[High School Level] - An introduction to spin 1/2 particles. I discuss states, bra/ket notation, measurements, probabilities, and quantum measurement collapse.
This is a life saver! I have a quantum mechanics exam at 9am tomorrow and this cleared up all those gaps in my knowledge that my lecturer wasn't able to fill. Thank you so much!
It's hilarious to me that this is high school level for some people... I have a BS and this is right at my level of understanding 😁 Thank you for explaining it this way! I learn really well from nice, thorough explanations like you have done here. I'd like to make a resource like this one day that incorporates math, classical and quantum mechanics, and modern chemistry into one self-contained resource, so people can teach themselves with no more prior understanding than an (average!!) high school education. I can't wait to watch the rest of the series--I've always wanted to understand spin better. Thank you!
I’ve looked at many references on this subject. This is by far the best explanation which I was able to follow step by step. Really appreciate the time you put into preparing this video.
Thank you so much for the video! I've been struggling in my quantum mechanics class because the professor never breaks down the things that should be simple like you did in this video! Cleared a lot up for me thanks so much
That sounds like 75% of my college engineering/physics teacher. They always assume that the calculations are more important than concepts, and they always assume that you know more than you actually do.
Thanks I have seen this whole playlist. let me explain : I failed my quantum physics study and I abandonned education because I thought I would never be able to grasp concepts like spin. Thanks to you two decades after my failure, I got to my attic to take back my books and for the first time I understood and even manage to get the easy to medium exercices done. Really, you gave me my self esteem back. God Bless you. 😉 Greetings from France and sorry for my probably broken English
You made me see through the different notation and now it suddenly is understandable. Before that I had the feeling I was never going to get it. Thank you so much ! ❤️
Is this correct?: To describe the result of measuring the spin of an electron in any arbitrary direction \vec{n}, we need to specify the spin state in some specified direction \vec{m} (specifying \vec{m} involves 2 numbers). The spin state along \vec{m} involves 2 numbers (the two angles on the Bloch sphere). So we get a total of 4 numbers to be specified compared to the three for the angular momentum of a top.
I’m missing something basic. In the 3-dimensional room where you do an experiment, the choice of what axis is the x axis is arbitrary. So how can the numbers that make up the matrices for measuring something on the “x, y and z axes” be different from each other?
You could ask the same question of spatial vector without reference to spin. If the x,y,z directions are all a matter of convention, how can we write i=(1,0,0), j=(0,1,0), k=(0,0,1)? The unit vectors i,j,k are a matter of convention, yet in components they are all different. The answer is that once you DO pick a coordinate system, you need for i,j,k to all be different vectors, because what is important is their relationship to each other. The same exact thing is true for quantum spin.
It seems that using (1 0) (0 1) vectors to represent up down in z axis is a merely a convention (35:00), is this the case? Or is there another reason for this choice?
@@matthewtoews6523 conventionally, the z axis is drawn going up and down, so it makes things easy to visualize when up is (1 0) and down is (0 1). Quantum chemistry illustrates atomic and molecular orbitals from this perspective
Professor, I hope you could read this comment to the end. Very helpful! You are a very good teaching professor. How old are you? I think your videos are short, simple, at the right speed for me to visualize the concept without seeing the video two or more times to understand, and there straight to the point. Thanks a lot for your teaching style. I have a question and suggestion for one of your future videos: how has been the evolution of the maths for these quantum mechanics and particle standard model theories? Are the spin mathematical expressions in continuous adjustment to keep up with experimental data or are there principles of physical laws and symestries such conservation guiding physicists and mathematicians in developing new theories to be proved with experimentation? How these math expressions came out in the brilliant minds of theoretical physicists? I am 56 years old and I have always love physics.I have a degree in engineering but I am trying to re-learn the rules of linear algebra, matrices, vectorial algebra, and Einstein's summation. Visualizing abstract spaces used in physics theories is a challenge. I am sure you will be able to explain it in a different and easier way than others to people like me.
Super! Thanks! I’m always disappointed when mathematical models include complex numbers because I feel like it obscures any possible intuitive feel for what the underlying reality might be like. In this case, I feel that it obscures one’s ability to think about hidden variables, and why they are impossible as explanations of spin state.
This is nice coverage - I think you're really supplying it at a good level. I will be honest about one thing, though - I find all the "I know this is a lot, but..." and particularly the "grown ups do it like this" stuff wildly annoying. And there is *absolutely* a reason some of these things are written as column vectors and others row vectors - it's done because that makes the standard matrix multiplication process do the right thing for you. Vectors and dual vectors represent *different things*, and they need to be notated differently to keep that straight. It's the same idea as using subscripts for covariant tensor indices and superscripts for contravariant indices - there are two different types of things in play and you have to make that clear in your notation. But, I really mean this comment as praise - it's hard to find this stuff presented *well*, technically, on RU-vid, and I think you're accomplishing that, so... nice job, man.
Stating how they’re different is not the same as EXPLAINING how they’re different. And is it really the case that mathematical notation hooves are driven by their coolness, like cgi cityscapes in sci-fi movies?
why does the inner product use conjugates instead of just products? I know the dot product also uses conjugates but that it just doesn't matter, but why?
Why spin up and down (z components ) are anti parallel? Up and down are orthogonal states.which means their inner product is zero and they are linearly independent.z + and z - are not linear independent, they are linearly dependent.Please help me to clear this concept.By answering how they are different or proper of approaching it.
The inner product in quantum mechanics does not correspond to spatial overlap-- it is rather used to measure the probabilities of measurements. The fact that |+,z> and |-,z> are orthogonal simply means that if the state is spin down there is a 0% chance to measure it spin up. It doesn't mean that the z direction in space is orthogonal to the -z direction in space. Check out my video on the Bloch sphere for more details.
Thanks for your great video! I have a question regarding the last example (min 51+). Your Qubit State is 3/5 |*spin up regarding z axis"> + i4/5 |*spin down regarding z axis"> If the qubit state would be 3/5 |*spin up regarding y axis"> + i4/5 |*spin down regarding y axis"> how would this change the following 6 calculations? Can u make a example? I think i cannot just ignore the orthonormal basis change from z to y, right?
You can translate the state from a linear combination of the up and down spins in the y axis ("y basis") to a linear combination of spins in the z axis ("z basis"): |spin up y> = 1/sqrt(2) |spin up z> + i/sqrt(2) |spin down z>, |spin down y> = 1/sqrt(2) |spin up z> - i/sqrt(2) |spin down z>. Now you plug it in your qubit state: |psi> = 3/5 (1/sqrt(2) |spin up z> + i/sqrt(2) |spin down z>) + 4i/5 (1/sqrt(2) |spin up z> - i/sqrt(2) |spin down z>) Combining like terms we get psi represented in the "z basis:" |psi> = (3+4i)/(5sqrt(2)) |spin up z> + (3i+4)/(5sqrt(2)) |spin down z> Now we know the bra vectors corresponding to measurements of spin in the x,y,z direction written in the "z basis," so we can do the calculations: ||^2 = |(3+4i)/(5sqrt(2))|^2 = 1/2 ||^2 = |(3i+4)/(5sqrt(2))| = 1/2 ||^2 = |(3+4i)/(5sqrt(2)) 1/sqrt(2) + (3i+4)/(5sqrt(2)) 1/sqrt(2)|^2 = 49/50 ||^2 = |(3+4i)/(5sqrt(2)) 1/sqrt(2) - (3i+4)/(5sqrt(2)) 1/sqrt(2)|^2 = 1/50 We already know the calculations for the y axis, because we were given |psi> in the "y basis." We just have to square modulus the coeficients and we get ||^2 = |3/5|^2 = 9/25 ||^2 = |4i/5|^2 = 16/25.
I’m missing a few things. What is “spin”? What’s spinning? I assume it’s an electron, but what’s to differentiate an electron spinning right side up from upside down? Also, the subject of dual vectors was introduced, but what are they? Also, where did the spin matrices come from?
I know this is an old comment, but spin is a challenging concept. In order for something to "spin", it has to have more than one particle. Like imagine a spinning top. The reason we know it's spinning is because the particles on the outer edge are rotating around the center. But there has to be some particles on the outside in order for the concept of spin to really make sense. For a single particle, it can't really "spin" because there's nothing to spin around any center. There is no "center" or "outside" of a particle. The reason why it's called spin is because it acts like spin. It has angular momentum, and that angular momentum can be measured.
@DnB and Psy Production Hey, man... There is a difference between probability of different axis. And I can pick corner in other room. Your version don't work.
It is great the way you deliver concepts in a proper sequence. Thanks for doing this. One question though: the probability you calculate @ ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-r0nDhrAwskU.html along X axis when squaring complex numbers doesn't seems to be quite right. Is there other part of the theory that I missed?
Ah yes. It is not the square of the complex number but the square of the absolute value. That is, |a + bi|^2 = a^2 + b^2. This ensures that all probabilities are positive real numbers. I should have emphasized this more in the video.
z = a + bi, where z denotes a complex number. If b = 0 then all real numbers can be considered complex since "a" is a real number. Thus 17 = 17 + 0*i = 17, The real numbers are a subset of the complex numbers.
You didn't miss anything. It pops out of nowhere because I didn't explain it, and that's because it goes far beyond the scope of this video. The answer, as I alluded to, lies in a branch of mathematics called "representation theory." I'll say a bit about it below. I can't fully explain representation theory it in a youtube comment, but I'll just say a few words so that if you run across the subject later you'll have heard some of the vocabulary already. SO(3) is the group of rotations in three dimensions. A "representation of SO(3)" is a map from SO(3) to a set of matrices such that the multiplication law still works. Because quantum mechanical states are vectors, and we want them to transform appropriately under the action of spatial rotations, we should ask the question, "What are all of the projective representations of SO(3)?" ("Projective" means that the multiplication law can fail by an overall phase. Because an overall phase doesn't matter in quantum mechanics, this is okay.) You can then go and classify what all projective representations of SO(3) are. (This is usually done by studying all of the representations of the "Lie algebra" of SO(3), and then arguing that all of the representations of the group must match all of the representations of the Lie algebra.) It turns out that there is exactly one projective representation of SO(3) for each positive integer n, which maps SO(3) into n x n matrices which act on the vector space C^n. Instead of labeling the representations by the integer n, we instead, by tradition, label them by the half-integer s, where n = 2s + 1. Don't worry if you understand none of that. Just tuck the words in the back of your head for later.
@@noahexplainsphysics Wow~ I sometimes saw or heard the terminologies such as SO(3) in the literature or presentations but didn't understand its meaning. Now I wish to know the details about these and more. You inspired me~ Thank you! You are a very nice teacher👍
Why vectors are denoted as column vectors? There's conventional reason which comes from the linear algebra. And this is really becomes important in complex vector spaces.
I’m taking a university QM class now, and studying spin 1/2. I have about 5 books that cover the topic. I have watched my own professor’s lectures and Susskind and Binney online. After watching this video, for the first time I think I could solve a lot of practice problems. I do have one question. Once you have designated one axis as the z axis (I assume arbitrarily), how is it determined which axis is the x axis versus the axis?
Interesting math. Seems to be derived from classical spin and angular momentum concepts while incorporating probability theory. The probability theory incorporation attempts to replace actual quantities which cannot be measured with probabilistic measured guesses. Getting there? This approach is used because we know some elements in the situation but not enough to be.fully deterministic, so we make educated guesses like throwing dice which can be thrown a million ways but still get only one of six results, The question is how and why did quantum spin arise as a critical element in particle physics analysis? What does it tell us and what and how does it contribute? Is the parallelism just useful math or is there a more profound principle of physics in action?
That interpretation doesn't work. There is no universal "z-axis" by which all spins are up and down. The whole description of quantum spin is completely rotationally invariant, it's just that you need to define some sort of coordinate system as a convention in order to write down the states explicitly. In my video on the Bloch sphere I explain how to think about spins pointing in arbitrary directions.
@@noahexplainsphysics Ah i see. I was personally thinking about the directionality dependence with respect to the time dimension. I shall check out your suggestion soon. Thanks!
