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@Parth - Did you see the new video by Dialect called 'The Geometry of a Black Hole'? It taught me things I didn't know about black holes before. You've previously made very clear & interesting videos on BH e.g. 'Strange Properties of Spinning Black Holes'. Perhaps you could say similar things to what Dialect said but using your own words & style?
This video appeared in my suggestions stream. It gave me quite a shock as it put back into my mind things that had long since disappeared from my thinking. In the early seventies I used to deal with such things and was very familiar with them as I did quite a lot of work on the interaction of light with matter ( basically calculating absorption spectra). Indeed is was part of my PhD thesis as well as the whole my MSc dissertation. Fermi golden rule would have been familiar to me though I would have also thought of it as first order perturbation theory. But these things are no longer familiar to me and your video like seeing a ghost from my past. Quite sad really.
In Fermi's golden rule or any time dependent perturbation theory, you assume that the states of the reference Hamiltonian (H_original in this video) are approximately the true states of the system. However, these states are weakly coupled to each other by some perturbation (H') and Fermi's golden rule describes the transition rate between them due to this coupling. It doesn't describe the rate between the old (states of H_original) and new states (states of H_original + H') as said in the video. This is because the new and old states are linear combinations of each other and the (or final state) is some electromagnetic field mode. These are both states of H_original. The whole point of Fermi's golden rule is to avoid calculating the states of H_original + H', or new states as referred to in the video. In the example given it is impossible to calculate the new states as the number of electromagnetic field modes is practically infinite. One other issue, and maybe this is pedantic, but the video uses the term "strong coupling" but this has specific terminology in quantum mechanics that is different than used here and could potentially lead to confusion. In fact, strong coupling specifically refers to the case when the coupling between states of H_original is large enough such that time dependent perturbation theory fails to describe things in a meaningful way. I get that the video is saying that if the coupling is stronger the transition rate is stronger, which is 100% true, but in Fermi's golden rule, the states should by no means be strongly coupled. I would say this video gives the basic idea of Fermi's golden rule but for people trying to understand it at a mathematical level, for say a class, the issue with initial and final states could lead to significant confusion. I wanted to mention this, especially the first part because it's important not to loose truthfulness even when distilling things down and I think it could lead to confusions.
To do undergrad level QM, you need to be comfortable with differentiation, integration, differential equations, some complex analysis and linear algebra. I hope that helps
I see this topic one year ago in the university and you remembered me the fascinating that it is! Btw in general the Quantum Perturbation theory is very cool topic (for curios the "Zetilli - Quantum Mechanics" book have a very good description of this theory)
It's worth noting that this rule leads to important factors in modeling chemical reactions, in particular the HOMO/LUMO energy states predict the likelihood of reactions, be it excitations or bonding, directly because of this principle.
I’d like to wish you Happy New Year! You have enthusiastic, curious, paradox, deep, original approach to the physics and mathematics. It is very entertaining to follow your reasoning and it is very useful because you stay precise and deep enough in your mathematical “justifications”.
So, the Golden Rule is ... Treat other Hamiltonians as your Hamiltonian would want to be treated?? I'm so confused! (Just Kidding) Great video. Thanks!
Parth, I liked your example with the extra electron whizzing by your atom. Is it possible that the waves are probable due to chaotic and random interaction with other waves? 🌪
Spin is an intrinsic property of particles, essentially the spin being +1/2 or -1/2 makes an electron an electron, along with it's mass and electric charge. Other particles have different spins. For example a photon has a spin of +1 or -1. Spin actually comes from relativity. When you include special relativity to quantum mechanics the intrinsic spin of particles naturally arises. Mathematically this comes from something called the Dirac equation, which is like the Schrodinger equation but includes special relativity.
I had a prof named Mitch Golden, who took pains to explain that, although he was teaching it, and it was his name, he wasn't the actual creator of the Golden Rule. You should point out that it only works asymptotically, for finite times, the Golden rule is violated, according to the time-energy uncertainty. The final transition probability is caused by cancellations in phase between states that don't conserve energy.
It reminds me of entropy in macroscopic systems. If one thermodynamic system can exist in more states than another one with the same energy, then the first one will be found with a higher probability than the latter one. I.e. the more degenerate system, which has the greater entropy, will occur more often than the one with fewer ways of distributing the same amount of energy.
Actually, in the derivation of Fermi's golden rule it assumes the transition is from a single state to a very large number of states and the reason you can describe it as a rate is due to this. The population will essentially become trapped in the large number of states and not transfer back into the initial state. If you look at the coupling between just two states you will see quantum mechanical oscillations between the two states which can't be described by a rate law. So you are totally right, Fermi's golden rule really an expression of entropy increasing. Also, I think you mean the more microstates a system has, not the more degenerate. Degenerate means states of the same energy level, but that doesn't have to be the case for what you are referring to.
I put my question here because it is your most recent videos and I do think that you do not come back to comments of your older videos. The question is quite simple. The “wave function”. For each particle concerned with “its” function does the sum of all probabilities equal 1? I do not find an answer to this question and have nobody competent to ask it. Thank in advance. PS For the rest, I am really grateful for your videos. Each time you manage to cover a subject as one can only wish.
