06. Development of Schrodinger's equation

Slides and transcripts: drive.google.com/drive/folder...
ERRATA: at 1:28:24, rho(r,t) should read P_x(t)
0:00 Recap
5:25 Introduction
9:23 Minimization principles of Fermat and Hamilton
14:28 Action is phase
22:47 Deriving Schrodinger's equation
25:29 The hydrogen atom
44:06 Wave function as charge density
45:55 Multipole expansion
53:04 Schrodinger equation as an eigenvalue problem
58:50 Schrodinger equation and matrix mechanics
1:07:32 Dispersion
1:15:03 Time-independent perturbation theory
1:20:38 Perturbation theory and degeneracy
1:23:47 Oscillating perturbation
1:29:13 Successes and failures of Schrodinger's wave mechanics
1:31:42 Born's interpretation of the wave function (quantum scattering)
1:36:37 Philosophical considerations
1:44:00 Determinism, free will, morality, faith-based beliefs
2:00:10 A brief comment on interpreting quantum mechanics

Пікірлер: 14

  • @pavelrozsypal8956
    @pavelrozsypal89565 ай бұрын

    Systematic, beautiful and historically correct exposition ! Such a difference from many other "trivialized" explanations appearing even in respected textbooks crudely distorting actual historical development of physical ideas of great physical science founders. All your videos are deep, illuminating and very interesting. Great work !

  • @jackdaniel8763
    @jackdaniel876311 ай бұрын

    Than you, for your impeccable work. I wish I could study under your supervision.

  • @worldhaseverything
    @worldhaseverything11 ай бұрын

    Thanks for everything.💟❤ And why this channel is so underrated?

  • @declup
    @declup11 ай бұрын

    Outstanding. Thank you.

  • @paoloberra5141
    @paoloberra51414 ай бұрын

    Great video!! Really complete. I've seen you have uploaded the pdf slides only for Heisenberg matrixes. Could you also upload the slides of the other videos? It could be useful to study them better. Anyway thanks for your effort! Regards Paolo Berra

  • @josephmellor7641
    @josephmellor764111 ай бұрын

    This was a great video and you've been knocking them out of the park in this series. With that being said, I have a few notes on the Philosophy section. First, even if the laws of Physics are not deterministic, it wouldn't imply free will. Non-determinism is necessary, but not sufficient for free will. Second, even if the choices of what the electrons do are unpredictable, the statistical average of those choices is not. If you take a trillion trillion coins and flip them long enough, eventually you'll get around 50% heads and 50% tails. You couldn't really have free will hiding in the randomness on the scale of the brain. Third, the soul (or whatever object makes the choice) will have to at some point cause a Physical system to change and we should be able to observe that change. To go off the example you gave, a rock falling on someone should not be judged because it does not have a soul and didn't choose to fall on someone. But how do we know that it didn't choose to fall on someone? The answer I hear all the time is that "It was just following the Laws of Physics," but so do the neurons in the brain. If there were a soul, we'd be able to see a difference in the paths atoms and electrons in the cell take or reaction rates that we wouldn't see in a similar pile of organic matter or in the brain of an ant. www.preposterousuniverse.com/blog/2011/05/23/physics-and-the-immortality-of-the-soul/ Fourth, the soul would still have to follow the laws of Physics, including conservation of energy and momentum, which means that if the soul moved anything in the brain around, it would also have to move something else in the brain around so there would be no net movement or our heads would look like they were getting hit by an invisible force all the time. Having to move things around in such a way that there's no net momentum seems like quite a difficult problem. Fifth, the soul would then have to do trillions of trillions of complex quantum calculations every second to know that sending X electrons to this neuron would make someone offer a friend five bucks. If the soul could do that, why would it run everything through a human body? Sixth, the soul would have to be affected by physical processes like drug use or head injuries that leads to a personality change. en.m.wikipedia.org/wiki/Phineas_Gage Seventh, you don't need the kind of free will you're thinking of to hold people accountable. For anything bad happening in society, you find the people whose behavior would need to change and you change it. For example, if someone goes out and kills someone else for fun, you find the killer and put them on trial. On the other hand, if person A threatens to kill person B's family unless person B kills someone else, then person A is the main person to go after because changing person A's behavior will reduce the likelihood of bad things happening in society. kzread.info/dash/bejne/q3aW3M2LaNeTZcY.htmlsi=H8XjV85BIWnaEJZZ Eighth, I believe your reason for why we should believe in non-determinism falls into the category of arguments in the video below. The consequences of not being able to pass judgment on people who have done bad things is itself a good enough reason to pass judgment on people who have done bad things. I can tell that's what's going on because you named actions that have specific consequences in the real world that most people would consider harmful, but not things like not going to a religious service weekly or not following the five pillars of Islam or committing blasphemy. kzread.info/dash/bejne/k6aH1cuThMyulZs.htmlsi=zmgLD_Q6GOYCyiSN Ninth, moral relativism does not mean everyone becomes the most evil person. There are individuals with different moral codes. These individuals agree on a lot of core ideas because they grew up in societies that promoted moral ideas that prevented the society from falling apart. A society in which everyone thinks you can walk around and kill anyone you want for any reason wouldn't last that long, so there's been a lot of natural selection for societies that promote the bare minimum moral values needed to survive. All this is to say that we would end up making something on the same level as an objective morality regardless of whether one existed. If you don't have kids, then any genetic factor that made you not have kids (whether that be something like infertility or emotional desires) will not be spread while those with the genetic desire to reproduce will spread their desire. This idea can also work on a cultural level, where cultures that encourage having kids will have more kids than cultures that discourage having more kids, which could lead to the survival of one culture over another. Someone may want to kill other people, but other people probably won't want to be killed or even encourage killing people who want to be killed, so the killers are prevented from spreading their ideas or genes. Tenth, I dislike the argument that scientists have reasonable justification to tentatively accept a belief, therefore faith in the supernatural is completely justified and on the same level. As you pointed out in the video, the *early* scientists believing in determinism wasn't because they were like "Man, we hate the idea of free will and the soul," but because it had worked so well for the past several centuries. You would need a massive amount of evidence to throw out determinism since science doesn't destroy past theories, but absorbs them. It wasn't until the Bell tests that local hidden variables had been ruled out and even then there are still plenty of interpretations of Quantum Mechanics that are consistent with reality. The honest answer is that we don't know, not "I'm going to say that we technically don't know when pressed, but I'll actually pick the one that agrees with my worldview." Lastly, to make any subjective belief an objective belief, religious people just call it an objective belief. That's it. There's no reason to believ

  • @SanderKonijnenberg

    @SanderKonijnenberg

    11 ай бұрын

    Thank you very much for sharing this opinion. I anticipated that that part of video would be the most open for debate: I tried to keep it brief (since it shouldn't be the main subject of the video), and I haven't studied philosophy nearly to the same degree as I've studied physics. So by all means, feel free to challenge it. Having said that, I would like to respond to the points you made: I fully agree with your first point, and I don't think I ever claimed otherwise (feel free to let me know if I'm mistaken): I claimed that 'determinism' implies 'no free will', and therefore 'free will' implies 'non-determinism'. I didn't intend to claim that 'non-determinism' implies 'free will'. Regarding point 2: From the law of large numbers it indeed follows that predictability can arise from random events. At the same time, there are still random fluctuations among those predictable trends (there's the statistical average, but there's also the variance), and we know from chaos theory that small variations can result in a significant change in outcome. So it's not obvious to me that the law of large numbers prohibits a phenomenon such as free will somehow emerging from elementary non-deterministic processes, in the same way that something as complex as a human being somehow emerges from interactions between elementary particles. I think this is what Freeman Dyson meant when he said "Mind is already inherent in every electron, and the processes of human consciousness differ only in degree but not in kind from the processes of choice between quantum states which we call "chance" when they are made by electrons." A rather bold hypothesis perhaps, and at this point it should probably not be considered anything more than that, but I think it would be unfair to dismiss Freeman Dyson as a dilettante. Points 3 to 5: I don't think I, or anyone else, can come up with a sensible detailed description of how a 'soul' would precisely interact with matter. It is indeed a very valid and challenging question. However, I think it's an equally valid and challenging question how a belief in determinism is compatible with our experience of having free will. I don't think one question should be swept more easily under the rug than the other. Point 6: Also here I cannot intelligently comment on how a 'soul' would affect or be affected by matter. I would like to remark though, that physical processes can also affect our sense for logic, but I think many would agree that this does not imply that logic is untrue or non-existent. Therefore, I'm not sure what conclusions we ought to derive from the observation that our personality and sense of morality can be affected by physical processes. Point 7: I'm not sure I follow this point. As soon as you're talking about 'changing' something, it seems to me you assume non-determinism: in a deterministic world, one sequence of events cannot be 'changed' to another. There would be just one predetermined sequence of events that takes place. Moreover, somebody has to decide that something is 'bad'; that the likelihood of 'bad' things ought to be reduced; that there is a person they 'ought to go after'; that they should 'find and change' people. The way you describe the entire scenario seems to me profoundly infused with the assumption of free will and morality. Point 8: Also here I think you assume free will (and thus non-determinism) without acknowledging it, when you say there's 'a good enough reason to pass judgment'. If everything is deterministic, our actions (such as the passing of judgment) cannot be a choice we make based on any sort of reasoning. You also assume some sort morality when you speak of 'bad things' and 'good reasons'. Point 9: Indeed, moral relativism does not mean everyone becomes evil, for the simple reason that 'evil' is undefined (or non-existent) in such a worldview. As you continue to describe, actions are not good or evil, they merely result in the propagation of your genetic material or they don't. You say we end up making 'something on the same level as an objective morality', but 'something on the same level' is not actually an objective morality. It would be an illusory belief that somehow benefits evolutionary fitness, which makes it a useful (but not objectively true) religion at best. And what then do we make of religious beliefs that are not objectively true? May we take them seriously? May we enforce them on others? Or should we actively try to suppress them? Point 10: I'm certainly not trying to claim that any and all beliefs (supernatural or not) are equally valid. In Born's quote, he did call belief in the incredible objectionable, and I agree with that, even though 'incredible' can mean lots of things. I also agree that ignorance should not be confused with evidence for your most favorite worldview. However, I disagree with the sentiment that believers in determinism are somehow more objective or more intellectually honest than believers in non-determinism. I also disagree with the idea that determinism always 'worked better' than indeterminism. Determinism 'worked better' only in a very restricted sense, namely in the natural sciences, engineering, etc. I don't think it's obvious that determinism 'worked better' in the humanities, and I don't think it's obvious that, as humans, we should take the humanities less seriously than the natural sciences. When believers in determinism/materialism confidently claim that free will wouldn't contradict their worldview, or that no belief in free will (or anything immaterial) is necessary to make sense of their beliefs and actions, I can't help but sense the same cognitive dissonance as when creationists claim that all the paleontological evidence is fully consistent with their worldview. Believers in materialism and determinism may roll their eyes when a religious apologist once again points out that in their worldview there would be nothing morally wrong with murder/rape/etc. But honestly, it seems to me like a simple, straightforward, and correct logical conclusion, which has already been explicitly articulated by Nietzsche. Historical events that followed (scientists promoting eugenics, emergence of secular authoritarian regimes, support of pedophilia by French intellectuals in the 1970s) demonstrate that the issue of morality is not trivially solved by rationality, applied to the worldview of determinism and materialism. The mental acrobatics that I've heard in response to this observation (e.g. 'they were not truly rational', 'secular authoritarianism is actually a religion') seem to me no less impressive than those performed by religious fundamentalists. Moreover, I agree that science doesn't destroy past (successful) theories, but absorbs them. It's exactly what quantum mechanics and its probabilistic interpretation did. It didn't 'destroy' past theories, in the same way that relativity didn't 'destroy' the old physics. In both cases, the old physics can be recovered from the new physics. Newtonian physics wasn't 'destroyed' by the conclusion that time is relative: Newtonian physics is an approximate case of the more general relativistic physics. Neither was deterministic physics 'destroyed' by Born's probabilistic interpretation of the wave function. As you indicated in your second point, some degree of determinism arises thanks to the law of large numbers. If, as you propose, we're going to admit ignorance, and not pick our favorite worldview, then we also should give no preference to a deterministic worldview.

  • @davidhand9721

    @davidhand9721

    3 ай бұрын

    I almost like your ideas circling the issue of the soul. One of the things that made me a strict materialist was studying the brain and seeing that there are drugs that change one's perceptions without changing the contents of their senses. The soul must relate to the brain somehow; it must receive information from the brain (senses) and it must transmit information to the brain (choices), so there is a flow from physical to spiritual and back again. If we can physically manipulate the brain to affect some function, then that function _must_ be performed either before or after the spiritual component has its turn in the physical brain. We are quite certain about the first and last steps, i.e. sensory and motor, being brain functions, but if higher-level functions are affected by drugs that do not change the relevant sensory functions, the role of the soul is narrowed to exclude those higher functions. Prozac doesn't make you see happy things, and it doesn't make you act happy when you do not feel happy, so your mood is determined right there in your brain. If we go through a good list of drugs, the soul has nothing left to do! Even if it exists, it's irrelevant, though I lean towards it not existing. The contradictions imposed by free will and QM then both disappear when we accept that we are physical. My brain is what decides what I do, and I am my brain, therefore I decide. Determinism has no conflict once I accept that I am a subsystem, not a magical being whose existence is independent of the rest of the universe. Simultaneously, when I accept that _my_ state is part of the wavefunction like everything else physical, determinism is finally restored to physics; there exists a me for every supposedly random outcome, but I am causally disconnected from all but one of those outcomes.

  • @sarkknuckle6029
    @sarkknuckle60299 ай бұрын

    you have said, action is phase, can you please provide some source to this. I have searched but have not found anything saying about the physical significance of the wave function's phase. Or, if it is related to action

  • @SanderKonijnenberg

    @SanderKonijnenberg

    9 ай бұрын

    You can go through Schrodinger's original papers to see how he made the connection between classical action and the quantum mechanical wavefunction. The relation between action and phase is also articulated explicitly in Feynman's path integral formulation en.wikipedia.org/wiki/Path_integral_formulation#Feynman's_interpretation

  • @sarkknuckle6029

    @sarkknuckle6029

    9 ай бұрын

    @@SanderKonijnenberg Thanks for the response. Your work in these videos, the way you have organized the topic, is so much appreciated. Hope it reaches as many people as possible.

  • @davidhand9721
    @davidhand97213 ай бұрын

    Sure, but compatibleism. My judgment here has nothing to do with morality, either. The brain is a physical object which evolves according to all relevant natural laws, so it is deterministic if physics is deterministic. But you _are_ your brain, and your brain is you; this is an observational fact. Your brain determines what you do, and you are your brain, therefore you determine what you do. End of story. That being said, my preference for determinism in physics is based instead on information and the principles of science. If the evolution of the universe dU/dt is not wholly a function of U then it must have additional parameters beyond U. There is literally not enough information in U(t) to calculate U(t+dt). But U is the state of the universe, which can only be defined as everything that exists, so if there's information outside of U, then we clearly aren't really talking about U. Furthermore, the basis of scientific reasoning is that we can infer functions of U by meticulously controlling and observing U itself, i.e. discover natural laws that apply at all t. If U does not include all of its own parameters, then it's no longer reasonable to use the scientific method to draw any general conclusions. The results of our experiments are contaminated with extra information that can never be interrogated out of U. We used the scientific method to discover quantum mechanics, so if this particular non-deterministic aspect of QM is true, then there is no reason to believe that QM is true. I choose to believe that truth can be known and science is valuable. Finally, determinism is restored by the Everett interpretation. You can have a deterministic view of QM so long as you acknowledge that you are a part of the wavefunction yourself, not some kind of magical god-like observer on the outside, beyond the influence of physics. The difficulty there is in fact the same mess that arises in free will when you assume dualism. Yes, of course the laws of physics apply to you, but that doesn't mean you are powerless, and it doesn't mean you have to twist your brain in knots with non-determinism.

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