Matrix formulation of quantum mechanics
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- เผยแพร่เมื่อ 31 ก.ค. 2024
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📚 Kets, bras, and operators can be written as column vectors, row vectors, and matrices. This leads to the formulation of quantum theory known as matrix mechanics, in which the usual operations of quantum mechanics can be written using the usual rules of matrix multiplication. We first explain how to write this formulation, and then we work through a number of examples to see it in action.
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⏮️ BACKGROUND
Dirac notation in state space: • Dirac notation: state ...
Operators: • Operators in quantum m...
Representations: • Representations in qua...
⏭️ WHAT NEXT?
Eigenvalues and eigenstates: • Eigenvalues and eigens...
Pauli matrices: [COMING SOON]
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Director and writer: BM
Producer and designer: MC
this video series should be incorporated into the curricula of every junior/senior level undergrad QM course
Wow, thanks! You can help us spread the word ;)
There's only so much time to cover basic concepts in undergrad QM.
Criminally underrated channel. Keep it up!
Haha, thank you!
Very well and clearly taught. Thank you so much for all your effort in explaining, recording, and sharing with us!
Thanks, I am glad you find them useful!
Thanks. After many years, I finally get Matrix Mechanics. You are a great teacher!
This is great to hear! :)
many thanks, been watching much of your clips this morning, and my Quantum Mechanics course makes a lot more sence now!! Thanks again, will suggest this to my friends!
Thanks for your support!
Goated video series. Sub + like well deserved. Keep up the good work!
Thanks for your support!
Your video's really helped me out! Thanks a lot bro!
Glad it helped!
THANKS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!, You are saving a lot of time of students like us who wants to learn quantum mechanics quickly for quantum computation otherwise books seems daunting.
Glad you find this helpful! :)
Man, these videos are really underrated!!
Glad you like them! :)
Worth watching
Thanks for your support!
clear and concise video!
Glad you like it!
such a good video that it can be understood at a high school level
Thanks for watching!
Thank you so much for this.
Glad you like it!
You should be given a nobel prize❤❤
Hahahaha, this is not quite how it works ;)
Thank you very much
Thanks for your continued support!
Nice.thank you so much
Thanks for watching!
ty
Your videos are awesome, congratulations! I have a question about the fifth expression in 9:49 of the video: if phi and psi (as states) are the same, we have that would be the expected value of A in the state psi (is it right?), but if psi and phi are different states, how do we interpret it physically?
You are correct that is the expectation value of A in state |psi> (preparing a video on this that should come out in the next few weeks). If they are different as in , the interpretation is not so clear. You can think of it as first A|phi> = |chi> giving you a new state, and then = tells you how similar the new state |chi> is to the state |psi>.
Thankyou Sir
Glad you like it!
Thanks for the video. I think "matrix mechanics" and "wave functions" are literally *the same thing* in finite dimensions, since either way your Hilbert space is just C^n. It's when you have to infinite dimensional cases when they are seemingly different things, but which are actually equivalent.
You are correct that the two approaches are ultimately equivalent. And the distinction may boil down to language, but I usually think of "wave functions" as the representation of a state in an infinite continuous basis (e.g. the position basis).
Hey, one question: Why should be the matrix entry A_ij? For example if you have a set of orthonormal basis-vectors: |b_1>=(1,0,0), |b_2>=1/sqrt(2)*(0,-i,1), |b_3>=1/sqrt(2)*(0,i,1), then the outer product of i.e. |b_3>
It feels like the best playlist for postulates of QM that I have ever come across...
In the "WHAT NEXT" section of the video description "Pauli Matrices" are written as "COMING SOON"... When can we expect that...? It's an really essential part... Waiting for that with excitement...
Glad you like it! We are hoping to do a full series on spin soon that will include this video on the Pauli matrices. However, the next few videos will cover the hydrogen atom and a few other bits and pieces, so it may still take a few months to get there...
Can you kindly suggest any book that takes the approach that you took... Bcoz your approach is till now the most intuitive one that I have found... it feels like everything is explained at proper place and time as it should be... No sudden out of context practical analogies inside in depth mathematics to make it more complicated... Everything is built step by step... Terms are introduced only when the need arises... And it is explained why we need this, what background is necessary and all things essential required to understand a topic... Are there any websites or blogs or other lecture videos of yours to follow... I am taking a Quantum Computing course this year which I have to complete by December and therefore I am in dire need of simple yet effective approaches to understand QM... It would be very polite of you if you could suggest anything nice...
@@MrAnuragprasad Thanks for your support! Unfortunately, we think there is no resource like what we are doing (hence the reason why we are doing this in the first place!). However, a set of books that we particularly like because they also cater to the learners needs with key sections and then additional sections that one may skip on first reading are the "Quantum Mechanics" books by Cohen-Tannoudji. I hope this helps, and good luck with your course!
I will try persuing the book you suggested... Can't thank you enough for your help...
Please make some videos on Hilbert spaces and their importance in quantum mechanics
Thanks for the suggestion!
I’m taking Landau’s book to learn QM. This vídeo saved me, becouse without a motivation and this bra-ket notation, it’s hard to understand this part.
Great to hear! :)
I don't get how one could represent things like position and momentum here. Those are continuous basis. Energy is fine, just a really long unit vector of coefficients for all posible energies. Do we just work in discrete basis like in matrix mechanics? If so, whats the form the hamiltonian will take in matrix form in the energy basis? What about the position operator in the energy basis? (To calculate useful properties like the expectations value)
Dear sir, I have two questions [1] pertaining to the foundation of QM physics, where both the wave mechanics and matrix mechanics, (related to this operator algebra stuffs) overlap together, is that called 'State Space Formulation'? I am a bit obscure here!
[2] While forming the structure of the outer product in terms of matrix formulation, if I write (c_i d*_j) under summation as (d*_j c_i) then will they still represent the A_ij matrix element? I think then I should take the transpose of elements under the summation sign!
correct me if I am wrong
TIA
I would say:
[1] The first important thing to note is the difference between "formulation" and "representation/basis". Formulations are about which mathematical objects you choose to do the maths. The "state space formulation" uses bras/kets/operators in Dirac notation, whereas the "matrix formulation" uses matrices and vectors. They're different but equivalent mathematical languages.
A representation, however, is a choice of basis: instead of leaving it abstract, you can choose a specific basis to write your states and operators This is the same you do with 3D vectors when you write them as v (abstract) or as v=(a,b,c) where "a,b,c" are the components in some basis. If you choose the position basis, what you get is precisely wave mechanics. This basis has an important place historically, but it is entirely equivalent to choosing another basis, for example the momentum basis. You can find more details in the video on wave functions: th-cam.com/video/2lr3aA4vaBs/w-d-xo.html
[2] A few things here. First, d*_j and c_i by themselves are just scalars, so you can exchange their order in any term in the sum without changing the result of the sum. Second, if you want the transpose of the matrix A_ij with matrix elements (d*_j c_i) then you need to exchange i with j, which gives (d*_i c_j). Third, another related question is: what is the matrix expression for the adjoint of the outer product? In the video on operators (th-cam.com/video/pNFna7zZbgE/w-d-xo.html) we find that (|psi>
@@ProfessorMdoesScience thank you indeed
I have a question: I learnt that the Hilbert space associated with the observable quantities of quantum mechanical systems, namely position and momentum is the space of complex square integrable functions. But the momentum eigenstate is a monochromatic wave of infinite extent, while the position eigenstate is a Dirac delta distribution. How can morph those to be compatible with its Hilbert space?
This is a subtle but very important point. To address it in detail we would need a full video, and we actually plan to release such a video in the future. But for now, let me just say that you are correct and plane waves, which are the eigenfunctions of the momentum operator, are not square integrable, so they don't belong to the Hilbert space of physically allowed states. However, we can construct square integrable functions by a superposition of plane waves (e.g. in a wave packet). For this reason, plane waves are an extremely useful mathematical tool, that although technically not part of the Hilbert space, can be used as "intermediaries" to construct valid states in the Hilbert space. I hope this helps!
@@ProfessorMdoesScience it helped thank you very much, I am waiting for the video release!
Hi, so basically I have a doubt, we know that the inner product of any two states results in a scalar (complex number in case if the field is complex) so how are we able to represent the inner product in terms of a column vector at 1:51 ?
Good question! You are correct that the inner product of any two states is a scalar. But note we do *not* represent the inner product of two vectors as a column vector, we represent a list of inner products , and i runs over the full list, as a column vector. What the column vector represents is the full quantum state, characterized by its coefficients c_i in the basis spanned by {|u_i>}. I hope this helps!
Thank you so much for the great work.
I have a question that I would appreciate your opinion in: is applying matrices fundamental to the theory or just a calculational trick?
In other words, do the laws of matrices (multiplication, addition, etc..) actually describe the behavior of the particles? or is it just a tool to compute the consequences of the laws of QM?
The physical world is described by mathematics, and matrices are a possible mathematical construct to describe quantum systems. So in this sense, matrices describe quantum particles just like the differential equation in Newton's second law describes classical particles.
@@ProfessorMdoesScience thank you so much for the reply.
Do you have any thoughts on Eugene Wigner's unreasonable effectiveness of mathematics since I have been thinking about it for a while, and your video about matrix mechanics is a concrete example on this unreasonableness, since matrix algebra was a purely mathematical construct by Cayley and others, and yet it is a very accurate description of the behavior of elementary particles?
Interesting ideas, but I'm afraid I'm not familiar enough with these more philosophical concepts to be able to comment!
@@ProfessorMdoesScience Thank you
Perhaps this is a strange question, because its a bit hard to state but here goes - A vector can be expressed as the single summation over an index of N elements because we have a geometric sense of how this "summing" represents a resultant N dimensional vector. But in the case of a matrix, in what sense does the double summation over indices "sum" these elements into a "matrix" object as opposed to just producing an array of elements?
Interesting points, but not sure I fully follow your argument. I would think of a vector not as a sum over N elements, but of an N-element object. For example, a 2D vector v=(v1,v2). The concept of sum comes when you perform operators like a scalar product of two vectors. For example, you can consider the square magnitude of the vector as the sum of the squares of the components:
v^2=v1*v1+v2*v2
But note that once you've done this, you don't end up with a vector anymore, but instead with a scalar. Similar ideas apply to matrices. I hope this helps!
❤ I like this video ❤
Glad you like it!
Nice succinct
Glad you like this!
Sir i have a question. Why do you keep changing i to j without any reason like in representing bra you change the basis to uj from ui. Is there a reason for that?
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!
@@ProfessorMdoesScience thanks
how do operators as matrices connect to operators as differential operators? Are the matrix elements just the differential operators?
Using matrices or differential operators to represent observables largely depends on the basis (representation) in which you are describing the state space in which the quantum system lives. If we have a finite state space basis (e.g. the spin degree of freedom of a spin-1/2 particle), then we can represent observables using finite matrices. If we have a countable infinite state space basis (e.g. that spanned by the energy eigenstates of the quantum harmonic oscillator), then we can represent observables using "infinite" matrices. If we have a continuous state space basis (e.g. the position or momentum representations, and note there are some caveats to these are these are not proper bases of a Hilbert space, but practically very useful), then we end up with differential operators. A full answer would require much more than a comment, but I hope this helps!
Please tell how delta ij replace double sum to single sum??
Let's take the example of the scalar product, where we have the double sum:
= sum_{ij} c*_i d_j delta_{ij}
The delta_{ij} is called the Kronecker delta, and is defined as follows:
delta_{ij} = 1 if i is equal to j
delta_{ij} = 0 if i is not equal to j
In the double sum over i and j, all terms for which i is not equal to j will vanish. Put another way, the only non-zero terms will be those with i=j, which turns the double sum into a single sum. I hope this helps!
bravo
Thanks, glad you like it! :)
❤
What physical properties do each element of those matrices represent? What physical action does an operator represent or is it just a measurement?
My biggest complaint about how QM is taught is that once the math in this video is introduced, linear algebra disappears almost entirely.
For example, tunneling is often introduced early in the first semester, but it is explained using braket notation. What does it look like using nothing but linear algebra?
I suspect I may find my answer in your videos, so I will keep watching.
(:
In quantum mechanics, physical properties are represented by operators, and in the matrix formulation it is the full matrix that represents the operator, not individual elements. Whether one uses the matrix formulation or not for a given quantum mechanical problem depends on the problem (as the difficulty of the maths can vary). For example, a problem that is very suitable to the matrix formulation is the study of spin. We don't yet have videos on spin, but we hope to publish some soon.
Regarding the physical action of an operator, it is *not* a measurement. Measurements in quantum mechanics are rather subtle, and we have a few videos dealing with this:
th-cam.com/video/u1R3kRWh1ek/w-d-xo.html
th-cam.com/video/odLwUXKY0Js/w-d-xo.html
I hope this helps!
@@ProfessorMdoesScience
Thanks! Understanding basic QM is on my bucket list.
(:
A question/comment about indices. I am very poor in keeping track of indices. Is it necessary to use different indices for each new operator/ket/bra? Actually I tried to prove last identity i.e. A^B^ = matrix. Intuitively I can see that it is correct. However, in the middle of the proof I am feeling messed up with all the a, b, c ...., z I used as indices. Sorry in advance as the question/comment is not a scientific one rather it just shows my incompetency.
I just figured out that I am forgetting a fundamental rule that matrix multiplication requires that number of columns of first matrix should be equal to number of rows of the second matrix. Any other rule of thumb would be appreciated.
Not sure what else to add from a rule-of-thumb point of view. However, it is definitely worth becoming familiar with the use of indices, as they provide a very compact way of dealing with matrices (and higher order tensors), and also provide a nice link to algorithmic implementations of these concepts.
@@ProfessorMdoesScience Thanks, the last point about algorithmic implementation is really good one as I have also started learning Fortran. It would definitely help.
How can we represent the momentum operator as a matrix
You have to calculate the matrix elements in the basis in which you want to write the operator. For example, if you work in the basis spanned by the eigenstates of the momentum operator, then all matrix elements are zero apart from the diagonal matrix elements, which are the momentum eigenvalues.
For a discrete basis, the matrix elements are A_ij and you can easily build the matrix from them as done in the video. For a continuous variable like momentum, writing the operator as a matrix is trickier because you now have an infinite and continuous number of entries. So in this case, rather than explicitly writing out the matrix, I would recommend to directly work with the matrix elements, which go from A_ij for a discrete basis labelled by i and j, to being functions A(alpha,beta) of two continuous variables alpha and beta.
Professor M does Science sorry but I’m still a bit confused. I’m not sure how you would write a basis spanned by the eigenstates of the momentum operator. Can you do a video of dealing with real operators like momentum and energy?
One last question. Can you solve the schrodinger equation by just doing matrix mechanics? As in can I write the Hamiltonian as a matrix acting on a ket vector and make it equal to the energy value times that ket vector and then solve for the ket vector and energy eigenvalue by just solving for the eigenvalues of the matrix?
I am planning videos on solving specific problems in the next few months :)
To your second question: yes, what you are describing is in fact what computer programs written to solve the equations of quantum mechanics do. You typically need to use supercomputers to solve these equations for applications in chemistry, physics or materials science because you end up with very large matrices, but the strategy is as simple as what you described.
Professor M does Science thank you very much. I love your channel and I know that it will grow in popularity due to your high quality videos. You should make a paypal so that people would be able to donate. Keep it up man 👍
am i meant to be ablke to just watch this and have epiphanies as an undergrad? took me a good 2hours to get everything, but atleast i understand better than when Miller did it.
For context, the level of these videos corresponds to a second course on quantum mechanics, after having already covered an introductory one. At Cambridge we teach this content in 2nd-3rd year undergraduate. However, our videos also aim to be self-contained, so if you follow the playlists you should have all information to be able to follow the content. This particular video falls within the first playlist I would recommend watching, which is the one on the "postulates of quantum mechanics": th-cam.com/play/PL8W2boV7eVfmMcKF-ljTvAJQ2z-vILSxb.html
This is math without physics !
Indeed, we use multiple videos on the playlist "the postulates of quantum mechanics" to introduce the maths necessary to explain quantum theory, and matrices are a key component of this. You can find plenty of videos that focus more on the physics in our channel!
can I communicate with you
You can find our contact email in the "about" tab.
I can't find it, can you send it to me?
@@user-ij8yu6fj3q professor.m.does.science@gmail.com
Anyone else from QGSS 21?
If you find our videos useful, it would be great if you spread the word with others at QGSS 21! ;)
@@ProfessorMdoesScience Yeah, I have shared it to their discord server!😊
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The video explains that called a matrix element of the operator A?
How is this helpful? For it seems that we have to know the matrix A to get to calculate an element of A?
Can you please elaborate using a concrete and simple example of an operator A and then work out to illustrate what exactly do we achieve?
Writing down an operator A in matrix form is a mathematically convenient way of working with operators. To do so, you need to know the operator you are working with (say A), and then you need to decide in which basis you are going to represent it. In the example, you use the {|u_i>} basis, and in this basis the matrix of A will take a specific form, with entries . In a different basis {|v_i>}, the same operator A would take a different matrix form, with entires . As to examples, we have a few scattered examples in the existing videos, but a very simple example is that of spin 1/2 particles. In this case, the state space is 2-dimensional and the matrices associated with operators are only 2x2, which makes everything very clear. We will publish a full series on spin 1/2 particles soon, after we finish our series on the hydrogen atom. I hope this helps!