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Probability Amplitudes : A curious case!

Well, I was trying to read QM from my favorite book- Feynman Lectures but there was one thing either he missed or I missed. The latter one is more probable. Actually, I wanted to know why we need probability amplitudes. What's the connection of them with nature(or mathematics) that make them easier to find than probability? I am asking this because I think they must have been in use because they are much easier to find than the all more important probability.

I really couldn't guess its use. Is it much more close to the "harmonic motion"(which I know it isn't, but still "amplitude" is used, there must be some significance)?

My guess: It might be that there is interference of waves and during that, the significant thing is the amplitude and the probability becomes proportional to its absolute square. But then again I am stuck.

So, in a nutshell, I want to know how to develop a quantum-mechanical thinking. Quantum physicists can witness the results but a reader - how can he predict what's happening or what'll happen? I think there is no way other than believing the facts and the math.

Note by Kartik Sharma
2 years, 2 months ago

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Let's start off with a long-running controversy in 19th century physics--is light a particle or a wave? Issac Newton in his book Principia Mathematica described it as "corpuscles" or particles, and yet in the 19th century, with James Maxwell's unification of electromagnetism, light is seen to be a wave phenomenon. In 1905, Einstein showed the quantum behavior of light, re-igniting the debate.

Meanwhile, Werner Heisenberg developed matrix mechanics, an early form of quantum theory, in an effort to explain or predict spectral lines hydrogen, which he felt to be connected with probabilities of "jumps" between "electron orbits". But he realized that ordinary probability amplitudes failed to describe them. He ended up using matrices in place of ordinary probability amplitudes, which worked--and then he was not sure if he should publish his own findings. Concurrently, Erwin Schrodinger independently developed a wave equation which essentially were based on equations in classical mechanics, but was converted into a wave equation by the inclusion of the imaginary number--thus being able to use principles in classical mechanics to analyze and predict wave behavior. Eventually, both Heisenberg's and Schrodinger's different approaches was proven to be mathematically equivalent. About this time, the Born rule was developed, initially as a practical "interpretation" of Heisenberg's work analyzing spectral lines and Schrodinger's wave equation. In a nutshell, the State Function as described by Schrodinger's wave equation is a complex probability amplitude, and the probability of a particular event happening is the absolute value of it. This has no counterpart in classical mechanics or regular probability theory, which doesn't deal with complex amplitudes.

It is as if instead of playing with real cards, we are playing with imaginary cards, and such cards, when played, involves probabilities that have imaginary basis. What's more, when you are holding a hand of cards, you are actually holding all possible hands--until the time you're asked to show your cards, and suddenly you do not have all possible hands. Complex probability amplitudes helps predicts what will be put down on the table.

It's best to think of this as Alice entering through a mirror into a world that is nothing like what we know about.

If you are unable to make much sense of complex probability amplitudes, you're in excellent company. A great many top physicists have tried to do that for the past century. As far as I know, nobody's ever been able to offer a "simple intuitive model" of how it works. It just does. It works extremely well, and it serves to constantly remind us that the quantum world is really a strange place.

Edit: The best mathematical analogy to complex probability waves and absolute values of it is how it's done in electrical engineering, where AC sinusoidal waveforms are often described as a complex amplitude, in phasor notation. The analogy is appealing, but don't take it too far in trying to apply this to quantum physics.

Edit 2: You can start with something simpler, the quantum "qubit", which, like the bit used in computers, has two states. But the behavior of the quantum qubit is more like what happens when polarized light passes through a polarizing filter--and you're forced to look at this as single photons passing through, instead of a Maxwellian EM wave. Then use of complex probabilities would be the appropriate thing to use. Conventional probabilities will not work. Of course, "quantum computers" are based on qubits, not bits, and billions are already being spent on developing such quantum computers, because of their promise.


If you can grasp the principles of qubits, it will make it easier for you to grasp probability amplitudes in general in quantum mechanics.

Edit 3: As a matter of fact, I remember now that I once made up 3 cards with polarizing filters, and I would show to people how two of them stacked together would completely block light and you couldn't see through them, and yet if I inserted the third one between the first two, you could see through all of them. This is an example of why ordinary probabilities won't work in describing this, if you assume photons are particles. Michael Mendrin · 2 years, 2 months ago

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@Michael Mendrin It was nice reading to all the "Origin of Quantum Mechanics". So, it is actually just the results we have got while experimentation that led to quantum mechanics and further research which has expanded by relying on mathematics, right?

Well, what I think(and know) is that there were experiments conducted by physicists(Heisenberg etc.) mainly based on the Slit Experiment. And then the probabilities were calculated for an electron(or "anything else") to come from a slit. The probabilities for the 2 slits showed interference when we tried to calculate the total probability giving us a hint that there must be some "complex variable" functioning the probability.

But my question would be why is it easy to find the complex probability amplitude and not the probability. How can we actually find the probability or probability amplitude? What I see is just the Boltzmann's Law. However it comes out that amplitude is equal to (in "most simple cases") \(a{e}^{-2\iota\pi \left(E/h \right )t}\)which is of course just \(a{e}^{-\iota\omega t}\) but why? Because light is a wave and a particle or what?

Another question: How do quantum physicists think? Kartik Sharma · 2 years, 2 months ago

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@Kartik Sharma Actually (and this is the reason for giving "origin of QM", etc), the true origin of complex probabilities is in Heisenberg's obscure "matrix mechanics" which he devised in order to make predictions about hydrogen atom spectral lines. I also gave an example of polarized filters to try to illustrate why ordinary probabilities wouldn't work. Let me warn you, the question of "what do complex probabilities mean in nature, and why do we need them" is one of the hardest challenges still facing physicists today. In fact, an entire discipline, called, "Quantum Interpretations", has arisen in an effort to try to make any intuitive sense out of this. How do quantum physicists think? They don't understand why, they simply get used to it. Work with the mathematics of quantum mechanics long enough, one simply gets used to thinking that way. Get used to going through the mirror into the strange world.

Quantum Interpretations

But most QM professionals don't even bother to pretend to know why it works.

Let's try something as simple as polarized light passing through polarized filters. This has a complete explanation using Maxwell's EM equations--as long as you believe that light is an electromagnetic wave. But it works even when a single photon is emitted at a time. A single photon is not a EM wave, Maxwell's equations cannot explain or describe single photons. What to do? Well, how about if Maxwell's EM waves are "really" mere "probability waves", an abstract thing, not real EM waves, with the provision that the probability of that one photon being somewhere is exactly \(1\). Then the expected value of finding that photon is found by "filtering the state function with an observable", an example of which would be passing photons through a polarizing filter. In order for this scheme to work out, i.e., pretending that light behaves like a wave when you want it to, and like a particle when you want it to, one has to resort to complex probabilities. This is the price for the wave-particle duality of matter.

Try this as an exercise. Given a source of light that emits one photon at a time. I have 3 polarizing filters, 1) vertical 2) 45 degree angle 3) horizontal. If I stack 1) and 3) together, the photons are blocked 100%. But if I stack 1), 2), and 3) together, the photons are not 100% blocked. Try explaining this using regular probabilities. Work on this for a while, and see if you can avoid resorting to complex probabilities. Whatever you come up has to work for a variety of other ways polarizing filters can be employed. Of course, the mathematical apparatus of quantum physics handles this one just beautifully, which is why physicists keep using it. Michael Mendrin · 2 years, 2 months ago

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@Kartik Sharma I would try to read about Qubit - they seem interesting! Thanks for introducing them here. Hope that will enhance my QM thinking! Kartik Sharma · 2 years, 2 months ago

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@Michael Mendrin Wave-Particle Duality Animation

Also, that's *Erwin Schrodinger ;) John Muradeli · 2 years, 2 months ago

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@John Muradeli It's Edwin Schrodinger in that parallel universe. But, all right, for this universe, I've corrected it. Michael Mendrin · 2 years, 2 months ago

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@Michael Mendrin @John Muradeli and others. Kartik Sharma · 2 years, 2 months ago

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@Kartik Sharma @Steven Zheng I'd love to listen your views too. Kartik Sharma · 2 years, 2 months ago

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@Kartik Sharma We are your top two huh?? Very pleased. However, "others" don't exactly get notifications if you don't directly mention them, so you probably should ^.^

Well I'm very sad to inform you but I'm of no clue on the topic. I only know some "popular media " quantum mechanics concepts, if you will, but I'm not seriously into it. But no worries, the Brilliant brain force will be here in no time.

Good luck! John Muradeli · 2 years, 2 months ago

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@John Muradeli And I always give a thousand-word-commentary when a hundred would do. But, between us, we have to carry the water. It would have been nice if anyone else could have jumped in, since that's an interesting topic. "Why do we need to have complex probabilities?" Michael Mendrin · 2 years, 2 months ago

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@John Muradeli Well, yeah I should have tagged "others" as well. But now you can see that the community has grown and grown a lot and one didn't usually get the names when they are required. So, it would be my appeal to the Brilliant community to ease group tagging - either there should be a group of physicists(:P) or a group of "people you follow".

Well, that was just trivial dreams because it has many other consequences. Anyways, "Popular media" quantum mechanics sounds interesting but how can you know "advanced" results of QM happening around the world while you say you're of no clue on this topic? I didn't get it as when I try to understand those popular media QM talks, they are just like a particle which went through an "invariant"(in which all the spins are "open"). Kartik Sharma · 2 years, 2 months ago

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@Kartik Sharma I guess it is an understatement when I say I have "no clue," but I've never had any formal education on the topic and know zero mathematics or fundamental principles behind anything I'm saying. For example, I know an electron can "jump" from one side of the mountain to the other. I also know that the region around the atom is not empty, but filled with probability clouds. Now, do I know why or how any of those things work? NO IDEA!!! I'm just a really good expert on conceptual ball physics. (Though, one could argue, all physics is the study of the ball ;)) John Muradeli · 2 years, 2 months ago

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