This is amazing, the amount of time people forget that they are teaching to people who DON'T know it yet it staggering, sometimes all you need is someone to tell you what each term is and it starts to come together.
@@ProfessorMdoesScience I have a question. How did you get (1/ sqr root 1 (1,1)) for the eigenvector? I understand v_2= v_1. But after that I'm lost how you found the vector.
This channel is amazing! I would like to study quantum mechanics with your videos as a main resource. Should I just watch videos in the order of old videos to new videos? or is there any sequences or order you would recommend?
Glad you like it! The best way to watch the videos is by following the different playlists. And amongst those, a good way to watch them is as follows: 1. The basic principles of quantum mechanics are described in the "postulates" playlist: ua-cam.com/play/PL8W2boV7eVfmMcKF-ljTvAJQ2z-vILSxb.html 1.5. A good complement to the postulates is to explore operators in more detail: ua-cam.com/play/PL8W2boV7eVfnb10T_COKPozxEYzEKDwns.html 2. A first example of using the postulates in (mostly) 1D is the quantum harmonic oscillator: ua-cam.com/play/PL8W2boV7eVfmdWs3CsaGfoITHURXvHOGm.html 3. You can then move to 3D with angular momentum: ua-cam.com/play/PL8W2boV7eVfmm5SZRjbhOKNziRXy6yIvI.html 4. A good example of 3D systems are central potentials: ua-cam.com/play/PL8W2boV7eVfkqnDmcAJTKwCQTsFQk1Air.html 5. And then you can complete your "first" course with the hydrogen atom: ua-cam.com/play/PL8W2boV7eVfnJbLf-p3-_7d51tskA0-Sa.html The above provides a list of the "basics" to get started. You can always complement those videos with extra topics such as uncertainty principles (ua-cam.com/play/PL8W2boV7eVfl3HQzTBZyGfrpfrv7YY3zN.html) or time evolution (ua-cam.com/play/PL8W2boV7eVflUqUY3dLhQdYuZjlbXi0mU.html). We are currently working on a series of videos on spin 1/2, which will also be a nice complement to the above. After the basics, we have some playlists with more advanced topics, including: 1. Density operators: ua-cam.com/play/PL8W2boV7eVflL73N8668N0EQUnID1XaEU.html 2. Identical particles: ua-cam.com/play/PL8W2boV7eVfnJ6X1ifa_JuOZ-Nq1BjaWf.html 3. Second quantization: ua-cam.com/play/PL8W2boV7eVfnSqy1fs3CCNALSvnDDd-tb.html We plan to extend each of these topics in multiple directions (e.g. quantum statistical mechanics, quantum condensed matter, etc) but not quite there yet. We are also working on a website to go with the channel where all of this will be summarised. I hope this helps!
Your videos clear up so many misconceptions i get from class! keep up the great work! could you possible do a video or set of videos for spin 1/2 particles?
Glad you find our videos useful, may we ask where you study? And yes, we are working on a series on spin 1/2 particles at the moment which we'll hopefully publish soon!
@@ProfessorMdoesScience I am about to finish 2nd year at Temple University in Philadelphia, Pennsylvania. I study chemistry and physics there. I finished up my introduction to quantum mechanics course and your videos have been very helpful through the semester! Can’t wait to see your new videos!
I got used to X,Y,Z but simply writing them as 1,2 ,and 3 is better and more general. Plus it makes it easier to remember they need the hbar/2 to become Sx, Sy, Sz.
I quite like the 1,2,3 notation as it is more natural for writing compact expressions such as the ones we use for commutation and anticommutation relations :)
Consider a quantum system comprising of electron spins. We know that the Pauli matrices X and Z do not commute => X.Z ≠ Z.X This means that we cannot simultaneously measure Spin along X and Z directions of an electron. However if we consider the operators (Z × Z) and (X × X) which are formed by tensor products of Z's and X's; then these operators do commute; i.e. (Z × Z)(X × X) = (X × X)(Z × Z). If we consider a system of two electrons, then - (Z × Z) is equivalent of measuring Z observable (i.e. spin along Z direction) on two electrons, and (X × X) is similarly equivalent of measuring X observable (i.e. spin along X direction) on two electrons. If this interpretation is correct, then logically it seems counterintuitive that if we take a system of two electrons, then we can simultaneously measure spins along Z and X directions for both!
This is the normalization constant. For example, if we look at the eigenvectors of sigma_1, we fidn that v_2+ must be equal to v_1+. This means that the eigenvector is such that the two entries are the same, and they could have any value. But if we insist that the eigenvector is normalized, then their value must be 1/sqrt(2). I hope this helps!
Hey! Don't know if I'm wrong, but I'm pretty sure the eigenvalue = +1 for sigma 2, the corresponding eigenvector should be (1 neg i). I double checked my math and it seems like the signs are flipped (as in, the v neg vector matches what I have for lambda = 1 etc) unless I'm wrong!
Remember that there is some freedom in how one chooses eigenvectors. For example, if v is an eigenvector a matrix A, then -v is also an eigenvector with the same eigenvalue. I hope this helps!
Sorry for the delay, normal work has taken over recently... But we are working on the Pauli matrices series (which will include spin 1/2) and others, hopefully soon!
Question is, are these (matrix)entries fixed, or do they change under some circumstances, as long it preserves some algebraic properties. For example, in moved or accelerated reference frame. Something like: Light is red, because source moves away Light is red, because electrons in moved atoms make other transitions, ... Is there some research going on?
We'll explore a related question in the upcoming videos on spin, where we find that the spin operators are proportional to the Pauli matrices, and then you can think of these operators as written in the basis of the third spin component (proportional to the third Pauli matrix). You could then imagine changing basis, and changing the matrix representation of the corresponding spin operators. I hope this helps!
Heisenberg was indeed one of the creators of the matrix mechanics formulation of quantum mechanics, and you can learn more about it in our series on the postulates of quantum mechanics. The Pauli matrices are a special instance of matrix mechanics which are very useful to study 2-state quantum systems. I hope this helps!
I would recommend you give this a go yourself to check. But broadly speaking, the commutator of sigma_1 and sigma_3 will be a matrix that has no "i" overall. However, the matrix sigma_2 does have an "i", so when we want to write the commutator of sigma_1 and sigma_3 in terms of sigma_2, we need to add an extra i. You can also view it in the opposite way: 2i * sigma_2 will give you an overall matrix without an "i" (as the "i" infront will multiply the "i" in the sigma_2 matrix), and this is then equal to the commutator of sigma_1 and sigma_3 which indeed has no "i". I hope this helps!
This is an choice you can make, both are valid eigenvectors. In general, if v is an eigenvector of a matrix A for a given eigenvalue, then so is -v. I hope this helps!
a) The eigenvectors of each Pauli Matrices is different but all Pauli Matrices share the same eigenvalues.Does this imply a degenerate case ? (even if experiments shows the same eigenvalues in all directions) b) As Pauli matrices represent the components of the total spin momentum vector, is this a form of projection Operator? c) Is there any mathematical theorem to show that it is acceptable to formulate a 2 dimensional spin state in a 3 dimensional complex space state x,y,z . My intuition says that the eigenvectors should be the same in x,y and z. If one reformulates the eigenvector in x direction (1,1) as (1,0) + (0.1) and claims that the basis are the same as in the z direction, does this imply the x basis and z basis are not independent basis ? d) I am missing the purpose of Pauli matrices in Physics here other than analogy to angular momentum to explain fine structure spectrum of atoms. Even in measurement, only the eigenvalues are obtained but we have no clue on the actual eigenvectors. Any help ? cheers
a) We typically talk about degenerate eigenvalues of a given operator. Here, each Pauli matrix is a different operator. We are preparing a full series on spin 1/2, and these should help clarify questions (b), (c), and (d). We are hoping to publish these soon!
They can be used to describe any two-state quantum system, which could be anything from the spin of particles like the electron to the qubits of quantum computers. More videos coming up soon looking at some of these examples!
Remember that "dagger" means that you transpose the matrix, so for sigma_2 you exchange the -i and the i off-diagonal terms... and then you calculate the complex conjugate, so -i turns into i and i turns into -i. This gives you sigma_2 back. I hope this helps!
We are working on a series on spin at the moment, and we have plans for a few subsequent series after that. But we do hope to get to QFT at some point! :)
Out of curiosity I wanted to know since we know that ocean water contains salt and we know salt water conducts electricity fairly well Then what if we insert high voltage electrodes in th ocean would it conduct electricity and if yes then till how far can it conduct As it seems absurd to assume that we can conduct electricity through an ocean across continent but can find a scientific reason why not
This is amazing, the amount of time people forget that they are teaching to people who DON'T know it yet it staggering, sometimes all you need is someone to tell you what each term is and it starts to come together.
This channel is truly a gem..thank you for making these videos.
Glad you enjoy it!
Never understood Pauli matrices with this much clarity before. Fantastic video. A true Gem. ❤❤❤❤
Glad you found our video useful!
I wish I I've found this channel in previous semester to understand quantum mechanics correctly 😢and with passion and interest.
Great to see you've found us now ;)
@@ProfessorMdoesScience I have a question. How did you get (1/ sqr root 1 (1,1)) for the eigenvector? I understand v_2= v_1. But after that I'm lost how you found the vector.
@@Blank-bq3jo This is because we choose to work with a normalized eigenvector, and the 1/sqrt(2) ensures that the lenght is 1. I hope this helps!
Happy to make the first comment in one of my most favorite UA-cam channels.....thanks to the team for yet another highly worthwhile video.....kudos👍
Thanks for your first and nice comment! :)
Thank you mam. It clears all doubts related to the Pauli matrix.
Glad this was helpful!
Can't wait for the next videos ....
Much awaited
We'll try to publish them as regularly as possible :)
The way of explanation is impressive.Thank u for making this lecture .Please make a lecture on similarly transformation.
Glad you like it! And we do cover unitary (similarity) transformations in this video: ua-cam.com/video/baIT6HaaYuQ/v-deo.html
I hope this helps!
@@ProfessorMdoesScience yeah 👍
This channel is amazing! I would like to study quantum mechanics with your videos as a main resource. Should I just watch videos in the order of old videos to new videos? or is there any sequences or order you would recommend?
Glad you like it! The best way to watch the videos is by following the different playlists. And amongst those, a good way to watch them is as follows:
1. The basic principles of quantum mechanics are described in the "postulates" playlist: ua-cam.com/play/PL8W2boV7eVfmMcKF-ljTvAJQ2z-vILSxb.html
1.5. A good complement to the postulates is to explore operators in more detail: ua-cam.com/play/PL8W2boV7eVfnb10T_COKPozxEYzEKDwns.html
2. A first example of using the postulates in (mostly) 1D is the quantum harmonic oscillator: ua-cam.com/play/PL8W2boV7eVfmdWs3CsaGfoITHURXvHOGm.html
3. You can then move to 3D with angular momentum: ua-cam.com/play/PL8W2boV7eVfmm5SZRjbhOKNziRXy6yIvI.html
4. A good example of 3D systems are central potentials: ua-cam.com/play/PL8W2boV7eVfkqnDmcAJTKwCQTsFQk1Air.html
5. And then you can complete your "first" course with the hydrogen atom: ua-cam.com/play/PL8W2boV7eVfnJbLf-p3-_7d51tskA0-Sa.html
The above provides a list of the "basics" to get started. You can always complement those videos with extra topics such as uncertainty principles (ua-cam.com/play/PL8W2boV7eVfl3HQzTBZyGfrpfrv7YY3zN.html) or time evolution (ua-cam.com/play/PL8W2boV7eVflUqUY3dLhQdYuZjlbXi0mU.html). We are currently working on a series of videos on spin 1/2, which will also be a nice complement to the above.
After the basics, we have some playlists with more advanced topics, including:
1. Density operators: ua-cam.com/play/PL8W2boV7eVflL73N8668N0EQUnID1XaEU.html
2. Identical particles: ua-cam.com/play/PL8W2boV7eVfnJ6X1ifa_JuOZ-Nq1BjaWf.html
3. Second quantization: ua-cam.com/play/PL8W2boV7eVfnSqy1fs3CCNALSvnDDd-tb.html
We plan to extend each of these topics in multiple directions (e.g. quantum statistical mechanics, quantum condensed matter, etc) but not quite there yet.
We are also working on a website to go with the channel where all of this will be summarised. I hope this helps!
Your videos clear up so many misconceptions i get from class! keep up the great work! could you possible do a video or set of videos for spin 1/2 particles?
Glad you find our videos useful, may we ask where you study? And yes, we are working on a series on spin 1/2 particles at the moment which we'll hopefully publish soon!
@@ProfessorMdoesScience I am about to finish 2nd year at Temple University in Philadelphia, Pennsylvania. I study chemistry and physics there. I finished up my introduction to quantum mechanics course and your videos have been very helpful through the semester! Can’t wait to see your new videos!
2nd year of undergraduate*
@@JoseMendez-ud6hj Good luck with your studies!
Thanks for this wonderful video,love from India ❤
Thanks for watching!
Amazing 👏 Explanation
Glad you like it!
I got used to X,Y,Z but simply writing them as 1,2 ,and 3 is better and more general. Plus it makes it easier to remember they need the hbar/2 to become Sx, Sy, Sz.
I quite like the 1,2,3 notation as it is more natural for writing compact expressions such as the ones we use for commutation and anticommutation relations :)
Wonderful explanation dear madam. I need all the topics from Quantum Mechanics from Ur Class.
Glad to be helpful!
The usual excellent presentation...
Thanks for your support!
Thank you very much, it is very clear and interesting :)
Glad you like it!
Consider a quantum system comprising of electron spins.
We know that the Pauli matrices X and Z do not commute => X.Z ≠ Z.X
This means that we cannot simultaneously measure Spin along X and Z directions of an electron.
However if we consider the operators (Z × Z) and (X × X) which are formed by tensor products of Z's and X's; then these operators do commute;
i.e. (Z × Z)(X × X) = (X × X)(Z × Z).
If we consider a system of two electrons, then -
(Z × Z) is equivalent of measuring Z observable (i.e. spin along Z direction) on two electrons,
and (X × X) is similarly equivalent of measuring X observable (i.e. spin along X direction) on two electrons.
If this interpretation is correct, then logically it seems counterintuitive that if we take a system of two electrons, then we can simultaneously measure spins along Z and X directions for both!
Nicely explain ❤
Glad you like it!
Spin hype!
Yeah! We'll start with general two-state quantum systems and then specialize the discussion to spin :)
Thank you!
Thanks for watching! :)
I'm tempted to think this is fake because I never see scientists that can both explain things well AND have legible handwriting 🤣
Good one! :)
My thoughts are the same as yours. Plus attractive looking as well.
how does the coefficient 1/root2 comes from
This is the normalization constant. For example, if we look at the eigenvectors of sigma_1, we fidn that v_2+ must be equal to v_1+. This means that the eigenvector is such that the two entries are the same, and they could have any value. But if we insist that the eigenvector is normalized, then their value must be 1/sqrt(2). I hope this helps!
this was great
Glad you liked it!
Hey! Don't know if I'm wrong, but I'm pretty sure the eigenvalue = +1 for sigma 2, the corresponding eigenvector should be (1 neg i). I double checked my math and it seems like the signs are flipped (as in, the v neg vector matches what I have for lambda = 1 etc) unless I'm wrong!
Remember that there is some freedom in how one chooses eigenvectors. For example, if v is an eigenvector a matrix A, then -v is also an eigenvector with the same eigenvalue. I hope this helps!
@@ProfessorMdoesScience This makes sense! I think I realized that a day or so after I left this comment. Thanks so much for the helpful video!
Sticking with you. real good stuff. Dan USA
Glad you like it!!
Long since we've heard from your channel, when is the next video coming? Eagerly waiting........
Sorry for the delay, normal work has taken over recently... But we are working on the Pauli matrices series (which will include spin 1/2) and others, hopefully soon!
Can you please make a video about spin 1/2 systems.
Your videos help a lot❤
We are working on a series of videos on spin 1/2 particles, so we'll hopefully publish them soon!
Question is, are these (matrix)entries fixed, or do they change under some circumstances, as long it preserves some algebraic properties.
For example, in moved or accelerated reference frame. Something like: Light is red, because source moves away Light is red, because electrons in moved atoms make other transitions, ...
Is there some research going on?
They stay the same, the eigenvalues change
We'll explore a related question in the upcoming videos on spin, where we find that the spin operators are proportional to the Pauli matrices, and then you can think of these operators as written in the basis of the third spin component (proportional to the third Pauli matrix). You could then imagine changing basis, and changing the matrix representation of the corresponding spin operators. I hope this helps!
I always thought matrices were a Heisenberg thing.
Heisenberg was indeed one of the creators of the matrix mechanics formulation of quantum mechanics, and you can learn more about it in our series on the postulates of quantum mechanics. The Pauli matrices are a special instance of matrix mechanics which are very useful to study 2-state quantum systems. I hope this helps!
Hello, can you pls make a video on spin 1/2 systems?
We are working on a series on spin 1/2 systems, so stay tuned! :)
When you calculate de commutation between Pauli´s matrices 3 and 1 ([3,1]), ¿where does "i" come from?
I would recommend you give this a go yourself to check. But broadly speaking, the commutator of sigma_1 and sigma_3 will be a matrix that has no "i" overall. However, the matrix sigma_2 does have an "i", so when we want to write the commutator of sigma_1 and sigma_3 in terms of sigma_2, we need to add an extra i.
You can also view it in the opposite way: 2i * sigma_2 will give you an overall matrix without an "i" (as the "i" infront will multiply the "i" in the sigma_2 matrix), and this is then equal to the commutator of sigma_1 and sigma_3 which indeed has no "i".
I hope this helps!
Why is the eigenvector corresponding to lambda=-1 of sigma_1 = (1 -1) and not (-1 1)?
This is an choice you can make, both are valid eigenvectors. In general, if v is an eigenvector of a matrix A for a given eigenvalue, then so is -v. I hope this helps!
a) The eigenvectors of each Pauli Matrices is different but all Pauli Matrices share the same eigenvalues.Does this imply a degenerate case ? (even if experiments shows the same eigenvalues in all directions)
b) As Pauli matrices represent the components of the total spin momentum vector, is this a form of projection Operator?
c) Is there any mathematical theorem to show that it is acceptable to formulate a 2 dimensional spin state in a 3 dimensional complex space state x,y,z . My intuition says that the eigenvectors should be the same in x,y and z. If one reformulates the eigenvector in x direction (1,1) as (1,0) + (0.1) and claims that the basis are the same as in the z direction, does this imply the x basis and z basis are not independent basis ?
d) I am missing the purpose of Pauli matrices in Physics here other than analogy to angular momentum to explain fine structure spectrum of atoms. Even in measurement, only the eigenvalues are obtained but we have no clue on the actual eigenvectors. Any help ?
cheers
a) We typically talk about degenerate eigenvalues of a given operator. Here, each Pauli matrix is a different operator.
We are preparing a full series on spin 1/2, and these should help clarify questions (b), (c), and (d). We are hoping to publish these soon!
Thank you.
Thanks for watching!
What do these things physically mean?
They can be used to describe any two-state quantum system, which could be anything from the spin of particles like the electron to the qubits of quantum computers. More videos coming up soon looking at some of these examples!
Dear mam I'm asking a question
How segma two dagger is equal to segma ? Please give me your answer
Remember that "dagger" means that you transpose the matrix, so for sigma_2 you exchange the -i and the i off-diagonal terms... and then you calculate the complex conjugate, so -i turns into i and i turns into -i. This gives you sigma_2 back. I hope this helps!
nice
Glad you like it!
❤
Please do qft gamma matrices.
We are working on a series on spin at the moment, and we have plans for a few subsequent series after that. But we do hope to get to QFT at some point! :)
Thapar guys ☠
Out of curiosity I wanted to know since we know that ocean water contains salt and we know salt water conducts electricity fairly well
Then what if we insert high voltage electrodes in th ocean would it conduct electricity and if yes then till how far can it conduct
As it seems absurd to assume that we can conduct electricity through an ocean across continent but can find a scientific reason why not