@@ProfessorMdoesScience your channel is such an asset. I teach some courses in undergraduate QM, and am constantly reviewing my own QM. Almost every day I understand something just a little better than before. I've had some great "aha" moments with your videos. It's very nice that they are short and focused. I've also worked through James Binney's course several times, given to undergraduates at Oxford. He is very clear too. Quite a daunting course (if you have the thick set of course notes AND work the problems!!)
@@vladimirkolovrat2846 Thanks for your kind words and glad to be helpful! Following on your comment on problems, we have actually recently launched some problems + solutions (more coming) to go with the videos, just in case they are useful: professorm.learnworlds.com/
This is fascinating. It's like a whole new way to describe systems. While in classical physics the differential equations used to be the main tool, in quantum mechanics, it's like linear algebra became much more used than differential equations.
Indeed! :) But we can actually do quantum mechanics in a variety of ways. For example, the theory still largely reduces to differential equations when working in the position representation.
Excellent video! It's mind boggling to me how non rigorous quantum mechanics was taught to me. This video and your other videos have really answered my concerns when bridging the vector representations in quantum mechanics to the standard wave function formalism I have been so used to.
Please provide link to some exercises regarding this video or any other video, so that we can get comfortable with all these concepts. You can also refer us to some exercise from any book. That would be quite helpful. Thanks!
At 10:11, Operation A is described in terms of the matrix element of A w.r.t. to u_i, u_j which in turn again depends on operator A. How does this calculation work? The operator A is in what form before we calculate matrix element and convert to u_i, u_j representation?
The way in which an operator A is defined usually involves its action on quantum states. For example, the identity operator 1 is the operator that acts on an arbitrary quantum state and leaves it unchanged: 1|u_i>=|u_i>. As another example, the projection operator P on state |psi> is the operator that "projects" an arbitrary quantum state along the reference state |psi>, and therefore P|u_i>=|psi>. Once you have these definitions of the various operators and their action on quantum states, you can build their representation in a given basis. Here I have a list of videos in which we describe how various operators are used in quantum mechanics, which should provide a nice set of examples: * Projection operator: th-cam.com/video/M9V4hhqyrKQ/w-d-xo.html * Translation operator: th-cam.com/video/978mMgGYs1M/w-d-xo.html * Time evolution operator: th-cam.com/video/zqmU4dW03aM/w-d-xo.html * Permutation operator: th-cam.com/video/mgqxywZMTjs/w-d-xo.html * Density operator: th-cam.com/video/DQEtg8pWT8E/w-d-xo.html I hope this helps!
Great clip. After finally and reluctantly looking into some basic Linear Algebra concepts, I finally see how all these fits into Quantum mechanics description in a Hilbert Space. My only lesson learnt is that I should have spent some time earlier on understanding Linear Algebra ( just need to find a friendly textbook.....it is not very difficult).
At around 9:23, ouermost Ket u-i is moved to left of outermost u-j, which is then interpreted as cross-product. Why not move Ket u-i just to the right of u-j & interpret it as scalar product instead ?
The first expression before moving anything has three terms in this order: |ui>. Both and are scalars, so we can move |ui> wherever we want around these two terms. If I understand your point correctly, then you can indeed also move it to the middle to end up with: |ui>. The final answer will be the same once you identify =Aij.
Representations became clear for me after this video, thank you. Is there a book where everything is explained in the same manner? Books I read were hard to understand.
Some books I like include those by (i) Shankar, (ii) Cohen-Tannoudji, and (iii) Sakurai. I am thinking about preparing some videos reviewing these books (and others). Do you think that would be helpful?
@@ProfessorMdoesScience I have been thinking about doing this for a few books. If there is a plan in place and you need some extra hands ( even editing or double checks) I would be willing to contribute. Just let me know! I think this is the way QM should be taught . I am a postdoc in Theor. And Computational Chemistry..
The term that is usually used in this context is "resolution", but I can imagine how, for example, "decomposition" could be used to describe the same thing. Where have you encountered these other terms?
Hello professor, the video was excellent but may I ask you to please recommend a book dedicated to only these mathematical tools. You already recommended me some but those are detailed theory and mathematics included but I want only these mathematical part. Thank you.
There are a number of books on the mathematics used in physics. One that is used often is: "Mathematical Methods for Physics and Engineering" by K.F. Riley, M.P. Hobson, and S.J. Bence. I hope this helps!
We re-write the identity in a particular basis as a mathematical trick to get to the required expressions. It is in the same spirit to a derivation where you add "0" to an expression in a convenient way (an example in this video on commutators: th-cam.com/video/57xgSIV9PY0/w-d-xo.html ). I hope this helps!
So is this just tensor algebra? Operators are tensors on the state space? |u1 x u2| is the tensor product? In that case, do we _need_ the basis to be orthonormal?
This is indeed standard vector space mathematics. For a typical physics student, the novel aspects associted with quantum mechanical state spaces is that the vector space is in general complex and can be infinite-dimensional. And you are correct that there is no fundamental requirement to have an orthonormal basis, but we typically use them becase they are very convenient to work with. I hope this helps!
@@ProfessorMdoesScience Thank you very much. I've been wondering if people sometimes use the Dirac notation for other tensor related topics(like General Relativity). It seems like a convenient alternative to index notation.
After grasping the concepts of inner product, outer product and matrix element I was wondering whether product of two kets e.g. |psi>|phi> (or equivalently product of two bras) also means something in Q.M?
This is sometimes used as a shorthand for "tensor product", which is an extremely powerful tool in quantum mechanics. You can find all the details about this in the following video: th-cam.com/video/kz3206S2B6Q/w-d-xo.html I hope this helps!
Perhaps I am doing something wrong here - but I was trying to expand Paul Matrix σ ͯ in terms of the |+> |-> basis, but not sure where am I going wrong. Is there a video somewhere that can explain how the operator [ [0 1] ͭ [1 0] ͭ ]; which is the Pauli operator σ ͯ in Standard basis |0> and |1> , can be expanded in terms of the |+> |-> basis. Guidance would be much appreciated.
Ah, I initially thought this was going to be about like, the representations of an algebra of operators ( or of a group) on a vector space. (That’s what I’m looking at atm so I was like “oh, nice!”) Of course, this topic is also very important as a foundation for things.
Dear sir, as we know lets say deep square well potential problem, we find there discrete set of eigen functions that span the hilbert space. my questions are---> 1. this set can be used as a energy representation. but individual states are represented in position representation. How is this reasonable? Can you please disentangle this concept! 2. In position representation hilbert space has continuous basis set. But in Hamiltonian eigenstates as a basis it follows a discrete set of basis elements. How are these two compatible with each other? Sorry for my naive understanding :)
Thanks for watching! Here are some thoughts: 1. Let us label the energy eigenstates of a system (say the infinite square well) with |n>. These are written as kets. We can re-write these energy eigenstates in any basis, for example in the position basis to get the wave function. To do so, we write: psi_n(x)=. This is the wave function of an energy eigenstate. 2. If I understand this correctly, you are asking how is it possible that we can use the position eigenstates (infinitely continuous) and also the energy eigenstates (infinite but discrete) to span the (apparently) same Hilbert space of, say, the infinite square well potential. This is a rather tricky question whose answer is quite subtle, but here I will provide some thoughts. The position eigenstates (delta functions) are in fact not valid functions in the Hilbert space of square-integrable functions (just like the momentum eigenstates, plane waves, are not either). So, although we use these bases for our mathematical manipulations for convenience, in reality they contain some redundancy. The correct way to count states that resemble position eigenstates and get the same number of states as the discrete energy eigenstates would be to use square integrable functions that are approximately (but not quite) equal to delta functions. I realise this is only a very rough answer, but I hope it motivates you to look into this more deeply. Hope this helps!
No problem, there are quite a few videos and it is not always obvious how to navigate them. The playlists are meant to group them in "topics", if that helps!
Sir if are analogus to scalar product then how a bra is projected on ket as they belong to diffrent vector space? OTHERWISE YOUR VIDEOS ARE SMALL AND SATISFYING.
Roughly speaking there is a one-to-one relation between elements in the direct and dual spaces (obtained by turning a ket into the corresponding bra). I hope this helps!
@@ProfessorMdoesScience yes sir that thing understood but I want to ask that how a ket element of vector space has projection on bra element of dual space(another)?
Just like in usual vectors, we have basis's like Cartesian coordinate, Spherical coordinate etc. Can you give me some examples of basis's in quantum mechanics
Bases in quantum mechanics are typically associated with the eigenstates of a Hermitian operator. Examples include the "position" basis, very useful when we study the motion of particles in 3D space, and you can find details in this playlist: th-cam.com/play/PL8W2boV7eVfnHHCwSB7Y0jtvyWkN49UaZ.html Another very useful basis is the "energy" basis, very useful when studying time evolution: th-cam.com/play/PL8W2boV7eVflUqUY3dLhQdYuZjlbXi0mU.html I hope this helps!
This is a concept that is typically introduced in the study of group theory. While this is a very important topic for quantum mechanics, it is somewhat more advanced of what we are covering at the moment in our videos. We do hope to publish some future series on the mathematical foundations of physics, including group theory, where we would cover these ideas. But these will not appear for a long while because for now we are focusing on the basics of quantum mechanics...
Sir please be consistent with your choice of i j you sometimes say ci is equal this projection and in the next slide you repesent that same thing with cj its very confusing Everything else is fabulous
See my answer to your other comment, but copied here again for completeness. This is very common practice: i and j are dummy indices, and depending on the situation we often re-label them to facilitate the maths. This is a strategy used widely when manipulating mathematical objects that have multiple components that can be indexed (either discretely or continously). I hope this helps!
This is easily one of the most important videos I have seen on youtube so far.
Thanks for watching, this type of comment really motivates us to keep going! :)
Agreed
I have been attending quantum mechanics lectures for 3 months now, and this is the first time everything started making sense. Thanks!!
This is great to hear! May we ask where you study quantum mechanics?
I am a master's student in Germany and in this semester I have a course on applied quantum mechanics.
Whenever he goes "so what does this mean?" I get so happy cause I'm usually asking myself the same thing at the same time.
Glad you find those useful! :)
Finally someone answers to the question "what is a representation?"
You saved my life, sir.
Glad to be helpful!
this actually saved my life
I'm glad it helped!
How did it save your life ?
I also want to know
Probably was based on a test of some sort that was important
This channel is a gem. University level, crystal clear explanations. Well done.
Glad you like it!
@@ProfessorMdoesScience your channel is such an asset. I teach some courses in undergraduate QM, and am constantly reviewing my own QM. Almost every day I understand something just a little better than before. I've had some great "aha" moments with your videos. It's very nice that they are short and focused. I've also worked through James Binney's course several times, given to undergraduates at Oxford. He is very clear too. Quite a daunting course (if you have the thick set of course notes AND work the problems!!)
@@vladimirkolovrat2846 Thanks for your kind words and glad to be helpful! Following on your comment on problems, we have actually recently launched some problems + solutions (more coming) to go with the videos, just in case they are useful: professorm.learnworlds.com/
@@ProfessorMdoesScience that's very kind of you, thank you for sharing these! I'm currently on a roll, watching your videos... much gratitude to you.
You guys summed up my whole semester in 2 3 hours video. Thank you.
Glad to hear the videos were useful! :)
so brief and precise. TH-cam requires such clear contents. thanks!
Glad you liked it!
This is fascinating. It's like a whole new way to describe systems. While in classical physics the differential equations used to be the main tool, in quantum mechanics, it's like linear algebra became much more used than differential equations.
Indeed! :) But we can actually do quantum mechanics in a variety of ways. For example, the theory still largely reduces to differential equations when working in the position representation.
Thanks!
Wow! Much appreciated!
Excellent video! It's mind boggling to me how non rigorous quantum mechanics was taught to me. This video and your other videos have really answered my concerns when bridging the vector representations in quantum mechanics to the standard wave function formalism I have been so used to.
Glad to hear you've found them useful!
Excellent series on quantum stuff
Many thanks!
Please provide link to some exercises regarding this video or any other video, so that we can get comfortable with all these concepts. You can also refer us to some exercise from any book. That would be quite helpful. Thanks!
We are working on a website where we will provide problems (and solutions). The hope is to launch it over the next half-year or so, so stay tuned!
At 10:11, Operation A is described in terms of the matrix element of A w.r.t. to u_i, u_j which in turn again depends on operator A. How does this calculation work? The operator A is in what form before we calculate matrix element and convert to u_i, u_j representation?
The way in which an operator A is defined usually involves its action on quantum states. For example, the identity operator 1 is the operator that acts on an arbitrary quantum state and leaves it unchanged: 1|u_i>=|u_i>. As another example, the projection operator P on state |psi> is the operator that "projects" an arbitrary quantum state along the reference state |psi>, and therefore P|u_i>=|psi>. Once you have these definitions of the various operators and their action on quantum states, you can build their representation in a given basis.
Here I have a list of videos in which we describe how various operators are used in quantum mechanics, which should provide a nice set of examples:
* Projection operator: th-cam.com/video/M9V4hhqyrKQ/w-d-xo.html
* Translation operator: th-cam.com/video/978mMgGYs1M/w-d-xo.html
* Time evolution operator: th-cam.com/video/zqmU4dW03aM/w-d-xo.html
* Permutation operator: th-cam.com/video/mgqxywZMTjs/w-d-xo.html
* Density operator: th-cam.com/video/DQEtg8pWT8E/w-d-xo.html
I hope this helps!
Great clip. After finally and reluctantly looking into some basic Linear Algebra concepts, I finally see how all these fits into Quantum mechanics description in a Hilbert Space. My only lesson learnt is that I should have spent some time earlier on understanding Linear Algebra ( just need to find a friendly textbook.....it is not very difficult).
Good insight! And yes, linear algebra is essential for quantum mechanics as well as many other areas!
At around 9:23, ouermost Ket u-i is moved to left of outermost u-j, which is then interpreted as cross-product. Why not move Ket u-i just to the right of u-j & interpret it as scalar product instead ?
The first expression before moving anything has three terms in this order: |ui>. Both and are scalars, so we can move |ui> wherever we want around these two terms. If I understand your point correctly, then you can indeed also move it to the middle to end up with: |ui>. The final answer will be the same once you identify =Aij.
@@ProfessorMdoesScience Got it now, thanks.
Great channel!! Is there a video ordering so that I can start at the beginning?
Nice explaination prof
Glad you like it! :)
Hope you make a short video on Dirac delta function. At 13:20 what about the integration when you just equate the L.HS. to C(beta)? Thank you
OK, I GOT IT. NO NEED FOR EXPLANATION.
Great that you got it!
Representations became clear for me after this video, thank you. Is there a book where everything is explained in the same manner? Books I read were hard to understand.
Some books I like include those by (i) Shankar, (ii) Cohen-Tannoudji, and (iii) Sakurai. I am thinking about preparing some videos reviewing these books (and others). Do you think that would be helpful?
@@ProfessorMdoesScience Yes, for sure it would be helpful. Thank you
@@ProfessorMdoesScience I have been thinking about doing this for a few books. If there is a plan in place and you need some extra hands ( even editing or double checks) I would be willing to contribute. Just let me know! I think this is the way QM should be taught . I am a postdoc in Theor. And Computational Chemistry..
@@ernek89 Thanks for the offer, we'll keep it in mind!
Thanks for clearing my doubt
Glad to be helpful!
Does the "resolution" of the identity in the U basis has similar meanings of "decomposition" or "factorizing" the identity by U basis?
The term that is usually used in this context is "resolution", but I can imagine how, for example, "decomposition" could be used to describe the same thing. Where have you encountered these other terms?
@@ProfessorMdoesScience Those are my words, and this is my first time to engage in QM formally. I want to check my understanding.
@@BruinChang Ok! So the usual term for this is "resolution", but what really matters is the mathematical expression :)
@@ProfessorMdoesScience got it, thanks!
Thank you very much
Thanks for watching!
Hello professor, the video was excellent but may I ask you to please recommend a book dedicated to only these mathematical tools. You already recommended me some but those are detailed theory and mathematics included but I want only these mathematical part. Thank you.
There are a number of books on the mathematics used in physics. One that is used often is: "Mathematical Methods for Physics and Engineering" by K.F. Riley, M.P. Hobson, and S.J. Bence. I hope this helps!
@@ProfessorMdoesScienceThank you sir.
Could you please provide a video conceptual representation problems from both QM as well as mathematics view point!
This is a good point! Once I finish with the series on the postulates, we will look at more conceptual ideas and see how we can use them.
@@ProfessorMdoesScience Very kind to you professor!
Why would you rewrite "1" as this identity operator? Why not keep it as "1"?
We re-write the identity in a particular basis as a mathematical trick to get to the required expressions. It is in the same spirit to a derivation where you add "0" to an expression in a convenient way (an example in this video on commutators: th-cam.com/video/57xgSIV9PY0/w-d-xo.html ). I hope this helps!
Thank you very much :)
Thanks for watching!
So is this just tensor algebra? Operators are tensors on the state space? |u1 x u2| is the tensor product?
In that case, do we _need_ the basis to be orthonormal?
This is indeed standard vector space mathematics. For a typical physics student, the novel aspects associted with quantum mechanical state spaces is that the vector space is in general complex and can be infinite-dimensional. And you are correct that there is no fundamental requirement to have an orthonormal basis, but we typically use them becase they are very convenient to work with. I hope this helps!
@@ProfessorMdoesScience Thank you very much. I've been wondering if people sometimes use the Dirac notation for other tensor related topics(like General Relativity). It seems like a convenient alternative to index notation.
@@narfwhals7843 We're definitely not experts in general relativity, but have not encountered this before.
So fascinating
Glad you like it! :)
After grasping the concepts of inner product, outer product and matrix element I was wondering whether product of two kets e.g. |psi>|phi> (or equivalently product of two bras) also means something in Q.M?
This is sometimes used as a shorthand for "tensor product", which is an extremely powerful tool in quantum mechanics. You can find all the details about this in the following video:
th-cam.com/video/kz3206S2B6Q/w-d-xo.html
I hope this helps!
@@ProfessorMdoesScience It is really encouraging that you give a good read to each comment and answer in sufficient detail, thanks again.
Perhaps I am doing something wrong here - but I was trying to expand Paul Matrix σ ͯ in terms of the |+> |-> basis, but not sure where am I going wrong. Is there a video somewhere that can explain how the operator [ [0 1] ͭ [1 0] ͭ ]; which is the Pauli operator σ ͯ in Standard basis |0> and |1> , can be expanded in terms of the |+> |-> basis.
Guidance would be much appreciated.
Please discard earlier comment. I figured out what I was doing wrong. Got it sorted now
Glad you resolved this, we are planning to publish a full series on spin 1/2 after we finish with hydrogen!
Ah, I initially thought this was going to be about like, the representations of an algebra of operators ( or of a group) on a vector space.
(That’s what I’m looking at atm so I was like “oh, nice!”)
Of course, this topic is also very important as a foundation for things.
We do hope to cover group theory and related topics in the future, but still on the basics of quantum mechanics for now! :)
Dear sir, as we know lets say deep square well potential problem, we find there discrete set of eigen functions that span the hilbert space.
my questions are--->
1. this set can be used as a energy representation. but individual states are represented in position representation. How is this reasonable? Can you please disentangle this concept!
2. In position representation hilbert space has continuous basis set. But in Hamiltonian eigenstates as a basis it follows a discrete set of basis elements. How are these two compatible with each other?
Sorry for my naive understanding :)
Thanks for watching! Here are some thoughts:
1. Let us label the energy eigenstates of a system (say the infinite square well) with |n>. These are written as kets. We can re-write these energy eigenstates in any basis, for example in the position basis to get the wave function. To do so, we write:
psi_n(x)=.
This is the wave function of an energy eigenstate.
2. If I understand this correctly, you are asking how is it possible that we can use the position eigenstates (infinitely continuous) and also the energy eigenstates (infinite but discrete) to span the (apparently) same Hilbert space of, say, the infinite square well potential. This is a rather tricky question whose answer is quite subtle, but here I will provide some thoughts. The position eigenstates (delta functions) are in fact not valid functions in the Hilbert space of square-integrable functions (just like the momentum eigenstates, plane waves, are not either). So, although we use these bases for our mathematical manipulations for convenience, in reality they contain some redundancy. The correct way to count states that resemble position eigenstates and get the same number of states as the discrete energy eigenstates would be to use square integrable functions that are approximately (but not quite) equal to delta functions. I realise this is only a very rough answer, but I hope it motivates you to look into this more deeply.
Hope this helps!
@@ProfessorMdoesScience Thank you, professor, it really helps me a lot!
can u plz make a video on Exception values?
If you mean expectation values, then we already have a video:
th-cam.com/video/rEm-Ejg5xek/w-d-xo.html
I hope you like it!
@@ProfessorMdoesScience thx
sorry for wasting ur time
No problem, there are quite a few videos and it is not always obvious how to navigate them. The playlists are meant to group them in "topics", if that helps!
@@ProfessorMdoesScience ok , I will refer the playlists for better navigation.
Thx , U are just awesome.
Sir if are analogus to scalar product then how a bra is projected on ket as they belong to diffrent vector space? OTHERWISE YOUR VIDEOS ARE SMALL AND SATISFYING.
Roughly speaking there is a one-to-one relation between elements in the direct and dual spaces (obtained by turning a ket into the corresponding bra). I hope this helps!
@@ProfessorMdoesScience yes sir that thing understood but I want to ask that how a ket element of vector space has projection on bra element of dual space(another)?
@@sahil-pu3cc This video may help:
th-cam.com/video/p1zg-c1nvwQ/w-d-xo.html
Just like in usual vectors, we have basis's like Cartesian coordinate, Spherical coordinate etc. Can you give me some examples of basis's in quantum mechanics
Bases in quantum mechanics are typically associated with the eigenstates of a Hermitian operator. Examples include the "position" basis, very useful when we study the motion of particles in 3D space, and you can find details in this playlist:
th-cam.com/play/PL8W2boV7eVfnHHCwSB7Y0jtvyWkN49UaZ.html
Another very useful basis is the "energy" basis, very useful when studying time evolution:
th-cam.com/play/PL8W2boV7eVflUqUY3dLhQdYuZjlbXi0mU.html
I hope this helps!
what do we mean by IRREDUCIBLE representation?
This is a concept that is typically introduced in the study of group theory. While this is a very important topic for quantum mechanics, it is somewhat more advanced of what we are covering at the moment in our videos. We do hope to publish some future series on the mathematical foundations of physics, including group theory, where we would cover these ideas. But these will not appear for a long while because for now we are focusing on the basics of quantum mechanics...
Sir please be consistent with your choice of i j you sometimes say ci is equal this projection and in the next slide you repesent that same thing with cj its very confusing
Everything else is fabulous
See my answer to your other comment, but copied here again for completeness. This is very common practice: i and j are dummy indices, and depending on the situation we often re-label them to facilitate the maths. This is a strategy used widely when manipulating mathematical objects that have multiple components that can be indexed (either discretely or continously). I hope this helps!
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