Great video! I remember in my quantum class my ears definitely perked up when we derived that sum of squares formula for the energy; I knew there would be some familiar number theory lurking there but sadly we didn't cover it haha
Haha, this is exactly how I got inspired to make this video! We covered it in class, and I immediately knew there would be some interesting number theory involved, especially because I self-studied some of modular forms theory with application to counting points on other lattices (like the E8 which I made a video about). So cool that this theory becomes involved with such a prototypical quantum system
Very nice. I'm firmly convinced that we are on the verge of a new wave in quantum physics where number theory and game theory will rewrite the paradigms completely, particularly p-adic and surreal number based analysis.
We have this exact same feeling in geometric algebra, category theory, Lie theory, noncommutative geometry, etc. I think the verge we are on is realizing just how many ways we can reformulate the same things in order to gain deeper understanding.
When two subjects in mathematics have a big correspondence with each other, you would be able to construct the same complicated objects. This is really rare, and it doesn't seem to be the case. And let's be real, thinking something can be done without actually doing anything about it is just a big waste of time. Even if you did turn out to be right, it doesn't unwaste this time.
The basic overview of the connection is that the Jacobi theta function has the q-series 1 + 2q + 2q^4 + 2q^9 + ..., so it is the q-series generating function for the number of integers with a particular norm. Therefore, its square is the generating function for the number of vectors in the 2d grid lattice with a particular norm. Using the combinatorics argument from 8:08, it's not too hard to see the q-series generating function for the degeneracy would be (theta^2 - 2 * theta + 1) / 4. Getting from this q-series generating function to the partition function from statistical mechanics just involves associated the inverse temperature with a substituted complex variable (like a Wick rotation if you've heard of it). It's something I worked out on my own, so I haven't found a source yet. I also am tentatively planning to make a video on this connection between the 2d box's partition function and the Jacobi theta function because I think it's deeply interesting and obscure. However, I can send several sources about the connection to the Dirichlet characters using the Jacobi theta function if you're interested.
@@worldequation thank you for the answer. Yes I am interested . I think the video you are planning to do (about the connection) is a good idea. I wonder if the average energy E=(d/dв ln(Z(в)) will be asymptomatically equivalent (for T very big or on the contrary T very small) to a nice formula (maybe E~C*KT ? where K is the bolzmann constant). If it's the case, we could have nice interpretation
After some test, I think that for d dimensional box, E~d/2*kT for large T. (Because -d/dx ln( EllipticTheta[3,exp(-x)])~1/2x for x~0). This have a nice interpretation because this is the same energy of a classical system with d degree of freedom. Therefore, each dimension is analogous to one degree of freedom. Maybe my maths are wrong, I used Wolfram alpha to test my conjecture
@@rainbow-cl4rk Wow, that's a super interesting route to take. I had been thinking a lot about average energy for this system because the logarithmic formula suggests to me that the Jacobi triple product identity should be used. I would have to think about it more, but I bet the asymptotics of the average energy would allow the use of that identity and give a clean result. I am absolutely going to look into this more
@@worldequation you could use the Poisson summation formula, For example in the 1d case: the sum of exp(-ak²) transform to sqrt(pi/a)*the sum of exp(-k²/a) which is asymptomatically equivalent to sqrt(pi/a). Likewise, using the Poisson summation for k²exp(-ak²) you show that this is asymptomatically equivalent to sqrt(pi/a)*1/(2a). Therefore the mean energy is asymptomatically equivalent to 1/(2a). since a=1/(kT), it shows that this is equivalent to 1/2*kT. Now for the d dimensional case, Z~cst*(sum exp(-ax²))^d, therefore E is equivalent to d/2*kT. Note that this is only true for T->∞. (But the approximation is good for T>>Tc=E0/K)
Awesome video! Thank you. Although it was really difficult to hear your words even with the maximum volume. I had to rely on the automatic close captions. Please raise the volume for the next videos.
Oh my God, this video is a masterpiece, what a beauty!! What genius!!! I think if we draw a path between physics and number theory, we will solve many problems stumbling blocks Thank you from the heart❤
As summary: a degeneracy is a natural number of different states that participate a particular energy level and it's scaled by counting the number of vectors that has norm equals that energy level in 1st quarter of the space summing the values of the Chi function of the factors of the number that represents the certain energy level if this last is not square, otherwise we subtract 1 from the sum. But, before we find it, we need to check whether the energy is allowed or not in a specific value which if it is square and there exists a one prime factor such this factor congruences with 1 mod 4 or it's not square and no one of its odd powered prime factor doesn't congruence with 3 mod 4 then the energy is allowed otherwise it is not. The sequence of degeneracies is a function that is generated by another function called "Partition function" that encodes the probabilities of a system being at any energy level and it is formulated by 'Jacobi Theta function', These two functions can be used to calculated the number of the vectors. If I made a fault, please tell me in the comments. And thank you so much because you show me an importance of Number Theory that improve me more to study this field.
I like this summary a lot, but I have one small correction. Whether the energy is allowed or not does not have to be checked before finding the degeneracy. Because there is a simple way to check whether the degeneracy is greater than 0 (meaning there are states with that energy), it follows that we could use the method I showed to check whether an energy is allowed. It's a neat corollary of the degeneracy formula that can be leveraged to make a very hard problem (that of asking is this energy allowed) more feasible without resorting so much to brute force.
Great video !! Thank you so much 😊 Micro detail, norm and norm squared should be differentiated for the || λ x || = |λ| ||x|| axiom of a norm. Also, for a little complement, the solution to the number theory problem commences when one considers the irreducibles (primes) of Z[i], aka Gauss’s integers (they are like the complex numbers but with integer coordinates, one can see the correspondence with the lattice in the present video)
Oh, you're totally right. I see your point, and technically, what I called the 'norm' it seems is not a proper a norm. Though, I have seen math PhDs call this squared norm the norm, perhaps to be concise. I'm not sure, but thanks for the correction!
@@worldequation Yep, the norm of an algebraic number in number theory (en.wikipedia.org/wiki/Field_norm) and norm in functional analysis are different functions. So it depends on whihc math PhD you ask :)
Very well presented, such a beautiful connection! I wonder if there is a generalization of the counting problem to higher dimensional lattices? Also, I'd really appreciate if you can provide the resources you used ^^
Thankyou for the nice video. I'm trying to understand the actual wave functions as products. While the versions you show at 6:13 make sense, I cannot see the more complex one at 19:10 are simple products. Is there something I'm missing? Also, are there ways of dealing with "near" but not "exact" degeneracies in number theory as well? Can you provide a good review paper or book chapter.
About the different versions of the wavefunction, the technical detail is that the simple, static graphs are actually showing the real-valued, time-independent factor of the full complex wavefunction. There is an extra complex-valued factor which includes a time-varying phase. However, this factorization can only be done if the states is time-independent, so it is an eigenstate of the Hamiltonian (it has a definite energy). The more complicated animations such as at 19:10, are time-dependent because they are superpositions of several definite-energy eigenstates. Therefore, the actual probability distribution that I plot in those animations is changing over time. However, it doesn't actually fully matter that these states exist and are not accounted for when we count degeneracy. Basically, this is because we only ever observe a quantum system having a definite energy when we make the measurement. Very long explanation, but I hope that helped.
@@worldequation Thanks for the thorough reply. I guess what I meant for the second question was if many energy levels are close (m2+n2 close to x2+y2 for integers m n x y, but not exactly equal) they effectively act as one degeneracy...so although this is very beautiful math, it might be really getting into the weeds to demand exact energy equivalence for large systems, and I was wondering if distributions in number theory might help.
@@zackbarkley7593 I see what you mean. Certainly, I have seen systems like this get treated in a sort of classical limit, where the energy is viewed as a continuum since the quanta are small relative to the scale of the energy. For instance, this particle in a box example often gets treated this way when studying the Planck function for blackbody radiation. I often see people make a geometric argument to do this, acting as if the lattice has simply become a continuum and treating the density of states as being given in terms of circumferences in the momentum space. I like this view and think it's intuitive, but it could possibly be made more rigorous with a number theory argument. I'm not entirely sure. I would honestly imagine that the more interesting thermodynamic quantities could be understood asymptotically in the high temperature limit using the result that the Jacobi theta function is related to the partition function. This seems like a much more practical connection to number theory in my opinion, especially with the many identities, symmetries, and properties of the Jacobi theta function.
@worldequation 😊 thanks. Yes it would be interesting to see what deviations (or perhaps determination of some dimensionless constants) would result from using the details from number theory applied to statistics so those weeds added up to something interesting in the whole that couldn't be derived only from pure statistical arguments and assumptions.
I think the flowchart at 17:43 would be easier to understand had we used Z[i] and its complex prime factorization, however both approaches are equally valid.
Why do you think f should be proportional to n? Remember that f is the temporal frequency. The frequency of the wave that you see in the pictures is the spatial frequency. That is indeed proportional to n as you can see in the pictures: for n=1 the wave goes through half a cycle over the length of the box, for n=2 it's a full cycle, for n=3 it's 3/2 of a cycle, etc. And instead of being proportional to energy, the spatial frequency is proportional to momentum p. That means p is proportional to n. We consider the particle to only have kinetic energy, so therefore E=p²/m. So E is proportional to p² which itself is proportional to n².
Do you have any literature relating these two things in a deeper way? Like the sum of chi is an important factor behind proving non-vanishing of L(1,chi) for non-principal chi s, which ultimately proves Dirichlet's theorem. Is this related to some physics?
Unfortunately, I don't have any literature explicitly on the connection to physics as I worked much of this out by observing the connection this physics problem would have to number theory on my own. However, I did think about this potential connection to L functions and want to explore it as I learn more about analytic number theory
Nice video! One thing I wonder - since factoring large numbers is hard, is this method still practical for counting degeneracies for large energy levels? Would asymptotics be more useful for determining statistical properties of the system?
I absolutely think so. I have been exploring this since making the video. I thought it would be interesting to find an asymptotic for the probability that any energy is allowed, almost like the result from the prime number theorem. I haven't tried yet, but I think it could be done with not much more work than what was laid out in this video. On the other hand, most important thermodynamic quantities are given using the partition function, the generating function at the end. It may also be interesting to find asymptotics of energy, entropy, free energy, etc as another commenter mentioned.
@@worldequation I know of some corrections to the asymptotic pi*R^2 of number of lattice points within a disk of radius R, which intuitively I’d expect correspond to lattice points on the boundary (the correction decreases in R, something like R^{-1/2}). It involves some harmonic analysis, Stein and Shakarchi Functional Analysis chapter 7 has some stuff in this
@@kieransquared Very interesting! I presumed a result like that might exist, so thanks so much for sharing! I feel like the result in this video about the formula for counting vectors on the lattice could be used in proving asymptotics such as this one, so I might need to study this more deeply
It's a proof that was a little too long to include in this video, but my favorite proof uses a generating functions approach where the number of vectors with a certain squared norm n is the coefficient of q^n in a q-series. It's not difficult to show this generating function is the square of the Jacobi theta function, but that square of the Jacobi theta function is a very particular type of modular form. There is only one modular form of that type up to complex multiples, so it is easy to determine a formula for the coefficients simply by comparing the leading coefficient.
Cool video! I suppose if instead of a 2D box, we say “two distinguishable particles (of the same mass) in a one-dimensional box”, this would physically justify the two side lengths of the 2D box being exactly equal. If we change it to make the two particles indistinguishable (say, in a boson-y way), then, for all the energy levels other than 2 k^2 , it cuts the degeneracy in half, while for those of energy 2 k^2 it, uh, cuts it in half except rounded up? I wonder, how much of the connection between the Jacobi theta function and the partition function, would remain if we made the particles indistinguishable like this? For most of the coefficients in the partition function, they would just be halved, And so I guess if you subtracted half the original partition function from the version of the partition function obtained after making the two particles indistinguishable, You would be left with only terms for energies 2 k^2 , each with a coefficient of (1/2). So, I guess, the relationship would be pretty close? Uh, I forget if the partition function had exp(-t E) or exp(i t E). If the latter, I would be concerned as to whether this sum converges (for real t). But, for the former, should converge for positive t. And then could probably analytically continue I guess. So, seems like the answer would be “it still relates pretty closely”.
Yeah, that's very interesting to think about! I haven't thought about the situation with indistinguishable particles yet. However, I did think about how n particles in 1D is equivalent to a n-dimensional box, and there are really fascinating results that can be explored through that lens. Specifically, 4 particles in 1D (or a particle in a 4-dimensional box) would have energies that are a sum of four positive perfect squares. There's a famous theorem that every integer can be represented as the sum of four squares, and it's provable using generating functions and modular forms arguments on the Jacobi theta function, once again. I'm not entirely sure how the situation changes when you consider that none of the quantum numbers can be zero, though.
@EricDMMiller number theory and physics. You use math in physics But do you care that there exist atleast a prime between n^2 and (n+1)^2 for all n positive integers in physics? Physic use some parts of mathematics but not all of it and number theory is the subject that is not really used to solve physic problems (untill now lol).
@@aweebthatlovesmath4220O don't even know of any part of Mathematics that use all results from another branch of Mathematics. So it is a non issue if physics doesn't and it doesn't make math less 'real'. Sometimes it takes 2000 years for a construct in mathe to blossom, so who knows when must you use set theory to prove Something in physics. 😉
Awesome great work ! Am guessing that the sum of characters can never go negative wonder how you would prove that? Also wondering how this relates to fifferent degeneracy/ statistics for fermions , bosons etc. Maybe subjects for another video ? 😊
Thank you! Yeah it doesn't seem obvious that the sum would never be negative, but once you put it into the product form, none of those factors can be negative interestingly. Definitely expect some more videos on statistical mechanics from me as it's one of my favorite physics topics
If a quantum system reaches thermal equilibrium with a reservoir at a temperature T, then we can consider the probabilities of a quantum system exhibiting a state given that it is in equilibrium with a reservoir of temperature T. This is the temperature that I refer to in this video. Statistical mechanics is hardly just a description of classical uncertainty, and there are sensible ways in which quantum mechanics gives rise to thermodynamics as well. This is the field of quantum statistical mechanics. For instance, it’s not hard to determine the thermodynamic properties of the quantum harmonic oscillator. Let me know if there’s something I’m missing
@@worldequationI am sorry for the previous tersely formulated comment, I just got triggered by your use of T=300K around 4:55 for a state that I would hazard to guess is a pure state (which technically isn't at any temperature). The actual state at that temperature would be of the form e^(-H/300), a density matrix I honestly wouldn't have the slightest idea how to visualise. The rest of the video is really good! On the wider point, statistical mechanics whether, classical or quantum, works iff you assume classical uncertainty (or lack of knowledge). It is a very effective description of large scale systems and temperature appears in such as an effective phenomenon. However, (probably due to a personal distaste for probability theory) I have always disliked temperature being viewed as fundamentally physical, since it emerges iff we deliberately smear out a perfectly deterministic system into a probabilistic one, which does not happen in "reality". Physical systems should be viewed as being in pure states (most of which do not have a temperature associated with them other than ground states) and they never reach anything like thermal equilibrium since their evolution is deterministic/unitary.
I see what you’re saying. To address the graphic at 4:55, this might have been a poor way of showing what I was trying to show. Basically, I just used the particle in a box graphic I’d been using throughout the video to have a visual representing putting that quantum system into thermal equilibrium with a reservoir at a temperature. I just chose room temperature because this is a temperature I naturally think of and know kT. I wasn’t choosing this temperature and the graphic to match, so sorry for the confusion. You’re absolutely right that such a low energy state would not be probable at room temperature. I also sympathize with the concern for how temperature gets viewed as fundamental when it isn’t in actuality. If I went more into the thermodynamics, I would have been more careful with this idea and what I mean by this to avoid assigning a temperature to a single particle, which is certainly not a good way to think about this. So thanks for mentioning this because it’s definitely important to get straight. I see how somebody could misinterpret that part of the video as saying that the particle has an inherent temperature or that the graphic is an accurate depiction of the system at room temperature or even that the probability density from the wavefunction is somehow associated to the purely statistical probability (all misconceptions I want to avoid). Thank you for giving me things to be more conscious of!
QM classicalized in 2010. Juliana Mortenson website Forgotten Physics uncovers the hidden variables and constants and the bad math of Wien, Schrodinger, Heisenberg, Einstein, Debroglie,Planck,Bohr etc. A proton is a collection of 1836 expanding electrons and add a bouncing expanding electron makes a hydrogen atom. Electron mass (9.11) multiplied by 1836 equals the proton mass (1.67). All atoms and atomic objects are expanding at 1/770,000th their size per second per second constant acceleration. Multiplied by earth’s radius equals 16 feet etc.: gravity.” The Final Theory: Rethinking Our Scientific Legacy “, Mark McCutcheon.
I find all this a little bit artificial: Oh look, my physical problem has some rules, I have no idea why, but as I know a math domain which ensure these rules, there must be a connection between physics and some math domain. Oh look: I have a set of small cubes that make a big cube and I want to make 2 smaller cubes without letting some on the side, but I can't ! So number theory must fundamentaly be connected to the real world because Fermat-Wiles theorem for n=3 also apply in the real world with true boxes made of wood. Or stones. Whatever. And oh, look, if I take 2 stones and 4 sticks, I can put all my items in two bags with the same number of pieces in each bag. Isn't that fundamentally incredible ? That the worlds know about mod 2 aritthmetics ?! That maths work with physics is not a mistery at all, whatever Wigner might have said. This is a false paradox and a fake astonishment. If the "real world" is consistent, then it must, by definition of consistency, follow any language based on consistency. Mathematics are nothing else but a self-constructed language with consistency as its unique requirement. The mistery is not that "maths work baby", the mistery is that the physical world is consistent. They don't work more with the 2d quantum box and Dirichlet characteristics than they worked 5000 years ago with classical stones in a bag and simple addition and multiplication. As long as the world will follow rules, any rules, then maths will work by definition of what it means to follow rules. If they didn't, would'nt you say there is an error in your mathematical demonstration and start to look for that error until you found it ?
Would you point me to a source on the proper pronunciation? I have seen it pronounced several ways and stuck with one for this video. It seemed to me that there might be several acceptable pronunciations, but I could very well be wrong.
How is it possible the Pythagoras rears his ugly head in QM. That should be illegal. Or, more likely, point masses are approximations of quantum particles meaning spherical coordinates are advantageous.
@@worldequation for the 2D particle box the possible energy levels are Pythagorean vectors where the norm squared is the result. This has a polar representation r = C where C is the energy level squared. Bumping this up to 3 dimensions yields a sphere with a radius rho = C. My thought is that this is a consequence of the polar nature of a point mass. The possible states are the intersection of all of the pure state circles/outputs of the chi function.
@@dominicellis1867 I would suggest it is probably more accurate to think of these vectors in the quantum number space as being in something like a momentum space. Therefore, a magnitude in this space is really the magnitude of the particle's momentum upon measurement.
Yeah, I noticed that after I uploaded it and unfortunately can't make it louder after it's uploaded. It's loud enough for me if I turn the volume up, but sorry about that
As Eugene Wigner once put it, mathematics is “unreasonably effective” in the natural sciences. This is part of its beauty.
Great video! I remember in my quantum class my ears definitely perked up when we derived that sum of squares formula for the energy; I knew there would be some familiar number theory lurking there but sadly we didn't cover it haha
Haha, this is exactly how I got inspired to make this video! We covered it in class, and I immediately knew there would be some interesting number theory involved, especially because I self-studied some of modular forms theory with application to counting points on other lattices (like the E8 which I made a video about). So cool that this theory becomes involved with such a prototypical quantum system
Fascinating video! Love how you connected the two subjects!
Fantastic explanation of the law of quadratic reciprocity was my take away.
This is the connection I was looking for.
Cool video! Look up quantum carpets, another example of algebra/number theory entering QC
Wow, this looks incredible! I've never heard of this before, but it seems somewhat related. You've given me a new thing to read about, thanks!
Very nice. I'm firmly convinced that we are on the verge of a new wave in quantum physics where number theory and game theory will rewrite the paradigms completely, particularly p-adic and surreal number based analysis.
We have this exact same feeling in geometric algebra, category theory, Lie theory, noncommutative geometry, etc.
I think the verge we are on is realizing just how many ways we can reformulate the same things in order to gain deeper understanding.
When two subjects in mathematics have a big correspondence with each other, you would be able to construct the same complicated objects.
This is really rare, and it doesn't seem to be the case.
And let's be real, thinking something can be done without actually doing anything about it is just a big waste of time.
Even if you did turn out to be right, it doesn't unwaste this time.
Do you have any source for the link between the partition function and jacobi theta function ?
The basic overview of the connection is that the Jacobi theta function has the q-series 1 + 2q + 2q^4 + 2q^9 + ..., so it is the q-series generating function for the number of integers with a particular norm. Therefore, its square is the generating function for the number of vectors in the 2d grid lattice with a particular norm. Using the combinatorics argument from 8:08, it's not too hard to see the q-series generating function for the degeneracy would be (theta^2 - 2 * theta + 1) / 4. Getting from this q-series generating function to the partition function from statistical mechanics just involves associated the inverse temperature with a substituted complex variable (like a Wick rotation if you've heard of it). It's something I worked out on my own, so I haven't found a source yet. I also am tentatively planning to make a video on this connection between the 2d box's partition function and the Jacobi theta function because I think it's deeply interesting and obscure. However, I can send several sources about the connection to the Dirichlet characters using the Jacobi theta function if you're interested.
@@worldequation thank you for the answer. Yes I am interested . I think the video you are planning to do (about the connection) is a good idea. I wonder if the average energy E=(d/dв ln(Z(в)) will be asymptomatically equivalent (for T very big or on the contrary T very small) to a nice formula (maybe E~C*KT ? where K is the bolzmann constant). If it's the case, we could have nice interpretation
After some test, I think that for d dimensional box, E~d/2*kT for large T. (Because
-d/dx ln( EllipticTheta[3,exp(-x)])~1/2x for x~0).
This have a nice interpretation because this is the same energy of a classical system with d degree of freedom. Therefore, each dimension is analogous to one degree of freedom.
Maybe my maths are wrong, I used Wolfram alpha to test my conjecture
@@rainbow-cl4rk Wow, that's a super interesting route to take. I had been thinking a lot about average energy for this system because the logarithmic formula suggests to me that the Jacobi triple product identity should be used. I would have to think about it more, but I bet the asymptotics of the average energy would allow the use of that identity and give a clean result. I am absolutely going to look into this more
@@worldequation you could use the Poisson summation formula,
For example in the 1d case:
the sum of exp(-ak²) transform to sqrt(pi/a)*the sum of exp(-k²/a) which is asymptomatically equivalent to sqrt(pi/a).
Likewise, using the Poisson summation for k²exp(-ak²) you show that this is asymptomatically equivalent to sqrt(pi/a)*1/(2a).
Therefore the mean energy is asymptomatically equivalent to 1/(2a).
since a=1/(kT), it shows that this is equivalent to 1/2*kT.
Now for the d dimensional case,
Z~cst*(sum exp(-ax²))^d, therefore E is equivalent to d/2*kT.
Note that this is only true for T->∞.
(But the approximation is good for T>>Tc=E0/K)
Awesome video! Thank you. Although it was really difficult to hear your words even with the maximum volume. I had to rely on the automatic close captions. Please raise the volume for the next videos.
Thanks! And sorry about the volume. I plan to fix it in the future
Oh my God, this video is a masterpiece, what a beauty!! What genius!!!
I think if we draw a path between physics and number theory, we will solve many problems stumbling blocks
Thank you from the heart❤
As summary: a degeneracy is a natural number of different states that participate a particular energy level and it's scaled by counting the number of vectors that has norm equals that energy level in 1st quarter of the space summing the values of the Chi function of the factors of the number that represents the certain energy level if this last is not square, otherwise we subtract 1 from the sum. But, before we find it, we need to check whether the energy is allowed or not in a specific value which if it is square and there exists a one prime factor such this factor congruences with 1 mod 4 or it's not square and no one of its odd powered prime factor doesn't congruence with 3 mod 4 then the energy is allowed otherwise it is not. The sequence of degeneracies is a function that is generated by another function called "Partition function" that encodes the probabilities of a system being at any energy level and it is formulated by 'Jacobi Theta function', These two functions can be used to calculated the number of the vectors. If I made a fault, please tell me in the comments. And thank you so much because you show me an importance of Number Theory that improve me more to study this field.
I like this summary a lot, but I have one small correction. Whether the energy is allowed or not does not have to be checked before finding the degeneracy. Because there is a simple way to check whether the degeneracy is greater than 0 (meaning there are states with that energy), it follows that we could use the method I showed to check whether an energy is allowed. It's a neat corollary of the degeneracy formula that can be leveraged to make a very hard problem (that of asking is this energy allowed) more feasible without resorting so much to brute force.
So we use degeneracy to check whether the energy is allowed or not. Thank you so much for correcting@@worldequation
Great video!
Very well presented. Thank you.
Great video !! Thank you so much 😊
Micro detail, norm and norm squared should be differentiated for the || λ x || = |λ| ||x|| axiom of a norm.
Also, for a little complement, the solution to the number theory problem commences when one considers the irreducibles (primes) of Z[i], aka Gauss’s integers (they are like the complex numbers but with integer coordinates, one can see the correspondence with the lattice in the present video)
Oh, you're totally right. I see your point, and technically, what I called the 'norm' it seems is not a proper a norm. Though, I have seen math PhDs call this squared norm the norm, perhaps to be concise. I'm not sure, but thanks for the correction!
@@worldequation Yep, the norm of an algebraic number in number theory (en.wikipedia.org/wiki/Field_norm) and norm in functional analysis are different functions. So it depends on whihc math PhD you ask :)
@@СтепанНестеров-р2р Thanks for the clarification 😄
This was a great video!
Very well presented, such a beautiful connection! I wonder if there is a generalization of the counting problem to higher dimensional lattices? Also, I'd really appreciate if you can provide the resources you used ^^
Wonderful video!
I m french and thank a lot for your work, I love your videos but I can t auto translate the subz but it s ok because you speak very clearly
Thankyou for the nice video. I'm trying to understand the actual wave functions as products. While the versions you show at 6:13 make sense, I cannot see the more complex one at 19:10 are simple products. Is there something I'm missing? Also, are there ways of dealing with "near" but not "exact" degeneracies in number theory as well? Can you provide a good review paper or book chapter.
About the different versions of the wavefunction, the technical detail is that the simple, static graphs are actually showing the real-valued, time-independent factor of the full complex wavefunction. There is an extra complex-valued factor which includes a time-varying phase. However, this factorization can only be done if the states is time-independent, so it is an eigenstate of the Hamiltonian (it has a definite energy). The more complicated animations such as at 19:10, are time-dependent because they are superpositions of several definite-energy eigenstates. Therefore, the actual probability distribution that I plot in those animations is changing over time. However, it doesn't actually fully matter that these states exist and are not accounted for when we count degeneracy. Basically, this is because we only ever observe a quantum system having a definite energy when we make the measurement. Very long explanation, but I hope that helped.
As for the "near" degeneracies, I'm not sure what you mean, but could you clarify?
@@worldequation Thanks for the thorough reply. I guess what I meant for the second question was if many energy levels are close (m2+n2 close to x2+y2 for integers m n x y, but not exactly equal) they effectively act as one degeneracy...so although this is very beautiful math, it might be really getting into the weeds to demand exact energy equivalence for large systems, and I was wondering if distributions in number theory might help.
@@zackbarkley7593 I see what you mean. Certainly, I have seen systems like this get treated in a sort of classical limit, where the energy is viewed as a continuum since the quanta are small relative to the scale of the energy. For instance, this particle in a box example often gets treated this way when studying the Planck function for blackbody radiation. I often see people make a geometric argument to do this, acting as if the lattice has simply become a continuum and treating the density of states as being given in terms of circumferences in the momentum space. I like this view and think it's intuitive, but it could possibly be made more rigorous with a number theory argument. I'm not entirely sure. I would honestly imagine that the more interesting thermodynamic quantities could be understood asymptotically in the high temperature limit using the result that the Jacobi theta function is related to the partition function. This seems like a much more practical connection to number theory in my opinion, especially with the many identities, symmetries, and properties of the Jacobi theta function.
@worldequation 😊 thanks. Yes it would be interesting to see what deviations (or perhaps determination of some dimensionless constants) would result from using the details from number theory applied to statistics so those weeds added up to something interesting in the whole that couldn't be derived only from pure statistical arguments and assumptions.
I think the flowchart at 17:43 would be easier to understand had we used Z[i] and its complex prime factorization, however both approaches are equally valid.
2:46 why is it quadratic in n? Because in frequency, I think it should be E = hf if f denotes the frequency and the frequency is proporional to n.
Why do you think f should be proportional to n? Remember that f is the temporal frequency. The frequency of the wave that you see in the pictures is the spatial frequency. That is indeed proportional to n as you can see in the pictures: for n=1 the wave goes through half a cycle over the length of the box, for n=2 it's a full cycle, for n=3 it's 3/2 of a cycle, etc. And instead of being proportional to energy, the spatial frequency is proportional to momentum p. That means p is proportional to n. We consider the particle to only have kinetic energy, so therefore E=p²/m. So E is proportional to p² which itself is proportional to n².
Do you have any literature relating these two things in a deeper way? Like the sum of chi is an important factor behind proving non-vanishing of L(1,chi) for non-principal chi s, which ultimately proves Dirichlet's theorem. Is this related to some physics?
Unfortunately, I don't have any literature explicitly on the connection to physics as I worked much of this out by observing the connection this physics problem would have to number theory on my own. However, I did think about this potential connection to L functions and want to explore it as I learn more about analytic number theory
20:54 Is that Langland’s program lurking over theeere ? (waiting on that epic functor)
Can you suggest some book where I can explore more about the connection between number theory and quantum physics?
Can you recommend any text book to read more about this topic ?
Nice video! One thing I wonder - since factoring large numbers is hard, is this method still practical for counting degeneracies for large energy levels? Would asymptotics be more useful for determining statistical properties of the system?
I absolutely think so. I have been exploring this since making the video. I thought it would be interesting to find an asymptotic for the probability that any energy is allowed, almost like the result from the prime number theorem. I haven't tried yet, but I think it could be done with not much more work than what was laid out in this video. On the other hand, most important thermodynamic quantities are given using the partition function, the generating function at the end. It may also be interesting to find asymptotics of energy, entropy, free energy, etc as another commenter mentioned.
@@worldequation I know of some corrections to the asymptotic pi*R^2 of number of lattice points within a disk of radius R, which intuitively I’d expect correspond to lattice points on the boundary (the correction decreases in R, something like R^{-1/2}). It involves some harmonic analysis, Stein and Shakarchi Functional Analysis chapter 7 has some stuff in this
@@kieransquared Very interesting! I presumed a result like that might exist, so thanks so much for sharing! I feel like the result in this video about the formula for counting vectors on the lattice could be used in proving asymptotics such as this one, so I might need to study this more deeply
Sorry, why does the sum of this Dirichlet character give the number of gaussian integers with a given norm?
It's a proof that was a little too long to include in this video, but my favorite proof uses a generating functions approach where the number of vectors with a certain squared norm n is the coefficient of q^n in a q-series. It's not difficult to show this generating function is the square of the Jacobi theta function, but that square of the Jacobi theta function is a very particular type of modular form. There is only one modular form of that type up to complex multiples, so it is easy to determine a formula for the coefficients simply by comparing the leading coefficient.
20:24 oooooooohhhh I’de love to get into that 😮🤩 still need a lot of study until getting there hahahaha
Cool video!
I suppose if instead of a 2D box, we say “two distinguishable particles (of the same mass) in a one-dimensional box”, this would physically justify the two side lengths of the 2D box being exactly equal.
If we change it to make the two particles indistinguishable (say, in a boson-y way), then, for all the energy levels other than 2 k^2 , it cuts the degeneracy in half,
while for those of energy 2 k^2 it, uh, cuts it in half except rounded up?
I wonder, how much of the connection between the Jacobi theta function and the partition function, would remain if we made the particles indistinguishable like this?
For most of the coefficients in the partition function, they would just be halved,
And so I guess if you subtracted half the original partition function from the version of the partition function obtained after making the two particles indistinguishable,
You would be left with only terms for energies 2 k^2 ,
each with a coefficient of (1/2).
So, I guess, the relationship would be pretty close?
Uh, I forget if the partition function had exp(-t E) or exp(i t E).
If the latter, I would be concerned as to whether this sum converges (for real t).
But, for the former, should converge for positive t. And then could probably analytically continue I guess.
So, seems like the answer would be “it still relates pretty closely”.
Yeah, that's very interesting to think about! I haven't thought about the situation with indistinguishable particles yet. However, I did think about how n particles in 1D is equivalent to a n-dimensional box, and there are really fascinating results that can be explored through that lens. Specifically, 4 particles in 1D (or a particle in a 4-dimensional box) would have energies that are a sum of four positive perfect squares. There's a famous theorem that every integer can be represented as the sum of four squares, and it's provable using generating functions and modular forms arguments on the Jacobi theta function, once again. I'm not entirely sure how the situation changes when you consider that none of the quantum numbers can be zero, though.
We usually ignore how numbers represent everything in nature- Pythagoras.
I thought a lot about Kronecker's quote: "God created the integers, all else is the work of man."
brilliant, thank you!
Needs to be louder but really interesting video!
Wonderful video, it was a bit quiet though.
Two subjects that i thought would never work together...
It still is shocking to me
Math and physics?
@@EricDMMillerpshh, never...
@EricDMMiller number theory and physics. You use math in physics But do you care that there exist atleast a prime between n^2 and (n+1)^2 for all n positive integers in physics? Physic use some parts of mathematics but not all of it and number theory is the subject that is not really used to solve physic problems (untill now lol).
@@aweebthatlovesmath4220O don't even know of any part of Mathematics that use all results from another branch of Mathematics. So it is a non issue if physics doesn't and it doesn't make math less 'real'. Sometimes it takes 2000 years for a construct in mathe to blossom, so who knows when must you use set theory to prove Something in physics. 😉
Very surprising, oh yes!
Beautiful
Awesome great work ! Am guessing that the sum of characters can never go negative wonder how you would prove that?
Also wondering how this relates to fifferent degeneracy/ statistics for fermions , bosons etc. Maybe subjects for another video ? 😊
Thank you! Yeah it doesn't seem obvious that the sum would never be negative, but once you put it into the product form, none of those factors can be negative interestingly. Definitely expect some more videos on statistical mechanics from me as it's one of my favorite physics topics
Indeed very cool!!
Amazing! Would be cool in 3D too.
There is no temperature in (pure) quantum mechanics. It only appears if you introduce classical uncertainty
If a quantum system reaches thermal equilibrium with a reservoir at a temperature T, then we can consider the probabilities of a quantum system exhibiting a state given that it is in equilibrium with a reservoir of temperature T. This is the temperature that I refer to in this video. Statistical mechanics is hardly just a description of classical uncertainty, and there are sensible ways in which quantum mechanics gives rise to thermodynamics as well. This is the field of quantum statistical mechanics. For instance, it’s not hard to determine the thermodynamic properties of the quantum harmonic oscillator. Let me know if there’s something I’m missing
@@worldequationI am sorry for the previous tersely formulated comment, I just got triggered by your use of T=300K around 4:55 for a state that I would hazard to guess is a pure state (which technically isn't at any temperature). The actual state at that temperature would be of the form e^(-H/300), a density matrix I honestly wouldn't have the slightest idea how to visualise. The rest of the video is really good!
On the wider point, statistical mechanics whether, classical or quantum, works iff you assume classical uncertainty (or lack of knowledge). It is a very effective description of large scale systems and temperature appears in such as an effective phenomenon. However, (probably due to a personal distaste for probability theory) I have always disliked temperature being viewed as fundamentally physical, since it emerges iff we deliberately smear out a perfectly deterministic system into a probabilistic one, which does not happen in "reality". Physical systems should be viewed as being in pure states (most of which do not have a temperature associated with them other than ground states) and they never reach anything like thermal equilibrium since their evolution is deterministic/unitary.
I see what you’re saying. To address the graphic at 4:55, this might have been a poor way of showing what I was trying to show. Basically, I just used the particle in a box graphic I’d been using throughout the video to have a visual representing putting that quantum system into thermal equilibrium with a reservoir at a temperature. I just chose room temperature because this is a temperature I naturally think of and know kT. I wasn’t choosing this temperature and the graphic to match, so sorry for the confusion. You’re absolutely right that such a low energy state would not be probable at room temperature. I also sympathize with the concern for how temperature gets viewed as fundamental when it isn’t in actuality. If I went more into the thermodynamics, I would have been more careful with this idea and what I mean by this to avoid assigning a temperature to a single particle, which is certainly not a good way to think about this. So thanks for mentioning this because it’s definitely important to get straight. I see how somebody could misinterpret that part of the video as saying that the particle has an inherent temperature or that the graphic is an accurate depiction of the system at room temperature or even that the probability density from the wavefunction is somehow associated to the purely statistical probability (all misconceptions I want to avoid). Thank you for giving me things to be more conscious of!
Why is 50 not square and 9 square?
OH you meant the multiplication square
QM classicalized in 2010. Juliana Mortenson website Forgotten Physics uncovers the hidden variables and constants and the bad math of Wien, Schrodinger, Heisenberg, Einstein, Debroglie,Planck,Bohr etc. A proton is a collection of 1836 expanding electrons and add a bouncing expanding electron makes a hydrogen atom. Electron mass (9.11) multiplied by 1836 equals the proton mass (1.67). All atoms and atomic objects are expanding at 1/770,000th their size per second per second constant acceleration. Multiplied by earth’s radius equals 16 feet etc.: gravity.” The Final Theory: Rethinking Our Scientific Legacy “, Mark McCutcheon.
I find all this a little bit artificial: Oh look, my physical problem has some rules, I have no idea why, but as I know a math domain which ensure these rules, there must be a connection between physics and some math domain.
Oh look: I have a set of small cubes that make a big cube and I want to make 2 smaller cubes without letting some on the side, but I can't ! So number theory must fundamentaly be connected to the real world because Fermat-Wiles theorem for n=3 also apply in the real world with true boxes made of wood. Or stones. Whatever. And oh, look, if I take 2 stones and 4 sticks, I can put all my items in two bags with the same number of pieces in each bag. Isn't that fundamentally incredible ? That the worlds know about mod 2 aritthmetics ?!
That maths work with physics is not a mistery at all, whatever Wigner might have said. This is a false paradox and a fake astonishment. If the "real world" is consistent, then it must, by definition of consistency, follow any language based on consistency. Mathematics are nothing else but a self-constructed language with consistency as its unique requirement. The mistery is not that "maths work baby", the mistery is that the physical world is consistent. They don't work more with the 2d quantum box and Dirichlet characteristics than they worked 5000 years ago with classical stones in a bag and simple addition and multiplication. As long as the world will follow rules, any rules, then maths will work by definition of what it means to follow rules. If they didn't, would'nt you say there is an error in your mathematical demonstration and start to look for that error until you found it ?
You should really look into the pronounciation of Dirichlet.
Would you point me to a source on the proper pronunciation? I have seen it pronounced several ways and stuck with one for this video. It seemed to me that there might be several acceptable pronunciations, but I could very well be wrong.
How is it possible the Pythagoras rears his ugly head in QM. That should be illegal. Or, more likely, point masses are approximations of quantum particles meaning spherical coordinates are advantageous.
Would you explain what you mean?
@@worldequation for the 2D particle box the possible energy levels are Pythagorean vectors where the norm squared is the result. This has a polar representation r = C where C is the energy level squared. Bumping this up to 3 dimensions yields a sphere with a radius rho = C. My thought is that this is a consequence of the polar nature of a point mass. The possible states are the intersection of all of the pure state circles/outputs of the chi function.
@@dominicellis1867 I would suggest it is probably more accurate to think of these vectors in the quantum number space as being in something like a momentum space. Therefore, a magnitude in this space is really the magnitude of the particle's momentum upon measurement.
@@worldequation does that mean that potential energy is a function of position with electron and proton charge density as the position space?
The sound is so soft I can't hear a single word properly
Yeah, I noticed that after I uploaded it and unfortunately can't make it louder after it's uploaded. It's loud enough for me if I turn the volume up, but sorry about that
WOW
I cannot hear you