06. Development of Schrodinger's equation

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

[pavelrozsypal]

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 !

[jackdaniel]

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

[worldhaseverything]

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

[josephmellor]

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.


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.


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.


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.


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

[paoloberra]

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

[gcewing]

I don't think that non-determinism is necessary for free will. In fact, I would argue quite the opposite. When you exercise your free will not to murder someone, you're not just acting randomly -- you're (hopefully) thinking about what the consequences would be and acting accordingly. That kind of logical reasoning requires determinism at some level. To the extent that reasoning is being done in your brain, it's happening despite any underlying randomness, not because of it.

We do have the feeling that we are able to act randomly when we want to, but I think that's just a subjective feeling, and it tells us nothing about whether physics is fundamentally random or not. We have that feeling because we don't have full access to the state of our own brain, and are therefore unable to fully predict what we will think in the future. Just as we wouldn't be able to predict the weather more than a few days ahead, even if all the underlying physics were completely deterministic.

So it seems to me that the issue of free will and the issue of whether physics is fundamentally deterministic have nothing to do with each other, and attempting to connect them won't lead anywhere useful.

[sarkknuckle]

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

[davidhand]

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.