Bosons and fermions: the symmetrization postulate

[@ThomasDerIrre]

Kudos to Bartomeu. Thank you very much for doing this. For everyone else, I am a senior researcher in physics as well and find these videos extremely enjoyable. I wish my professor in Quantum Mechanics would have been that understandable.

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- Quantum mechanics. Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality! Commutators (Fermions) are dual to anti-commutators (Bosons). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation). The big bang is an infinite negative curvature singularity -- non null homotopic. Gaussian negative curvature is defined using two dual points. Singularities are dual:- Positive curvature (synchronic points) is dual to negative curvature (enchronic points) -- Gauss, Riemann geometry. Same (symmetry, summations) is dual to difference (anti-symmetry, differences). Bosons like to be in the same state, Fermions like to be in different states. "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero -- non null homotopic. Points are dual to lines -- the principle of duality in geometry!

[@nihargupte8032]

You are I think the best source on QM on KZbin. I am a 1st year PhD student currently and your videos have helped clear so many gaps in my knowledge!

[@ProfessorMdoesScience]

Great to hear! May I ask where you are doing your PhD? And in what topic?

[@monissiddiqui6559]

What an amazing series of videos on Identical quantum particles. I have a math background with exposure to many areas in applied and pure math and just started studying QM ( have done roughly half of MIT 8.04 on MIT OCW). I was watching some of your videos to supplement some of the basics on wave functions but quickly got sucked into lots of other videos on your channel. I really appreciate the focus on mathematical details! When I started QM I only expected to use some of my linear algebra and statistics and basic multi variable calculus. I was pleasantly surprised to find out that QM also heavily involves fourier transforms ( and mind was blown to find out that operator eigenstate bases somewhat generalize the notion of fourier transform!). Watching more of your videos and constantly seeing more concepts in math that I studied being applied to QM is such a joy :D. I didn't expect tensor products and group theory. Now I can't wait to find out what else haha. Thank you so much for these amazing videos and I will be sure to share these with anyone I know who is into QM ( seriously, who ISN"T??). I hope they will move beyond discussing morbid examples of whether cats are alive or dead to understanding WHY electrons can't have the same two quantum states. Cheers from Canada

[@ProfessorMdoesScience]

Thanks for your message, such kind words motivate us to keep going! And yes, maths is intimately related with quantum mechanics (and most of physics!). It is often the case that some of the mathematical details are skipped in physics courses, and this is one of the things we want to address in our videos :)

[@jbragg33]

Hi, thank you for the videos. At 11:38, why you do you write the kets of Psi and Chi with a subscript 1 and 2 ? I don't understand...

[@ProfessorMdoesScience]

The subscript indicates from which state space making up the tensor product state space those particular states originate. It may be helpful to follow the full playlist on identical particles, as this notation is explained there: kzbin.info/aero/PL8W2boV7eVfnJ6X1ifa_JuOZ-Nq1BjaWf I hope this helps!

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- Quantum mechanics. Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality! Commutators (Fermions) are dual to anti-commutators (Bosons). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation). The big bang is an infinite negative curvature singularity -- non null homotopic. Gaussian negative curvature is defined using two dual points. Singularities are dual:- Positive curvature (synchronic points) is dual to negative curvature (enchronic points) -- Gauss, Riemann geometry. Same (symmetry, summations) is dual to difference (anti-symmetry, differences). Bosons like to be in the same state, Fermions like to be in different states. "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero -- non null homotopic. Points are dual to lines -- the principle of duality in geometry!

[@phelipeantonie6638]

Perfect video!!! Cheers from Brazil

[@ProfessorMdoesScience]

Glad you like it! :)

[@bhoopendragupta4782]

This video series was awesome. I have learnt more from this KZbin channel than my college. We are so lucky to have this quality education provided us for free. This channel deserves more views and subscribers. Kudos 👍👍

[@ProfessorMdoesScience]

Great to hear you like our videos! May I ask what College you study at?

[@bhoopendragupta4782]

@@ProfessorMdoesScience I am from India and I am currently pursuing my m.sc in physics from one of the IITs.

[@micahshaw782]

Thank you so much for these videos, these videos are so great and helpful!

[@ProfessorMdoesScience]

Glad you like them!

[@robwindey9223]

Great explanation!

[@ProfessorMdoesScience]

Glad you liked it! :)

[@muhammadtanveer1747]

Thanks a lot

[@sayanjitb]

Dear sir, my first question, 1. Is V(N fold tensor product space) a direct sum of V_+ and V_--? 2. If I apply symmetrizer and antisymmetrizer operator on any kets in V_(\psi) then it can be projected onto V_+ as well as V_-. Then how can one categorize which states correspond to which particle unambiguously? Because from a general state I can construct a totally symmetric and antisymmetric state simultaneously. TIA

[@ProfessorMdoesScience]

Good questions: 1. No, VN is not in general the direct sum of V+ and V-. This means that in general there are states in VN that don't belong to either V+ nor V-. An exception is N=2, where VN *is* the direct sum of V+ and V-. 2. The symmetrizer and antisymmetrizer are different operators, so you can only apply one or the other. If you apply the symmetrizer, you end up in V+, and if you apply the antisymmetrizer, you end up in V-. V+ and V- are non-overlapping, so you can never have a simultaneous symmetric and antisymmetric state. I hope this helps!

[@salmanshahid2677]

I am a bit confused about what was said about exchange degeneracy in this video. To illustrate , I'll use the SG 2 particle system in your video on exchange degeneracy. Let's take the starting state as |up>|down>. I apply the permutation OPs to get a set of 2 vectors. The problem was that if we physically argued that these 2 permutations are the same, we'd have to say the same about their superpositions , and this was the problem. In this video, around 9:00, you say that we start with some state and symmetrise it- that's fine. You then say that we can apply a permutation on that same starting state and we show that we get the same result upon symmetrisation. This is all well and good but is not a complete answer to the exchange problem. If this starting state is |up>|down>, we've only shown that 2 states ( from all the possible combinations) symmetrize to the same state in V+ . I understand that proving the same for any arbitrary combo is straightforward too , but I just wanted to check if my concern here is valid.

[@ProfessorMdoesScience]

You are correct that what we've shown is that any permutation of a given state symmetrizes to the same state. If I then understand what you are saying, it is that we should have also said that this implies that any linear superposition of all possible permutations then symmetrizes to the same state. You are absolutely correct about this, and as you also say, it is a straightforward consequence of what we've shown. Thanks for pointing this out! :)

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- Quantum mechanics. Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality! Commutators (Fermions) are dual to anti-commutators (Bosons). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation). The big bang is an infinite negative curvature singularity -- non null homotopic. Gaussian negative curvature is defined using two dual points. Singularities are dual:- Positive curvature (synchronic points) is dual to negative curvature (enchronic points) -- Gauss, Riemann geometry. Same (symmetry, summations) is dual to difference (anti-symmetry, differences). Bosons like to be in the same state, Fermions like to be in different states. "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero -- non null homotopic. Points are dual to lines -- the principle of duality in geometry!

[@farzin393]

Thanks so very much for the great videos! I have a question: I am puzzled about the projector onto the symmetric subspace, or \hat{S}_{+}. Your definition is of course correct, but you expand it for binary group as 1/2(\hat{P}_{12}+\hat{P}_{21}). What are \hat{P}_12 and \hat{P}_21? can you write them explicitly? I might understand what you meant to say though, but doesn't the symmetric group consist of {1,\pi}, with identity 1 and \pi a transposition swapping the two elements? and then the projector would be 1/2(1+\pi); and of course \pi can be realized by a unitary. Thank you again!

[@ProfessorMdoesScience]

Thanks for watching! We are using the P_{12}, P_{21} notation as it allows us to more easily generalize to more complex situations. But you are correct that in the example of two particles, we could have used a simpler notation. Indeed, P_{12} is the identify operator and P_{21} is the operator that swaps the two particles. We go over the precise definition of the "P" operators in this video: kzbin.info/www/bejne/o5jUqaytj7KHoNU I hope this helps!

[@farzin393]

@@ProfessorMdoesScience Thank you very much for clearing it up.

[@themorrigan3673]

Just a question, hoping I could have some enlightenment, how would I be able to get the ground state for 3 noninteracting fermions in a given potential given that these 3 fermions are all in the same spin state

[@themorrigan3673]

Does it mean that each fermions must be in different energy state ?

[@ProfessorMdoesScience]

The 3 fermions need to be in a different state, but this doesn't necessarily mean a different energy, as we could have a degenerate energy with multiple distinct states available. I hope this helps!

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- Quantum mechanics. Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality! Commutators (Fermions) are dual to anti-commutators (Bosons). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation). The big bang is an infinite negative curvature singularity -- non null homotopic. Gaussian negative curvature is defined using two dual points. Singularities are dual:- Positive curvature (synchronic points) is dual to negative curvature (enchronic points) -- Gauss, Riemann geometry. Same (symmetry, summations) is dual to difference (anti-symmetry, differences). Bosons like to be in the same state, Fermions like to be in different states. "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero -- non null homotopic. Points are dual to lines -- the principle of duality in geometry!

[@shobhitkhajuria3120]

hi, nicely explained. I had questions regarding this. its troubling that it is an extra postulate! why cant we prove this using the 1. identical (permutation invarient)2. indistinguishable( entangled, wave functions cant seperate) . why other states which differ by a phase factor(upon permutation are exculded? is it because permmutation is hermitian and commutes with H so the maxiaml set of commutation observable thing is take into account? so only +1,-1 are taken? can you elobrate on this? because if all this works, it must not be a postulate and can be proven.

[@prathamhullamballi837]

Great video! I have a question: You applied the symmetrizer to the identical particles system state and showed that for psi and P(psi), the resultant vector is the same and are totally symmetric, and therefore as symmetrization leads to the same vector for any permutation operator P, we effectively solved the exchange degeneracy problem. If we do the same thing by applying the antisymmetrizer, for psi, we get S_(psi) but for P(psi), we get S_P(psi)=etaS_(psi), but these two aren't equal in general, so how do we say that totally antisymmetric case also solves the exchange degeneracy problem?

[@nitink4076]

Are V+ and V- subspace of VΨ?

[@ProfessorMdoesScience]

V+ and V- are subspaces of V.

[@hyperduality2838]

@@ProfessorMdoesScience V+ is dual to V- -- a dual space.

[@orionyeung1654]

These videos are great! I started at the one on the identical particles, so maybe I should have started earlier or missed something in that one. But I don't see why we couldn't have postulated that there are m fold symmetries, although I do see that it would be odd to have them. Could we have that P_alpha^m for a specific integer m be the identity instead of solely having m=2? We could keep unitarity by adding some phase that's a multiple of pi/m. It seems almost reasonable to have these other particles since the Fock space isn't covered by the totally anti/symmetric states. Would m have to be prime for particles to keep their new-term-here-ness? The projectors seem like they might be messy. I could keep going but I might just spin my wheels. If you have any resources like a textbook (or another one of the videos you guys have would be great! They're done very well!) please let me know.

[@ProfessorMdoesScience]

This is a great question! The answer is actually a topic of current research, and it turns out that it is possible to have something intermediate between fermions and bosons. Such particles are called "anyons". There are a few texts that cover them, for example "Anyons: Quantum Mechanics of Particles with Fractional Statistics" by Lerda, but a better starting point may be to simply Google the term. I hope this helps!

[@orionyeung1654]

Oh wow, thank you! I'll look around

[@rodrigoappendino]

I das searching for one particle operator and many particle operator. But it seems that you stopped with the identical particle videos. :(

[@ProfessorMdoesScience]

I think you found it now, but in general they would be in the same playlist :)

[@vaanff1942]

Nice

[@imaginarynumbers-e5n]

I do not understand how does thus prove that all particles are either fermions or bosons. Could someon explain it to me? Thanks

[@ProfessorMdoesScience]

In basic quantum mechanics (what we are covering here), we take this as a "postulate", we don't derive it. I hope this helps!

[@iotaphysics909]

You are making us not to attend our college classes as i get much more knowledge within your short videos🥰🥰🥰

[@ProfessorMdoesScience]

Glad to hear you find the videos helpful!! :)

[@hyperduality2838]

Knowledge is dual to according to Immanuel Kant -- synthetic a priori knowledge! The Schrodinger representation is dual to the Heisenberg representation -- Quantum mechanics. Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality! Commutators (Fermions) are dual to anti-commutators (Bosons). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation). The big bang is an infinite negative curvature singularity -- non null homotopic. Gaussian negative curvature is defined using two dual points. Singularities are dual:- Positive curvature (synchronic points) is dual to negative curvature (enchronic points) -- Gauss, Riemann geometry. Same (symmetry, summations) is dual to difference (anti-symmetry, differences). Bosons like to be in the same state, Fermions like to be in different states. "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero -- non null homotopic. Points are dual to lines -- the principle of duality in geometry!

[@nitink4076]

Correct me if I am wrong, “If we consider the spin degree of freedom of a system of two fermions, the system can either be in the state ½[|↑,↓> - |↓,↑>] or in the state ½[|↓,↑> - |↑,↓>]. Both of these states are equivalent and will lead to same physics (just like as in the case of the global phase factor example). It is because for probability, we are taking modulus square.”

[@hyperduality2838]

Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality! Commutators (Fermions) are dual to anti-commutators (Bosons). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation). The big bang is an infinite negative curvature singularity -- non null homotopic. Gaussian negative curvature is defined using two dual points. Singularities are dual:- Positive curvature (synchronic points) is dual to negative curvature (enchronic points) -- Gauss, Riemann geometry. Same (symmetry, summations) is dual to difference (anti-symmetry, differences). Bosons like to be in the same state, Fermions like to be in different states. "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero -- non null homotopic. Points are dual to lines -- the principle of duality in geometry!

[@xarhspapapadatos]

I don't get why you say that the symetrization postulate is required. Why don't we just say that the a,b factors depend on the preperation of the quantum mechanical system. Our professor showed that two identical particles must exist in a symmetric or an antisymmetric state and nothing else. He then showed that the mean value of the P12 operator does not change through time meaning that in a two particle universe, these two particles must have a specific, innate quality that determines their symmetry. What I find hard is proving the same thing for an N number of particles.

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- Quantum mechanics. Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality! Commutators (Fermions) are dual to anti-commutators (Bosons). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation). The big bang is an infinite negative curvature singularity -- non null homotopic. Gaussian negative curvature is defined using two dual points. Singularities are dual:- Positive curvature (synchronic points) is dual to negative curvature (enchronic points) -- Gauss, Riemann geometry. Same (symmetry, summations) is dual to difference (anti-symmetry, differences). Bosons like to be in the same state, Fermions like to be in different states. "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero -- non null homotopic. Points are dual to lines -- the principle of duality in geometry!

[@ProfessorMdoesScience]

The values of a and b don't depend on the preparation of the system, what the symmetrization postulate tells us is that they depend on the precise nature of the set of identical particles (whether they are bosons or fermions). I hope this helps!

[@xarhspapapadatos]

@@ProfessorMdoesScience Yeah I know it does and thanks for your answer. What confused we was that you said that not knowing the precise value of a and b could be a dead end if we didn't have the symetrization postulate. When it could easily depend on the preperation of the system. The reasoning is what I thought was flaud

[@ProfessorMdoesScience]

@@xarhspapapadatos Ok, this builds from the previous video where we did set up the system in a very specific manner, and even in that case unless we have the symmetrization postulate we cannot uniquely determine the values of a and b. I think we are on the same page now :)

[@xarhspapapadatos]

@@ProfessorMdoesScience Thanks for responding. It's clear now

[@JohnVKaravitis]

If dark matter is a thing, then it s/b a fermion, no?

[@ProfessorMdoesScience]

What do you mean by s/b? The supersymmetric partner of a boson? If so, then yes, it is a fermion!

[@JohnVKaravitis]

@@ProfessorMdoesScience You know exactly what I mean, and you've avoided answering my question directly.

[@mariak8480]

@@JohnVKaravitis actually he did answer your question very directly.

[@hyperduality2838]

@@ProfessorMdoesScience Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality! Commutators (Fermions) are dual to anti-commutators (Bosons). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation). The big bang is an infinite negative curvature singularity -- non null homotopic. Gaussian negative curvature is defined using two dual points. Singularities are dual:- Positive curvature (synchronic points) is dual to negative curvature (enchronic points) -- Gauss, Riemann geometry. Same (symmetry, summations) is dual to difference (anti-symmetry, differences). Bosons like to be in the same state, Fermions like to be in different states. "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero -- non null homotopic. Points are dual to lines -- the principle of duality in geometry!

[@prasadpawar7027]

Quantum Mechanics is saved and physicists still get to keep their jobs.

[@ProfessorMdoesScience]

Phew! ;)

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- Quantum mechanics. Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality! Commutators (Fermions) are dual to anti-commutators (Bosons). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation). The big bang is an infinite negative curvature singularity -- non null homotopic. Gaussian negative curvature is defined using two dual points. Singularities are dual:- Positive curvature (synchronic points) is dual to negative curvature (enchronic points) -- Gauss, Riemann geometry. Same (symmetry, summations) is dual to difference (anti-symmetry, differences). Bosons like to be in the same state, Fermions like to be in different states. "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero -- non null homotopic. Points are dual to lines -- the principle of duality in geometry!

[@DrMarcoArmenta]

OMG!

[@ProfessorMdoesScience]

I hope this is good! :)

[@DrMarcoArmenta]

@@ProfessorMdoesScience It is! You blew my mind!

[@geralt_rivia]

Talk too fast. I cannot follow

[@ProfessorMdoesScience]

Thanks for the feedback! We are trying to slow down in the more recent videos. But you can also try to use the KZbin speed function to slow the video down!

[@nitink4076]

Correct me if I am wrong, “If we consider the spin degree of freedom of a system of two fermions, the system can either be in the state ½[|↑,↓> - |↓,↑>] or in the state ½[|↓,↑> - |↑,↓>]. Both of these states are equivalent and will lead to same physics (just like as in the case of the global phase factor example). It is because for probability, we are taking modulus square.”