Eigenvalues and eigenstates in tensor product state spaces

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📚 Tensor product state spaces allow us to mathematically describe quantum systems of multiple degrees of freedom (e.g. particles moving in 3D, orbital and spin degrees of freedom, or multi-particle systems). In this video, we learn how we can build the eigenvalues and eigenstates of operators acting in tensor product state spaces without having to solve the full problem. Instead, these eigenvalues and eigenstates can be constructed by combining the eigenvalues and eigenstates associated with the individual state spaces making up the tensor product space.
⏮️ BACKGROUND
Eigenvalues and eigenstates: • Eigenvalues and eigens...
Tensor products: • Tensor product state s...
⏭️ WHAT NEXT?
3D quantum harmonic oscillator: • The 3D quantum harmoni...
Two interacting quantum particles: [COMING SOON]
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Director and writer: BM
Producer and designer: MC

[@supergravity66]

Your videos are the best on quantum mechanics: precise, thorough & well-organized!

[@ProfessorMdoesScience]

Thanks for the kind words! :)

[@sandippaul468]

Finally, it's so good to see you say "Si" instead of "Psi".

[@ProfessorMdoesScience]

Never late to learn ;)

[@novaflynn6361]

Totally Love these Videos... Recommended to anyone beginning multi particle quantum mechanics

[@ProfessorMdoesScience]

Glad you like them, and thanks for the recommendations! :)

[@PianoAndLife]

These videos helps a lot, thanks!

[@ProfessorMdoesScience]

Glad you like them!

[@erikdiaz-bautista3061]

I really love and enjoy your videos. Thanks guys! 👏 🤜🤛

[@ProfessorMdoesScience]

Really glad to hear this! :)

[@subhajitsadhukhan8521]

Eagerly waiting for the videos on H atom......

[@ProfessorMdoesScience]

Working on them, they'll be here over the next few months... :)

[@njyde]

Thank you for the clear explanations! What if the operator that acts on V cannot be written simply as the direct sum of an operators that acts on V1 and one that acts on V2. Can you make a video about this more general case?

[@ProfessorMdoesScience]

Thanks for the suggestion!

[@soheilcyrus9989]

I am waiting to see more videos about quantum in this channel, is there the possibility that you share videos about Wigner-weyl distribution and P-Q representations?

[@ProfessorMdoesScience]

Thanks for the suggestion! We'll add them to our list, although it may take a while to get there as we have many in the pipeline...

[@mdshaifullah9598]

Sir, Could you please make a video on translational symmetry and the Bloch theorem?

[@ProfessorMdoesScience]

Thanks for the suggestion! We are hoping to do a whole course on condensed matter physics, which would include these topics. However, we will first finish up the basic quantum mechanics course we are currently doing, so it make still take a while before we get there...

[@zeio-nara]

Thanks for an amazing video, it's very interesting to learn about these concepts. Could you please give some examples where applying these ideas for multiparticle system can be used in practice - is it applicable to predicting chemical properties of molecules / materials under certain conditions or for what?

[@ProfessorMdoesScience]

Glad you like it! The simplest example is when we have two particles, and we'll cover the hydrogen atom (proton+electron) over the next few months as an example of this. Beyond that, you may want to take a look at our series on identical quantum particles (kzbin.info/aero/PL8W2boV7eVfnJ6X1ifa_JuOZ-Nq1BjaWf) and then the one on second quantization (kzbin.info/aero/PL8W2boV7eVfnSqy1fs3CCNALSvnDDd-tb), which are the building blocks to study more complex multi-particle systems such as materials. We hope to explicitly cover materials (condensed matter physics) in the future, but it will take us a while to get there.

[@richardthomas3577]

Excellent!

[@ProfessorMdoesScience]

Thanks for watching! :)

[@MrLethalShots]

I just found your videos today and they are amazing! Have you ever considered doing any on statistical physics?

[@ProfessorMdoesScience]

Glad you like them, and thanks for the suggestion! After we are done with basic quantum mechanics, we do hope to move on to other topics including statistical mechanics and condensed matter :)

[@MrLethalShots]

@@ProfessorMdoesScience Awesome! I look forward to them!

[@nuriaurgell4379]

One question, if the multiplicity of λ_n is g_n and the multiplicity of μ_k is h_k. Can we compute the multiplicities of the eigenvalues of A+B?

[@ProfessorMdoesScience]

Interesting point! I am not sure there is a general answer to this question, as the overall multiplicity will depend on how the eigenvalue spectra of A and B compare (i.e. if there are any eigenvalues that are the same between the two). Coincidentally, our latest video covers an example of this: the isotropic quantum harmonic oscillator. I therefore encourage you to check it out here: kzbin.info/www/bejne/gWa8pKupfb16fsU I hope this helps!

[@dien6]

Is there any sort of possible explanation/derivation as to why so many tensor product spaces have operators which look like these sums (as you mentioned, the 3D HO for example). Is there any way to know the hamiltonian can be written as this sum? (in the video on the 3D HO it just gets assumed to take on this form).

[@ProfessorMdoesScience]

Good question! The Hamiltonians take this form when they are non-interacting, meaning that the various degrees of freedom are independent of each other. Even when the Hamiltonians appear to have interacting terms, it is sometimes possible to make a change of coordinates to make them non-interacting. We have an example of this in the video on two interacting quantum particles: kzbin.info/www/bejne/oYDYoXx7nb6Xj5o I hope this helps!

[@dien6]

@@ProfessorMdoesScience thank you for your answer, yes I got it now, thank you for the quick reply and also the great videos!

[@TheWingEmpire]

Awesome

[@ProfessorMdoesScience]

Thanks! :)

[@physdb]

Is there any good reason why the operator C takes that form?

[@erikdiaz-bautista3061]

It's because this is a way to express that the operator C acts on a state |X> that belongs to a space V, which is the tensor product of spaces V_1 and V_2, not neccesarily of the same dimension. Here, there is a video about that: kzbin.info/www/bejne/oauWY2NsiJd1bLM For example, A_1 in C will act only on a state belonging to V_1, not on a state belonging to V_2, while I_2 will act on a state belonging to V_2: C |X> = (A_1 × I_2) (|psi> × |phi>) = (A_1 |psi>) × (I_2 |phi>)

[@theopdiamond6226]

Hi, I have a qs. Here we showed that the tensor product of the eigenvectors is also an eigenvector Of C with eigen Values as the sum of the individual eigenvalues. However, this doesn't mean that any eigenvectors of C can be written in this way. we might have Xhi and Omega that are Still part of the spectrum of C but can't be written in this way. Is this true? If so Then this is problematic Since it means that we didn't find the whole spectrum of C.

[@ProfessorMdoesScience]

We actually have all possible eigenstates (and eigenvalues) in this way, a good way to see this is by enumerating them. Imagine V1 is n-dimensional, which means that a Hermitian operator will have n eigenstates. Similarly, let V2 be m-dimensional, with a total of m eigenstates for an operator acting on it. Then, the tensor product state space V=V1xV2 is nxm dimensional, and will have nxm eigenstates. If we build the eigenstates of C in V from those of A and B V1 and V2 as explained in the video, then we get nxm eigenstates, so we have all of them. I hope this helps!