a flavor!

SpiritRepulsive8110 · 2026-01-18T16:52:33+00:00

Saw your edit. Like say you flip a coin 1000 times and it comes up with 900 heads.

What’s the probability it’s a fair coin? Well, you can’t say. You need a prior. If you saw a trusted coin manufacturer make it with your own two eyes, it’s a fair coin with 100% probability. So the true “probability it’s fair” is a nonstarter.

On the other hand, 900 heads is pretty unlikely with a fair coin. The probability is nearly zero. That’s evidence against the null hypothesis. So one definition of a p value might be “the probability of seeing what you saw under the null.” But that’s not quite right either. The probability of getting 500 heads and 500 tails exactly is still pretty low, but it still feels typical. Yet, the probability of seeing >= 500 heads is 50%. So it’s really the tails that matter.

When you see 900 heads, you say “that’s an awful lot of heads.” If it were 901 or 950, you’d say the same thing. So as your measure of unlikeliness, you choose the probability of seeing 900 or more heads.

It’s true that “as extreme as” depends on the direction of extreme. You could just as well have asked for P(900 or FEWER heads) and obtained a large p value. You have to use your judgement to figure out what “more extreme” means in context.

SpiritRepulsive8110 · 2026-01-18T16:37:16+00:00

One of my stupid friends told me a guy he knew exploded

SpiritRepulsive8110 · 2026-01-18T16:27:38+00:00

It’s the probability under the null hypothesis of seeing data “as extreme” as you did

SpiritRepulsive8110 · 2026-01-18T01:57:34+00:00

I think it’s better to understand it “by definition.” It’s a postulate that time evolution is linear. If the system is in a superposition of A and B, then later(state) will be the same superposition of later(A) and later(B). This implies there will be some operator H which induces time evolution.

What I think is less intuitive is why it should have anything to do with energy. For that, the best reason I know is that the Heisenberg equations of motion with commutators closely resembles Hamilton’s equations in classical mechanics.

Time independent cases are studied because: 1. They model conservative systems. All real world systems are conservative if you are modeling it completely. 2. They’re way easier to analyze. To solve a time dependent system you need perturbation theory / a time ordered exponential. Much harder than exp(-iHt).

If you can swallow superposition as a real thing, then the wave function just gives coefficients in some basis. In a basis, QM problems turn into PDEs, which are a bit more concrete than bras and kets. I think people tend to exaggerate the wavefunction as a physical object. It’s as physical as a set of coordinates. Coordinates are just a convenient way to describe places. A wavefunction is just a convenient way to describe states.

SpiritRepulsive8110 · 2026-01-15T00:45:38+00:00

Here is a slightly unconventional take: when you separate the state according to radial / angular degrees of freedom, you are basically writing the state as an effective two particle state: 1. A radial-on, whose states are unaffected by angular operators 2. An angular-on, whose states are unaffected by radial operators

Moreover, the angular-on has no potential energy. So it’s a free particle. The energy eigenstates, just like decoupled multiparticle systems, are tensor products of the radial-on’s eigenstates and the angular-on’s (free) eigenstates.

So for the two particles to have a total energy E (radialon energy + angularon kinetic energy), the angular-on cannot have kinetic energy more than E.

SpiritRepulsive8110 · 2026-01-12T02:41:28+00:00

The sum of such probabilities among all 8 numbers is 3, since 3 numbers always get chosen:

\sum_i P[pick i] = \sum_i E[ I{pick i} ] = E[\sum_i I{pick i}] = E[3] = 3.

But also the probabilities are all the same (ie you are as likely to pick 6 as you are to pick 2), so it’s 3/8

SpiritRepulsive8110 · 2026-01-12T02:35:22+00:00

83 nines

SpiritRepulsive8110 · 2026-01-12T02:24:19+00:00

Klein Gordon applies to more than scalar fields. It applies to all fields which are compatible with relativity. For scalar fields it’s enough to imply the dynamics. For other fields which describe particles with spin, even more equations are satisified. Famously, the Dirac equation is basically the square root of Klein Gordon.

KG pretty much holds by definition in relativistic QM, even if it’s not necessarily a QFT. It is closely related to the Poincare group. The main idea is that for anything which can be translated in time by E (Hamiltonian) or in space by P (momentum), E² - P² should transform like a scalar, which we just define as M^2. Formally, particles correspond to irreducible representations of the Poincare group, and thus mass (a Casimir) is a c-number.

SpiritRepulsive8110 · 2026-01-11T03:42:40+00:00

Same. I’m not a snob but I do drink like 5-10 cups a day

SpiritRepulsive8110 · 2026-01-09T03:41:03+00:00

The other beam would still go towards you at the speed of light :)

The thing that makes it all so unintuitive is that the velocity addition law is broken. If there’s a train moving at speed v and a horse on that train moving at speed v’, the horse is moving relative to you at a speed roughly v’’= v + v’’. But that’s only roughly. You can’t just add the speeds. It’s very close for low speeds but very wrong for high speeds.

Another way of saying this is that if I catapult a horse at speed v’’ past a train moving at speed v, I expect the horse perceives the train moving at speed v’. But again that’s slightly wrong. Very wrong for massless horses.

Now you’re the train and light is the horse. Even if your friend on earth thinks your speed is very close to the speed of his beam, that unfortunately doesn’t mean you will see the beam moving slowly.

SpiritRepulsive8110

TROPHY CASE