The Born Rule
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- Опубликовано: 18 мар 2015
- The maths of quantum mechanics isn't actually hard. Plus you get to decide how far down the rabbit hole you want to tumble.
Error: at 3:55, one of the coefficients I wrote is 1/(4*sqrt(2)) + i/(4*sqrt(2)). Jiří Rožnovják pointed out that I did this calculation wrong, because as it stands, if you added the probability of each possible state, you don't get 1! The coefficient should be: 1/(sqrt(4*2)) + i/(sqrt(4*2))
The questions:
1. First a math one. If I have a complex number a+bi, use pythagoros’ rule to find the length of it. So what the probability of a state with this coefficient?
2. Then of course, an interpretations question. Many people have found the use of complex numbers in the wavefunction very significant, saying this shows that the wavefunction can’t actually be real then (literally. Ok, I need to stop.). In other words, if can’t be a real physical object. If so, what kinds of interpretations of the wavefunction would it rule out? But also, what do you think of this conclusion, that something described by complex numbers couldn’t actually represent ‘physical reality’? Are people taking the word imaginary too literally?
Can I just say, I've just stumbled onto this series and it's probably the best one for understanding quantum mechanics that I've found yet. thanks
This was indeed not the easiest video I've watched today, but it was by far the most interesting and enjoyable I've watched today. Thanks for the awesome work!
Watching through the playlist all in a row makes me laugh even harder every time you make a math pun. i loves imaginary jokes!
I'm an undergrad junior majoring in physics and I find your videos very clear and informative for understanding aspects of quantum. You do an amazing job in make these videos enjoyable to learn and comprehensive. I like your own interpretations on some of the implications of quantum as well and I can tell you really enjoy what you do. Thank you in advance and keep up the great work :)
Anthony Sosa It's so cool to hear from someone who's also studying physics :) Thanks a lot of your encouragement! Do you have any advice/comments about how the level I pitch these videos at suits you? I always hoped that they could still be somewhat helpful for someone actually studying quantum mechanics.
Looking Glass Universe Well I think you do a great job in simplifying and conceptualizing some of the quantum features. In my class we focus on the mathematics much more exclusively, and all I would suggest is perhaps introducing some more of basic formula's and equations actually used for quantum. But I believe you do do that in other videos, so continue what you're doing please :)
Such a beautiful and simple (and visual (: ) explanation of complex numbers, very easy to grasp.
I soo love this channel! :)
Yay! Thank you!
Love you're videos, love you're voice, love how you explain everyting so clearly, hope you can keep making this wonderfull videos but no rush do, cheers!!
Oscar_L Thanks so much :)! I'm working on the next one.... I'm really sorry for the wait.
I see the imaginary coordinate as a new "dimensional line" starting from your point going at 90° inside or outside the paper.
So it isn't physical ("real") from the number line perspective, but it remains as a usable dimension in my mind. Is that far-fetched?
Great video, as always, LGU!
I really like the way you put humour, good drawings and physics together. It really makes it visual and simple and very pleasing to listen to. Never would I skip one of your videos, I would surely be missing something.
Your alive/dead cat superposition equation draw is epic. It is simply the best way to make it crystal clear. Thank you.
(Sorry for my english)
Hugs from Belgium!
That's actually how I like to see it too :)
@@LookingGlassUniverse umm where is mathematics tho, I mean the thing here was super easy and you said it was not and now am confused.
Keep making those videos you are awesome. Gives me inspiration to dive into the QM world after the video introductions (that is learning about the mathematics).
Rafi Chai yay :D that's fantastic to hear!
Love your video format stuff. I'm accedently learning stuff.
Awesome video!
1) Using the Born rule, we first find the modulus of the number a + b*i, which is sqrt(a^2 + b^2). Then, we must square the modulus to find the probability, which is a^2 + b^2.
2) In my view, math is a tool that can describe many aspects of the world. Saying that imaginary numbers do not pertain to physical reality, but that real numbers do, seems illogical to me. Both real and imaginary numbers are purely conceptual; it's not like real numbers or imaginary numbers are true physical entities, they just represent parts of the universe when we think they should, like when we are using a ruler to measure lengths or counting the number of apples in a basket. In my mind, 1 is no more "real" than 2*i, because both real and imaginary numbers are necessary to model reality. Complex numbers are used in quantum mechanics (as you mentioned), electric circuits, and fluid flow, to name a few applications. In electric circuits, complex numbers are used in regard to the impedance of components, which ultimately allows one to calculate other real (i.e., non-imaginary) quantities relating to the circuit, like current. Also, at the moment, physics cannot adequately describe the flow of fluids, especially turbulent flow (in fact, a new set of mathematics may be needed to describe it, in the same way that calculus was needed to describe the motions of objects). However, with complex analysis, specifically conformal mapping, one can model fluid flow in a number of situations, which would have been impossible with real numbers alone. In short, I think that people are taking the term "imaginary number" too literally; the word "imaginary" does not mean that imaginary numbers have no bearing on reality. Since they both are just useful concepts and nothing more, I think imaginary numbers are just as "real" as real numbers.
Lovely :) I really enjoyed the examples you brought up to support your case :)!
I love the videos and about to start working on it :) (Good stuff and you're awesome)^2 :)
This is the exactly the type of video that I am always looking for. People just throw a bunch of mathematical expressions and attempt to explain term by term. They don't even summarise the equation and present it in a much more comfortable way
seriously,i find it amazing!!said this a few times! you seem to be an accidental teacher and that looks cool,believe me its fun watching these clips...
Thank you :D!
What a wonderful video. I would love it if you could explain stephen hawkings book a brief history of time. I read about quantum mechanics in it but i couldnt understand it but you helped me understand it. Please do some videos. Im sure a lot of people will love it
I wanna marry your voice. Just your voice.
If you'll swap me yours, this could work out. (Surely anything is better than sounding like a 12 yr old)
Looking Glass Universe
You sound like you're constantly smiling throughout your videos. It's like quantum mechanics *genuinely* makes you happy.
Ahaha! Thanks! QM is super cool :P I also really love teaching.
Looking Glass Universe Do happen to any tensor calculus by any chance? It's really hard to find good materials on that.
Nup :)
Can I say im not studying quantum mechanics or even physics at the least. I just finished highschool and have an interest in physics. I just found out about the born rule from a physics asylum video and thank you you helped me understand it! I already knew about complex numbers so it made it easier but even then you did a great job!
YAYAYAYAYAYAYAYAYYAAYYAYAYA I WAITED 3 WEEKS AND ITS FINALLY HERE LOVE YOU AND THANKS!
Awww!! Thank you!! I'm sorry for making you wait :(! I will try make them faster.
thats alright :) im just glad they arent stopping
Thanks a lot :) I don't intend to stop them
This is a wonderful series. Fantastic. Just the right level of complexity. Please keep it up. Here is a question that has bothered me about the wave function ever since I first saw it (which was about two weeks ago): the coefficients (a, b, c...) are discrete amounts. I think this could be called a cardinal series (forgive me it is 47 years since I last took a maths course and I do not recall the proper terminology). The series might be infinite but it would in principal be mappable onto the natural numbers. Yet the number of possible positions in the event of a wave function collapse would seem to be a continuous function of some kind, in the sense that for any possible positions x and y such that x > y, there is always another possible position z such that x > z >y. This is not mappable onto the natural numbers, so how could the wave function represent the set of all possible positions of a particle? I probably have used the wrong terminology, but I am sure you get my drift. If this does not make sense - no problem. Please do keep up the great work. PS I have the same question about the multiple universes hypothesis (or at least some descriptions of the hypothesis) since the number of universes, even if infinite, must be countable or whatever is the proper mathematical term. How can a countable set be spawned by a continuous set of probabilities?
I'm not 100% that this will answer you- but unlike numbers, space is not continuously divisible. It doesn't divide less than 10^-35 meters. So there is NOT always another possible position z such that x > z > y.
I really like your explanation of complex numbers as arrows that can point anywhere in the plane, not just left or right. That image goes well with the name "complex." And you did not use the word "imaginary" in the initial description -- a great gain, I think. The whole "square root of negative one" approach has never felt right to me. I remember a video of Feynman giving a popular lecture about QED, and he used the term "arrow" again and again.
I really appreciated Feynman's approach too! It really is a much better way to think of it isn't it? I mean, I feel like that's how I think of it usually anyway- not the square root of -1 business!
Good to see the Born rule being explicitly tackled. The heart of quantum mechanics is a probability theory of variables that don't commute. Probability "amplitudes" and their phases are dry for youtube, but a necessary part of any reasonable understanding of QM.
Yes exactly. I decided it needed to be done because I think phases are really crucial to QM theory. Now just to make it not overly dry....
Hello I have been watching your videos for a while and think they are understandable and interesting! I was wondering if you could ever make a video explaining the basic concepts/implications of String Theory, since I find it very hard to even get a basic idea of it. Thanks!
Isaac Toth Hi, thank you! I'm sorry, I probably won't do that for a while since I don't know string theory. Even if I learnt it, I'd only be comfortable teaching it if I was very very confident about what I learnt.
A presentation as nice as it is creative. The (square of the ) length of a complex number represents probability, however, only if that length is at most 1
I like what you did there with the Schrodinger cat bit
Thanks, great video!
The "imaginary" part of a number is poorly named. It's just as real as the "real" part. As in not at all. But both can represent real objects and measurable interactions.
"It's just as real as the "real" part. As in not at all" YES.
I love these comments. This attention towards the imaginary and not the real part might be do to the fact that we are "transforming" the complex number to a real number when taking the modulus, when a better way of looking at it might be that the real part of the complex number and its real modulus are on separate planes.
Don't say that to the Platonists.... lol
1) for the complex number a+bi, the length of the line from the origin to that point is √(a^2+b^2). therefore the probability is a^2+b^2.
2) I've never thought of the wave function as a physical object in and of itself, but rather as a mathematical representation of a physical state. I thought of it as a property the object had, like its spin or its charge. that said, I don't think needing to be defined with complex numbers necessarily means it can't be real as long as the actual output is real. like, the probabilities are real numbers, it's just the math behind them that gets wobbly. a lot of physics calculations involve infinities, but we don't worry about that because in the end they spit out finite results: I think the same principle applies here.
√(a^2 - b^2) because i squared is - 1 ...
Hey homework!
1: the lenght of the arrow is given by the square root of (a^2 +b^2) so the square of it is just (a^2 + b^2). Pretty easy to calculate!
2: Complex numbers are wierd! But this does not mean that they are less "real" in a sense that they CAN be used to represent physical properties. I think people tend to stick to much to names and so bash imaginary numbers as "fake" numbers not understanding how and where they can be used. It's like me as a kid trying to understand fractions and negative numbers. At first I was doubtful that these "numbers" meaned anything but studying math and seeing how they applied to the real world (pizza, and bank accounts) i now understand them better.
For exemple I haven't studied complexes that much but I know they can be used to describe more tangible thinks like the harmonic motion of a slink wich is pretty "real" as far as I'm concerned!
Hi, I really love your videos and your humor :) you have the very rare talent of explaining complex subjects in a simple (and fun!) way!And about the second question, I do believe that people take the word "imaginary" too literally (aren't "real" numbers imaginary too?).Anyhow, I stumbled upon a very interesting way to visualize the wavefunction while looking for an explanation about fractals a while back. Google: "How to fold a Julia fractal", you'll love it!Also try googling: "Quantum waves visualized in 3D" - very cool!
I just wanted to point out that there was a ket of a cat in this video, absolutely marvelous :D
1: Length = (A^2+B^2)^0.5, Probability = A^2+B^2
2: imaginary number is also used in AC current. In the case of AC, all properties are real. Engineers are just using imaginary number as a math tool to describe current and voltage phase mismatch. If we consider probability function as superposition of many waves then we can easily think of (or imagine) two type of fundamentally difference waves that our function is composed of (one call real one call imaginary). Since I do not even know what the 'real' one really represents, it is impossibility to say if the imaginary is real or 'fake'. All I can say is the 'imaginary' function is as real as the 'real' ones. However, both function may or may not have a physical meaning. If they do have meaning, may be you can makes videos about it?
Very interesting topic. Thanks for the videos :)
The problem is not that the wavefunction is defined in terms of the complex number system. That is merely one way to represent the underlying mechanism described by the wavefunction, and as it turns out, the asymmetrical properties of the real number line aren't perfectly matched to describing electromagnetic waves. The bipolar vibrations of EM waves are inherently symmetrical - each polarity is the mirror image of the opposite polarity. Representing the vibrations as alternately positve and negative is not quite right - because negative numbers are not symmetrical mirror images of positive numbers.
In particular, the square and square root operators defined on the real number line do not work the same way on negative numbers as they do on positive numbers. To fashion a number system that can properly represent electromagnetic waves, it's necessary to expand the real number line into the complex number plane. Cyclic vibrations can then be mapped as rotations about the origin, and the complex square root operator can be defined symmetrically, not only for positive and negative real numbers, but for all complex valued vectors in the plane.
In the context of electromagnetic waves, our intuitive notions of "positive" and "negative" are somewhat misleading. Since the vibrations are symmetrical, it would be more consistent to visualize them as alternating between something such as right-handed and left-handed polarities. The "negation" operator is then defined as transforming right-handed into left and vice versa, and the "left-hand square root" would be no more "imaginary" than the right-hand version.
The use of imaginary numbers reminded me of when we learned about impedance in the AC circuits coarse I took a while back. The resistance (from resistors) is the real component and reactance (from inductors and capacitors) is the imaginary component. These component show the impedance effect on the electric and magnetic fields, which when combined, produce the AC wave.
Yup, this is a great example of complex numbers being useful!
If "imaginary" numbers can't be used to describe real world values, such as length, then they are only a means to an end - a way to get from inputs to useful outputs that can be used in the real world.
Maybe complex numbers in wave functions are similar, in the way that we ignore what 'quantum particles' are actually getting up to and instead just know their quantum wave function and so how they are likely to behave when measured.
what is the difference between the coefficent and the probability of an action occuring
Complex numbers are real!! And important! Great job!!!
Are imaginary numbers useful for determining vectors? Or at least the probability of a vector? I ask cuz you seemingly have two pieces of information to determine one state. Well that's the simplest explanation of a what a vector is. I think... thanks for the awesome vids!!!!
Thank you so much for the reward :-)
This is where I agree with Euler: most of the confusion about complex numbers could be avoided by calling "imaginary" numbers "lateral" instead.
1. |a+bi| = (a+bi)(a-bi) = [a^2 + b^2]^(1/2)
2. Complex numbers can be interpreted as vectors with two real components and an additional multiplication operation (complex multiplication), or even as 2 by 2 matrices with real components. I see the wave function as an abstraction rather than as a concrete things. What would it even mean for a wave function to be concrete? Would it mean that quantum objects in superposition are just infinite clouds as opposed to a particle interacting with its superposition?
Hey, I have a QUESTION.... (about video 3, I posted it on the wrong video)
Isn't it weird that in the Double Slit experiment, the probability of the particle passing through each slit is fixed, given by its wavefunction?
I, for example, perform the experiment with a detector on one of the slits, which will result in the collapsing of the wavefunction of any particle passing through it. Given that I know the total number of particles, I could make out the number of particles passing through the other slit, and I have actually collapsed their wavefunctions without disturbing them the least by any measurements. Thanks!
(Sorry if the question is silly. I am a kid yet, and watched this quantum stuff just out of curiosity)
I dont know if i understood the question, but i think the wavefunction of the particle passing in the slit with no detector does not colapse, it doesn't make an intereference pattern because there is no other wave to interefere with.
The method of finding the probability described in this video is only accurate for normalized wave functions. For a non-normalized wave function squaring the coefficient would only give you the relative probability which you can compare to the relative probability of another wave function. To tell if a superposition of wave functions is normalized sum all the squares of the coefficients, if this is equal to one then the wave function is normalized. (this is still a simplified version of the real process, but will suffice for any superposition of normalized wave functions.)
when I was 14 I read Feynman's QED and I think he used arrows instead of complex numbers to try to explain the probability of photon paths too, but didn't realise the arrows represented complex numbers at the time!
Wait wait wait hold up, what do you measure the arrows with. Do you use feet or inches or what. Also when you do the notation when you put d+ei do you actully add it up or is it plain like that. And my final question is : for the examples how do you get those anwsers?
Being taught complex numbers in school: Takes two weeks and still don't understand it
Being taught complex numbers in the middle of a video about quantum mechanics: Takes 5 minutes and I understand it
NB: I'll do all the homework from all videos at once.. soon...
+Char lie Ahaha! I thought that that might be a little ambitious. I'm glad it worked for you though :)!
I failed geometry in high school then I repeated the subject in a 'remedial' class taught by one of the football coaches. Guess what grade I got....
Can someone please explain to me where the square roots of the probabilities in the wave function come from? That's the only part I conpletely don't understand because it seems so random...
The historical sequence was developing the wave function and later realizing that if you squared the coefficients in front of each possible outcome, you would get the probability of each outcome.
Here's the historical sequence: Schrodinger developed his equation to describe actual experimental results in quantum physics. To describe the results accurately, Schrodinger had to put square roots as coefficients before each outcome. Schrodinger's Equation is a partial differential equation. This is a type of calculus equation that is looking for a solution which itself is an equation, not a number like in algebra. The solution to Schrodinger's Equation gives you the actual results of quantum physics experiments. The solution IS what we call the wave function: all the possible outcomes (for example, particle goes thru Slit A or Slit B) with a coefficient in front of each outcome--just like in this video.
Later, Max Born realized that if you square the coefficients in front of each possible outcome, you get the probability of Slit A versus Slit B (the Born Rule). So, it wasn't until after Schrodinger developed the equation that it was realized that his coefficients are actually the square roots of the probabilities of each outcome.
In your example at 3:50 I get 13/16 instead of 1 when I add the probabilities together. What am I doing wrong?
What's the vertical axis, anyway?
Thanks for another awesome video! I love to learn something about the maths of QM.
Question: Do the probabilities always have to add up to 1? If yes, there's probably a mistake in your video.
In 3:55, the terms of the equation are 1/(4*sqrt(2)) + i/(4*sqrt(2)), i/2, and 1/sqrt(2). If you square their lengths and add them up, they add up to 13/16 instead of one.
Ah, crap. Thank you for pointing that out, you're right :'( My excuse is that I majored in pure maths- I really am appallingly bad with actual calculation (as you can see). Damn. I've put a note into the description (crediting you for figuring it out).
I'm so glad my viewers are smarter than me- stops me from embarrassing myself :)
Looking Glass Universe
It's okay :). I am majoring in applied math and CS, so for me it would be an embarrassement if I didn't notice it haha.
:D Ahaha! Thank you!
Complex numbers just could be generalisation of the set of real numbers, that is to say, the real number line is a special case.
"Because we can use imaginary numbers to describe this, it must be nonphysical."
Language strikes again.
I think that's silly. That we use two numbers and call one of them imaginary has no effect on how things actually are. The universe doesn't reform itself around the ways that we talk about it.
Nicely put!
I agree with you, ***** but are we really allowed (or did we need) to use the word "imaginary" if it wasn't a little non-physical?
If you define physical as anything that the universe does, no, imaginary in that sense goes into the definition of physical.
If you define physical as anything you can measure, in that case I think Imaginary is a good word :)
PositiveANegative
"Are we allowed to use the word imaginary if it wasn't a little non-physical?"
Yes.
Let me give an example: We could define a coordinate system over a 2-dimensional space, such as the floor of my bedroom, where each point on my floor is defined by a complex number. The real part of the number would represent position along one side, the imaginary representing position along the other.
I can assure you, both dimensions that my floor spans are equally "real."
I don't understand what you are saying in your second paragraph.
***** We use Imaginary not because we don't think it exists, but because we know we can't measure it, so we have to use our imagination to figure out how it looks like.
It is, at least, how I understand it.
PositiveANegative
Why couldn't you use real numbers on things you know you can't measure? Why can't you use imaginary numbers on things you can measure? Is there something wrong with the example I gave? And do you know that an imaginary number is just a real number multiplied by i?
1a. sqrt ( a^2 + b^2)
1b. (a^2 + b^2)
2a. It wouldn't rule out any interpretations of the wave function. It's possible to use something like two reals, but this makes things more complicated. It's probably possible to use all kinds of exotic math and number systems and come up with virtually the same models.
2b. The use of any given number system in a model is not an indication of how well that model adheres to reality, nor whether or not the idea being described represents a real physical phenomena or purely an abstract entity necessitated by the model.
I'm new to this specific part of quantum mechanics and I don't study it but I think about it a lot and it's puzzling paradoxes. What if instead of a probability wave of all superpositions, what if quantum states exist as a positioning of a 5 dimension structure that moves through reality like a kaleidoscope or spinning cube. Much like a ball pushing through a 2 dimension plane it's only observed when it's interacting with the plane and only 2 of its dimensions are observable for beings on the plane. Maybe there is no wave function each observation is just a still frame of this 5d object. As in, events that happen are 'printouts' of this constantly changing enigma and measuring it is the act of taking the picture.
In that case somebody had too much to drink. ;-)
Complex numbers do pop up in the real world. In electrical engineering for example, if the voltage and current waves of an AC signal are not in sync, there exists a complex power
I'm curious: how do you mean that you don't believe in the real numbers?
The probability, given as length in Pythagorean form of c/\2 = a/\2 + b /\2, where a is the real part and b is the imaginary part of the complex number to address the first part of the question. For the second part of the question for interpretations, let us consider John Napier logging an interpretation of Luca Pacioli's Double-Entry System of Accounting for particle charge, expressed by interpretive dance in an Olivia Newton-John (Max Born's granddaughter) Xanadu remake at the retirement house as if Lord Shiva or King David only it's David Hilbert as a Hilbert space interpretation, you know that David Hilbert with an imaginatively expressed a what the Fock state or number state may be interpreted imaginatively! Great channel the Looking Glass Universe, Cheers! People with schizophrenia demonstrate significant abstraction deficits. Take analogies or metaphors literally typically. The book An Imaginary Tale, by Paul Nahin, explains how the imaginary number is utilized as a tool in many fields from accounting, to electrical engineering and physics. I suppose for interpretation we must address for what application and how to account for abstraction deficits logging interpretations of 'imaginary' aspects of a mathematical formula or equation.
Fantastic
Complex numbers are central to Quantum Numbers. Also exp(ia) =cos(a) + i sin(a).
I do have a question... {it probably is really stupid, but if you could just..}
It's a complex plane... And every "arrow" has an absolute value.. But why do we always get a REAL absolute value... Isn't that a bit biased...
If [ i ] is the [ 1 ] of another "universe" ... And a complex number contains both.. Is there a layman /interpretation/ to why the abs. Value is always real...
Is it because the COMPLEX NUMBER is a 1-D Ray from the origin ?
P.S. I am watching this video an hour after midnight, it can't get crazier than that!!
Plus, I love the little "jokes" that you sneak in, keep it up!
Aditya Khanna Ah, I like your question! The answer, unfortunately, is that we made it that way. The abs represents the lengths of the vectors, and we always think of 'real' things- like lengths- to be represented by real numbers :P
Looking Glass Universe Aw, I would have really liked if it was a figment of the cosmic mathematical framework....
Like it can't be complex "as the universe prohibits it" ( I love using this statement :p)...
Thanks though!
If with all physical properties we can reorient our reference frame and get a negative to become positive on the real number line, when we ask what is the square root of negative one, we could be asking what is the square root of one in a different reference frame, thus imaginary numbers are just representations of opposite reference frames we could consider. Thus physical reality is preserved if we can reorient our reference frame an infinite number of ways. So then we can argue that we really can't reorient our reference frame an infinite number of ways because we can't reach infinity. Maybe we are constantly approaching the observation of physical reality so nothing is real until after it is observed and during or before observation everything is imaginary.
Looking Glass Universe I think you miss(didn't mentioned) something in the video. If you want to square of length of an complex number in complex plane. You should multiply it with its conjugate. Therefore we can say "Probability of a quantum statement is square of its eigenvalue for real numbers. But Probability of a quantum statement is multiplication of its eigenvalue and its conjugate."
By the way, thanks for video, I'm looking forward to seeing your next video ;)
Thanks for the comment :) Yes, that way is equivalent to the way I went with in the video. I thought that it was easier and more intuitive in terms of lengths. But you're right, this is another way to do it :)
So are you saying that the probability when the complex number is at "45°" is zero? lower than that angle is positive probability and higher than that is negative probability? I´m just deducting this from Pythagoras rule to find the length but now I´m incompletely confused.
TheRafaelacosta You're using that (-i)^2=-1 right? So you concluded the length is sqrt(a^2-b^2)? It's actually sqrt(a^2+b^2) for exactly the reason you pointed out- otherwise it can be negative. But the length of the arrow can't be negative!
why complex numbers in wave function?
Hey, I just found your channel (Veritasium mentioned it in his AMA here: www.reddit.com/r/IAmA/comments/38gekk/iama_guy_who_makes_science_videos_on_youtube/cruvgq9 ). I love the videos that I've seen so far, and I'll be watching more when I can! I'm looking forward to any future videos you make as well! :)
Honestly, I think that people really do take the labeled names of types of numbers too seriously. We named them a certain word simply because we didn't understand it at the time/still don't understand it. People freak out and and make assumptions based on a name. Last time I checked, we are the ones learning and discovering about these anomalies as we study on; the numbers do not fit into our rules. Our rules fit around our beautiful, mysterious number-friends.
I AM A BEGINNER IN QUANTUM MECHANICS AND I HAVE COMPLETED WATCHING ALL YOUR VIDEOS. I AM VERY MUCH INTERESTED IN QUANTUM MECHANICS AND I HAVE HALF COMPLETED REAFING THE BOOK "THE THEORETICAL MININMUM - QUANTUM MECHANICS "BY LEONARD SUSSKIND. I HAVE BECOME YOUR FAN.
I HAVE MANY CONFUSING AND IMPORTANT DOUBTS.
CAN YOU PLEASE FIND SOME TIME TO HELP ME?
Saying that a wavefunction can't be real just because it has complex numbers on it sounds like you can never use a rational procedure which takes irrational numbers into account. I'm not a mathematician or a physicist so maybe I can't appreciate the nuances of the thing, but don't we use that kind of thing all the time? When you have three oranges and two apples does that mean that you can never mix them, or rather that you need a more general category (fruit) to express both numbers as a single thing? Or, if we draw a square with side one, does that mean its diagonal will never be used in association with a rational thing because.. you know, it isn't rational. If nature finds complex numbers cooler than it seems to me like we have been counting oranges and apples all along when in fact we should be looking at fruits.
1: length = sqrt[a^2 + b^2], probability = a^2 + b^2
2: Wavefunctions aren't real in the sense that they are not a wave travelling through any physical medium. But then that would mean that electric fields are not real in the sense that they do not exist outside of the effects that we observe resulting from them (which then means that the electric fields ARE real physical things because they have effects). So, wavefunctions might be visualized as existing inside of an abstract probability space and since we see their effects (the probabilities of detecting an object ehre or there or etc.) then they too are real physical things. Anyone that argues that complex numbers cannot describe reality needs to stop and think about the nature of the "real" numbers that they might assert do describe reality.
Imaginary numbers have been a great concept that demonstrates impossible states. Anyone who has studied classical physics will recognize this.
I might be wrong, but I think the results from the wave function are all real. These complex numbers might just be a helpful structure for intermediate calculations, just like finding all zeros of a cubic function.
where are the answers to the hw??
That we get real values from complex numbers computations, I think, hints at the fact that in the physical world we see manifestations of some deeper structure. Not necessarily corresponding to our complex numbers math, but the complex plane is a better approximation of the generating function, whatever it may be. I love the puzzle that our universe is!
It's not a deeper structure. Quantum mechanics is a realization of Kolmogorov's axioms. We can realize them with real numbers, in which case we get probability theory (which is clearly a relevant description of physical phenomena) but we can also realize them with complex numbers (and then we get quantum mechanics) and with quaternions (in which case we get a slightly modified version of quantum mechanics that has not been analyzed in depth, as far as I know). My guess would be that one can also form all kinds of algebraic products of these structures. The density matrix, for instance, is such a product between ordinary probability distributions and quantum mechanical wave functions.
@@lepidoptera9337You did not understand the question.
@@matveyshishov I do understand the question but it's pointless. The structure of quantum mechanics is formed by two things: the first one is the assumption of statistical independence of individual quantum detections, which leads us to an ensemble theory. That's Kolmogorov. The second one is relativity, which requires the product structure of Hilbert and Fock space (otherwise entanglement would produce non-locality and FTL effects, which are not being observed). There is no "deep" structure here. It all follows trivially from a few observations that have been cast into a fairly trivial ensemble approximation. You don't even need that. You can just start with Poincare symmetry and start evaluating its representations. That automatically leads to quantum field theory.
@@lepidoptera9337 I don't think it's wise on your side to insist that you understand the question when the person who stated the question told you that you didn't.
No amount of textbook definitions will change that, even if you quote the full list of mathematicians ever lived.
This "IAmVerySmart exercise of theorem name soup cooking will simply stay here, online, for everyone to see, forever".
@@matveyshishov I understand the topic because I am an experimental high energy physicist. Measuring quanta was my daily bread for years. It's a nearly trivial thing to do and it illuminates the theory beautifully if you have to do it hands on. The reason why theorists are talking so much nonsense about this is because they don't know what the experimental layer looks like.
What you are doing here is basically telling the man who has been building single family homes for years that nobody understands how to build single family homes. That's just total bullshit, kid. ;-)
I think coefficient of wave function must be there for something else also that u haven't told us.....
Like there are lot of cases when even though the magnitude of complex number( consider the circle with centre at the origin) is same but they are different. So this difference might have an effect in wave function of that particle..
Please help me where I am wrong...
Next I have a huge doubt :
Real numbers for real axis ,
Complex numbers for both real(x axis) and imaginary (y axis) ..
What kind of numbers should we take for the 3rd axis (z axis)..
Moreover why can't we use vectors instead of complex numbers which would eliminate 3 rd axis problems...
Please recommend some books on quantum mechanics for beginners which are easy to understand (I am just finishing my high school)
You're absolutely (get the pun?) right! There's no point in using complex numbers so far... but all will be revealed shortly!
U didn't answer why we cannot use vectors instead of complex numbers and books for reference on quantum mechanics pleaseee reply....
ashwin gopal yes what you told is exactly correct !!
I think it is entirely real. The use of complex numbers in this case seems to just be a way of representing a 2 dimensional vector (I use it a lot like that anyway), but that does not make the vector it is describing not real. I feel like math is just a tool that humans wield to track the world around them. The complex numbers are just used to describe something that we cannot describe with ordinary numbers alone, so it does not make sense to me that someone would call the wave function imaginary because of it.
BTW: Thanks for these videos! I find them very useful in learning Quantum Mechanics!
Since we use numbers as a tool to describe the world, something that is mathematically abstarct does not make it physically abstract. We created the mathemtics of complex numbers before we had these applications for them, so we are just fitting the mathematics with the patterns we have observed in the universe. I personally do not like that imaginary numbers are named imaginary because that implies that they are less real then real numbers, which is false. They were just created after we had real numbers, and they are extremely useful in many scenarios outside of quantum mechanics.
1. has been answered often enough, I think, but
2. Not sure that it would actually rule out anything, but assuming that the wavefunction isn't "real" (whatever that means) might make a few interpretations more complex by having to introduce laws which state that the world behaves exactly _as if_ there's an underlying wavefunction that's "real".
I don't think that claim has any justification, though. Just because complex numbers are not ubiquitous in everyday life doesn't mean that they are any more or less "real" than real numbers.
Which makes me wonder, if, instead of having a "real" part and a "imaginary" part, the standard way of talking about complex numbers was by using an angle and a length, would people still claim that?
Often, "real" in the context of the nature of the wavefunction boils down to the claim that the wavefunction is a direct description of the external physical world, as opposed to an epistemic statement about an observer's knowledge of what they might observe of the external physical world.
This is a paper on quantum physics and human language that you might find interesting: iopscience.iop.org/1751-8121/40/12/S13 arxiv: arxiv.org/pdf/quant-ph/0610131v3.pdf
"Which makes me wonder, if, instead of having a "real" part and a "imaginary" part, the standard way of talking about complex numbers was by using an angle and a length, would people still claim that?"
Do you mean something like argand diagramns? Or bloch spheres? You could represent Schrodinger's cat using a bloch sphere, for example.
Morberticus
Thanks. I'll look at those papers.
By angle and length I just mean the argument and modulus of a complex number. You can, of course, visualize those in an Argand diagram. (But you can also visualize real and imaginary part in an Argand diagram.)
1. Well, sqrt(a^2+b^2) for length, a^2+b^2 for probability.
2. You can absolutely describe real things with imaginary numbers. A sine wave that moves over time, for example. To let the wave move, we'd have to calculate its velocity at every point, its acceleration, and all sorts of horrible things... Or we could use vectors that point in an imaginary direction to make it work. acko.net/blog/how-to-fold-a-julia-fractal/ explains it better than I ever could (see "travelling without moving")
Awesome :) That's right, complex numbers can be very useful for calculations.
The probability is the sum of squares of a and b that is a² + b². Listen +Looking Glass Universe, I wanna discuss a question. Imaginary numbers are no less real than Real numbers, they are just two dimensional and since the functions of complex numbers are 4 dimensional. What if the wave function may be 4th dimensional. If we can find the fourth dimensions other than x, y or z. I know that it is not possible to think about the 4th dimension but anyway, it can be possible that the wave function may be able to tell what a particle is doing, if the 4th dimension is somehow founded. Then the quantum mechanics will be complete. What do you think about that!!!!!?
Might help some people to say that that all state vectors lie on the unit n-sphere where n-1 is the dimensionality of the Hilbert space?
If you want you could do a video on quaternions and SU(2)
Stat thermo is all about probabilitites and does not describe the microstates, but it is just as real. Copenhagen yields probabilities for measurements and that's that. But of course metaphysically (in the philosophical sense, not the colloquial one) it is unsatisfactory.
I actually have a question - I can easily think of a basis where the basis vectors are linearly independent but not mutually orthogonal, but I cannot think of a physical example of some operator whose the eigenvectors are not orthogonal but linearly independent and the measurement that would yield this result. Any ideas?
A postulate of QM is that operators representing physical observables are hermitian. Say u and v are eigenvectors of some hermitian operator O, with eigenvalues u' and v', then
= u' = v'
which implies
(u'-v') = 0
which implies
must be 0, since (u'-v') is not zero
So (leaving aside degeneracy) the eigenvalues of hermitian operators are always orthogonal. Or are you asking why observables must be represented using Hermitian operators?
The main reason Hermitian operatrs are used is to guarantee that all their eigenvalues are real - however this is not a particularly strong requirement. What I am wondering is that we can think of a (either real or complex) vector space with linear operators and and with basis vectors that are not orthogonal.
Jason93609 Morberticus The proof up above shows that actually the hermitian property is very strong and does imply that the eigenvectors must be orthogonal if their eigenvalues are distinct. (The key in the proof there was that = u'*=u'* because the eigenvalues are real). But this doesn't imply all eigenvectors are are orthogonal. Consider the case when |u> and |v> have eigenvalue u. Then any combination of those two vectors is an eigenvector since: O(a|u>+b|v>)=u(a|u>+b|v>). So actually, if an eigenvalue has degree n, then there is a n-dimensional subspace of eigenvectors. However, the standard thing to do is to find an orthogonal basis (of n vectors) for this subspace for each eigenvalue. Then all these vectors are othogonal (by consruction if they have the same eigenvalue, and by the above proof if they have different ones). Now we have an orthogonal basis :D
Jason93609 By the way, that's for asking this awesome question! I was wondering the same thing myself not so long ago and happen to do a lot of thinking about it then... which is why I happen to know that. Because textbooks usually stop at the distinct eigenvalue case, which is frustrating..
Looking Glass Universe Morberticus That's all very well, indeed if we want all e'values to be real then we work with an orthonormal eigenbasis and if there are degeneracies then we can pick the eigenvectors to be orthogonal. But if we entertain the possibility of complex eigenvalues we don't need to restrict ourselves to Hermitian operators. I think it is interesting to speculate about that possibility. I.e. pick an non-orthogonal basis by construction. It will probably lead to notational complications, but I think it is still interesting.
Sorry this is from the last time (QM3). I'm still learning about your self:
So . . . I think the measurement at one gate means the condition pervades at all gate not just the one being measured? I think this cause the experiment IS a measurement. Also randomness means a coin flipped by a person or persons even if they construct a device . . . it's still human behavior. So a coin flip only shows human behavior is random? Interference fails to be illuminating cause it approximates as true . . . if a tree falls and no one is around to hear it does it make a sound (the true = yes). As an answer this misses the point of the 'question'. It's not a question at all!
U HAVE A VERY SENSUAL VOICE..
At first I was like "Arrows? But you're drawing vectors!" and then like "oh, so it's beginner-level..."
So, homework, SPOILERZ EVERYUONE!!!
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1)sqrt(a^2+b^2) is the length, a^2+b^2 is the probability
2)Can't answer the question itself (I didn't get it), but i have a comment on it: as complex numbers are real (that makes sense, trust me), I don't feel like anything might be just imagination. That's probably philosophical question more or less... If anything happens in our mind it is real, at least to ourselfs)
PS: Loving your complex-puns ;о)
One thing I'm always wondering is why we use the term "imaginary" if what the number ends up being is a vector. 4+3i can just as easily be described as (4,3), and have pretty much the same characteristics, unless of course I'm missing something
Yes, this is how I feel exactly! For a lot of the purposes we use imaginary numbers we could have just as well used a 2D vector space. Quantum mechanics is a great example of this.
On the other hand, complex numbers have one property that vectors don't: they can be multiplied, by virtue of the equation (i)^2=-1. This is the only 'imaginary' part to me. But in quantum for example, you don't have to use that property at all and so what you're working with really is just vectors.
Looking Glass Universe
If you represent two complex numbers as an angle and a length, you can multiply them by adding their angles and multiplying their lengths, no squaring of i required. In particular, this allows you to look at complex numbers without using axes and the labels "real" and "imaginary".
However, I do not think that QM would work with vectors instead of complex numbers. The first explanation of amplitudes in QM I read (lesswrong.com/lw/pd/configurations_and_amplitude/) uses the fact that i^2 = -1 heavily. Also, if you look at the equations for polarization for example, i crops up everywhere, especially in things like e^ix. As x varies, of course, you will go through 1, i, -1 and -i. This seems rather crucial for the whole thing to work.
(Of course, you can model this with little 2D arrows, if you "multiply" them like stated above, but it's just not something inherent to vectors.)
I you represent a complex number as a 2D arrow, I find that this makes them less weird:
multiplying by 1 doesn't change the direction. (i^0)
multipyling by i rotates the arrow a quarter turn to the left (i^1)
multiplying by -1 rotates the arrow half a turn to the left (i^2)
multiplying by -i rotates the arrow three quarters of a turn to the left (i^3)
NNOTM The fact that i^2 = -1 is used heavily because finding lengths is usually represented as taking dot products. Here it is easier, calculation wise, to take dot products by using i^2 = -1. But, as you pointed out yourself, that isn't necessary. The only other thing that 'requires' i^2 = -1 is changing the basis that the wavefunction is written in. This really is much much nicer, tidier, and easier when done with complex numbers... but still possible to be done without.
Yes, as you point out, you could just 'make' a multiplication rule for vectors (not necessary, but then you can use all the regular maths of QM). However note that, from an algebra point of view, that would make these vectors isomorphic to complex numbers. So in a sense it's the same... but you're right, it seems less weird :)
This is a really interesting discussion. If you're making vectors behave as complex numbers aren't you just putting an isomorphic mask over them, or is there a way that this can be derived without looking at the behaviour of complex numbers? In other words, are vectors a subset of complex numbers or just similar, and one is easier to work with, or are they both a derivative of some math above them?
Unfortunately, the word vector has two meanings. Under addition, complex numbers are a subclass of vectors- where here I mean 'vectors' in the most general axiomatic sense (in this sense, even functions can be considered a subclass of vectors). But then there are the little arrows that we usually think of as vectors (which are another subclass of general 'vectors'). They don't have a natural multiplication rule unlike complex numbers. In 2D, we can make one analogous to complex numbers and that's actually pretty natural, making them completely isomorphic when thinking about addition and multiplication, but in high dimensions it doesn't work. So really, they are distinct subclasses of vectors. In QM though, you only 'really' use addition and length/dot product properties of complex numbers though- and these properties are exactly the same for vectors. That's why I don't think that the imaginariness of i^2=-1 is important to QM. What do you think?
Ques 1 root over a squared minus bsquared
I watched some vids about this in urs is the easiest to understand
Gauss thought that "Imaginary" numbers should have been called "Perpendicular" numbers, if I am not much mistaken in my history.
How is it you know so much about quantum mechanics?
bashed my head on it for many years... but I still don't know a lot!
1)a) sqrt(a^2 + b^2 i)
b) a^2 ^ b^2 i
Answer to the second part. I built a structure with my absolute value of psi squared.
Not gonna post the HW but it's interesting that you point out the complex nature of the wave function. It would mean that the wave function is merely an abstract entity as it cannot represent physical measurable values because I am assuming one cannot measure a variable and get a complex number, I can't imagine (pun intended) it happening, but abs(wave function)^2 is a probability density. That's kinda weird; the object can't represent physical variables but the absolute value squared of the object can. I don't really know what that means but it works. That's not to say it's not interesting, perhaps it's a mathematics-ontology-physics problem.
Thank you, I think you're the first person to point out the crucial argument in the 'the wavefunction isn't real' view: "one cannot measure a variable and get a complex number".
BlueLightningSky There are things you can measure and which are expressed as complex numbers e.g. electrical impedance of a circuit.
The imaginary numbers are just an example of an algebra for pairs of numbers, with a number of desirable properties. The fact that they have the geometric interpretation of negative areas doesn't imply that they do not represent anything; after all we also have negative numbers, but no negative lengths can be observed. Similarly, there are algebras for quadruplets, octuplets, etc.
You can describe classical statistics using some of the formalisms of QM - with some important differences; in practice (although not necessarily in theory) all operators will commute and when you calculate conditional probabilities you sum the mod. squared of the state vectors, not the complex amplitudes. It would not be the simplest way of doing statistics but it is possible. All reals can be mapped onto moduli of complex numbers.
Let me play devil's advocate here: In electrical impendance, only the real part has physical meaning. It is convient to intro the complex part, it makes the maths easier and is simply a nice way to represent the phase of the (real) current. At the end of the calculation, you throw away the imaginary part only measure the real part.
Only because you make one measurement at a time. It is a consistent representation and for that reason it cannot be deemed unphysical.
Jason93609 I must admit I do not know much about electrical impedance. But let me be clear with what I was saying; you cannot make a measurement and have it yield a complex number. You can make measurements and express it as a complex number but you can never empirically obtain a complex number while performing a measurement. That is why I do not think you can associate complex variables to something physical. I believe in QM, the wave function is not what is called "observable" but rather the operators.
Also I find something wrong with this statement: "All reals can be mapped onto moduli of complex numbers." because all reals would include the negative numbers.
I'll answer the second question first, so that I don't spoil the answer to the first one for those who want to work it out for themselves.
As long as the probability ends up being a real number between 0 and 1, I see no reason why you can't have a complex coefficient in a wave-function. Is there anything that indicates otherwise? And what would make the coefficient be (1/4 + i/4) instead of simply 1/√2?
Now for the calculation of the first question:
_a²_ + _b²_ = a² + _b²_
As you noted in your comment and above, there aren't and can't be negative probabilities since the length or any complex number is positive.
To answer your question though, yup, wavefunctions can be purely imaginary! A bit weird right?
"When squaring a complex number, you usually (to be precise: in all cases where you have both a real part and an imaginary part) get a term that still includes i as a factor."
But she asked for the probability, not the square of a complex number. The probability, as she described it, was the square of the length of the arrow. IOW your equation for the square of a complex number is accurate, but it's not the probability.
"Is it possible for a wavefunction to only have an imaginary part?"
Yes, but it's really something more akin to a coordinate system. Just because you called it "i" in your system doesn't mean my coordinates for it will be imaginary as well.
Looking Glass Universe But what determines whether the coefficient is real or not? What is the difference between these coefficients: (1/4 + i/4), (1/4 - i/4), (i/4 - 1/4), (-i/4 - 1/4), (1/√2), (-1/√2), (i/√2), (-1/√2)
They all have the exact same probability, so what's the difference between them?
Forrest Fyre Yes, I figured that out eventually. Glad I got to learn something new today. =)
Nillie That's an awesome question, and basically what I want to answer next time. But I'll try explain it briefly here:
Imagine I have two states: 1/root(2)[ |up> + |down>] or 1/root(2)[ |up> - |down>], where up or down is the spin in the x direction. If you measure the spin in x, both states act the same. However, if you measure them in the z direction, they will act completely different from each other. One is always up in the z direction, the other always down.
Why did Matt Damon not play in The Born Rule?
In the end, you, LGU, thank people for viewing, especially because "it wasn't the easiest video". First of all, the thanks is on _our_ part. I don't see that you are getting compensated with piles of cash or anything. And secondly, the hardness of this vid (which it wasn't, really) is irrelevant. If the subject matter requires a certain level of hardness, then that level is what it is supposed to have.
A lot of people learning QM don't care how hard it will be. QM behavior brings to light the fundamental questions of the universe. Naturally, exploring it merits whatever effort needed.
Nehmo Sergheyev Yay :) Thank you, that's really cool to hear! My aim really is to give the tools to anyone who wants to put in the work to learn :)
2. A silly conclusion. The algebra of quantum mechanics is isomorphic to the algebra of complex numbers, but we could just as well use a certain type of 2x2 matrix as another contributor has said. It is merely a matter of experience that when we make measurements, we don't ever need to access individual components of the wave function.
1.
length: sqrt(a^2+b^2)
probability: sqrt(a^2+b^2)^2 = a^2+b^2
Can engineering students do PhD in quantum mechanics?
Nothing stops you from switching to the physics department by getting the necessary course credits. Will you be happy? I doubt it. It's most likely nothing like you imagine it is. ;-)
whats actually a wavefunction
Imaginary, imagery, image of a real event built up from infinite reflection in QM-TIMING.
Actually, it was demonstrated recently that the wavefunction is a real object...
I am not sure how it affects each interpretation, tho...
www.newscientist.com/article/dn26893-wave-function-gets-real-in-quantum-experiment.html
I'll have to read the paper. Published in Nature Physics... sounds good!
Yes, people are taking the word "imaginary" too literally, sadly.
Complex numbers are used in statistics If I recall correctly. And there is the Gamma function. So, complex numbers actually represent reality more accurately. We mostly use real numbers for convenience. Law of least effort, right?
Yeah, that's right. Complex numbers can be very useful!