(no subject)
It's weird, I said I wasn't going to return to LJ, and yet here's another post.
One of my classes this semester was Optics. Unfortunately the professor was complete shit and didn't teach us anything, but he did encourage us to give presentations of our own (so he could do even less -- he came right out and admitted as much) and the presentations have been very interesting. On Monday, my friend Brad is going to be covering nonlinear optics, which should be very cool. Matt did calculus of variations in a way that made much more sense than anything I'd seen before. And there were a few other more standard things like geometric optics and diffraction.
Mine was on Hamilton's analogy between geometric optics and classical mechanics: the gist of it is that, in the low-wavelength limit, light can be treated as consisting of rays instead of waves, but these rays behave exactly like particles from classical mechanics, so that we may treat them as formally identical. This allowed Hamilton to port over into mechanics an entire system he had developed in optics for treating complex systems of rays: and, in fact, this is what we know today as Hamiltonian mechanics. His optical system was largely forgotten for almost a century (but was recently revived.)
However, in the process of building this analogy, Hamilton pointed to parallels in variational principles such as Fermat's principle of least time and the principle of least action, and this is what DeBroglie later seized upon to form his eponymous relationships. (The derivation is elementary: if we formulate Fermat's principle as delta Int 1/v ds = 1/f delta Int 1/lambda ds = 0, where lambda is wavelength, v velocity and f frequency (which is regarded as a constant); and we formulate least action as delta Int mv ds = delta Int sqrt{2mT} ds = delta Int sqrt{2m(E-V)} ds = 0; and then we say that the two are physically equivalent, so that one integral must equal the other, we get that p = mv must be proportional to 1/lambda, and v/lambda = f must be proportional to E.)
Schrodinger then took the analogy to its ultimate conclusion and looked for a wave theory which reduced to classical mechanics in the low-wavelength limit, and that's where wave mechanics came from.
If I ever teach optics, I'm going to spend about 5 classes developing all the prereqs to that in detail (Snell's law/law of reflection, particle and wave theories of light, Huygens principle in detail, calculus of variations and Fermat, the Fresnel equations for reflection and transmission, geometric optics and the Eikonal equation, etc.) and then spend about 2 minutes at the end of the last lecture to derive DeBroglie and hint at Schrodinger. It would be awesome prep for a first course in quantum mechanics.
Alternatively, this analogy lets us use matter to do optics, treating potential energies as indices of refraction and using, for example, EM fields to create "lenses" and "mirrors" for electrons. This is the principle behind the electron microscope.
The analogy is also the basis for lots of semi-classical approximations in quantum mechanics, such as the Eikonal approximation and the WKB method.
And then you get into a lot of really cool things about the classical action being the phase of the wavefunction, so that the principle of least action is really just a manifestation of simple interference of matter waves, Feynman's path integral emerges from the Huygens-Fresnel principle, and the Bohm-Aharonov effect and other results involving geometric phase all become immediate.
Here are the slides, if anyone is interested: http://www.filefactory.com/file/1l6q6kazqanh/n/Hamilton_s_analogy_between_Optics_and_Mechanics_pptx
Other than that, I'm taking PDE 1, the math of quantum mechanics (mostly operator theory and functional analysis), measure theory, numerical analysis, and japanese 2. It's been a whirlwind tour of analysis and has completely changed the way I look at the subject.
Especially amazing has been the discovery of just how deep Fourier/Harmonic analysis goes. Before this semester, I had seen it in passing, and knew vaguely, mostly conceptually, what it was about (decomposing functions into superpositions of waves of different wavelengths, etc.), but had no real concrete understanding of its mechanics or, critically, of the vast ocean of underlying theory. Now, it sits as this beautiful gem in the center of a confluence of different questions and tools: functional analysis, partial differential equations, complex analysis, and even number theory. The unexpected way in which all this gets extended to more general topological groups and the plethora of inequalities and identities generated in exploring the consequences of the basic motivating ideas are nothing short of breathtaking. It's one of the richest subjects I've encountered yet, and I have a feeling I'll be learning it for years to come.
Anyways, I thought I'd post some actual math, to finish up: a short, pretty proof of the Poisson sum formula.
Suppose f is sufficiently smooth and decays rapidly enough at infinity (e.g. let f be in the Schwartz space S of R). We define F(x) = Sum_{n \in Z} f(x+n). Note that this is 1-periodic, and so we may define its Fourier series. (This is really clever!) Consider it's kth Fourier coefficient,
(F_k)-hat = Int_0^1 Sum_{n \in Z} f(x+n)e^{-ikx} dx = Sum_{n \in Z} Int_0^1 f(x+n)e^{-ikx} dx (where we can interchange integral and sum because f is so nicely behaved)
= Sum_{n \in Z}Int_n^{n+1} f(x)e^{-ikx} dx = Int_R f(x)e^{ikx}dx = (f_k)-hat
So, F's kth Fourier coefficient is just f's kth Fourier coefficient! Again, this is a very clever construction: because now we may write down F's Fourier series as F(x) = Sum_{n \in Z} f(n)-hat e^{inx}
But then F(0) = Sum_{n \in Z} f(n) = Sum_{n \in Z} f(n)-hat,
which is the Poisson sum formula. Tada!
If I make more posts, they'll probably be math or physics posts. I'm currently working through Siegel's proof of a functional equation for the Dedekind eta function (the second one they list.) This is a critical step in the calculation of Hardy and Ramanujan's asymptotic formula for the number of partitions of n. Maybe I'll post that when I understand all the details.
One of my classes this semester was Optics. Unfortunately the professor was complete shit and didn't teach us anything, but he did encourage us to give presentations of our own (so he could do even less -- he came right out and admitted as much) and the presentations have been very interesting. On Monday, my friend Brad is going to be covering nonlinear optics, which should be very cool. Matt did calculus of variations in a way that made much more sense than anything I'd seen before. And there were a few other more standard things like geometric optics and diffraction.
Mine was on Hamilton's analogy between geometric optics and classical mechanics: the gist of it is that, in the low-wavelength limit, light can be treated as consisting of rays instead of waves, but these rays behave exactly like particles from classical mechanics, so that we may treat them as formally identical. This allowed Hamilton to port over into mechanics an entire system he had developed in optics for treating complex systems of rays: and, in fact, this is what we know today as Hamiltonian mechanics. His optical system was largely forgotten for almost a century (but was recently revived.)
However, in the process of building this analogy, Hamilton pointed to parallels in variational principles such as Fermat's principle of least time and the principle of least action, and this is what DeBroglie later seized upon to form his eponymous relationships. (The derivation is elementary: if we formulate Fermat's principle as delta Int 1/v ds = 1/f delta Int 1/lambda ds = 0, where lambda is wavelength, v velocity and f frequency (which is regarded as a constant); and we formulate least action as delta Int mv ds = delta Int sqrt{2mT} ds = delta Int sqrt{2m(E-V)} ds = 0; and then we say that the two are physically equivalent, so that one integral must equal the other, we get that p = mv must be proportional to 1/lambda, and v/lambda = f must be proportional to E.)
Schrodinger then took the analogy to its ultimate conclusion and looked for a wave theory which reduced to classical mechanics in the low-wavelength limit, and that's where wave mechanics came from.
If I ever teach optics, I'm going to spend about 5 classes developing all the prereqs to that in detail (Snell's law/law of reflection, particle and wave theories of light, Huygens principle in detail, calculus of variations and Fermat, the Fresnel equations for reflection and transmission, geometric optics and the Eikonal equation, etc.) and then spend about 2 minutes at the end of the last lecture to derive DeBroglie and hint at Schrodinger. It would be awesome prep for a first course in quantum mechanics.
Alternatively, this analogy lets us use matter to do optics, treating potential energies as indices of refraction and using, for example, EM fields to create "lenses" and "mirrors" for electrons. This is the principle behind the electron microscope.
The analogy is also the basis for lots of semi-classical approximations in quantum mechanics, such as the Eikonal approximation and the WKB method.
And then you get into a lot of really cool things about the classical action being the phase of the wavefunction, so that the principle of least action is really just a manifestation of simple interference of matter waves, Feynman's path integral emerges from the Huygens-Fresnel principle, and the Bohm-Aharonov effect and other results involving geometric phase all become immediate.
Here are the slides, if anyone is interested: http://www.filefactory.com/file/1l6q6kazqanh/n/Hamilton_s_analogy_between_Optics_and_Mechanics_pptx
Other than that, I'm taking PDE 1, the math of quantum mechanics (mostly operator theory and functional analysis), measure theory, numerical analysis, and japanese 2. It's been a whirlwind tour of analysis and has completely changed the way I look at the subject.
Especially amazing has been the discovery of just how deep Fourier/Harmonic analysis goes. Before this semester, I had seen it in passing, and knew vaguely, mostly conceptually, what it was about (decomposing functions into superpositions of waves of different wavelengths, etc.), but had no real concrete understanding of its mechanics or, critically, of the vast ocean of underlying theory. Now, it sits as this beautiful gem in the center of a confluence of different questions and tools: functional analysis, partial differential equations, complex analysis, and even number theory. The unexpected way in which all this gets extended to more general topological groups and the plethora of inequalities and identities generated in exploring the consequences of the basic motivating ideas are nothing short of breathtaking. It's one of the richest subjects I've encountered yet, and I have a feeling I'll be learning it for years to come.
Anyways, I thought I'd post some actual math, to finish up: a short, pretty proof of the Poisson sum formula.
Suppose f is sufficiently smooth and decays rapidly enough at infinity (e.g. let f be in the Schwartz space S of R). We define F(x) = Sum_{n \in Z} f(x+n). Note that this is 1-periodic, and so we may define its Fourier series. (This is really clever!) Consider it's kth Fourier coefficient,
(F_k)-hat = Int_0^1 Sum_{n \in Z} f(x+n)e^{-ikx} dx = Sum_{n \in Z} Int_0^1 f(x+n)e^{-ikx} dx (where we can interchange integral and sum because f is so nicely behaved)
= Sum_{n \in Z}Int_n^{n+1} f(x)e^{-ikx} dx = Int_R f(x)e^{ikx}dx = (f_k)-hat
So, F's kth Fourier coefficient is just f's kth Fourier coefficient! Again, this is a very clever construction: because now we may write down F's Fourier series as F(x) = Sum_{n \in Z} f(n)-hat e^{inx}
But then F(0) = Sum_{n \in Z} f(n) = Sum_{n \in Z} f(n)-hat,
which is the Poisson sum formula. Tada!
If I make more posts, they'll probably be math or physics posts. I'm currently working through Siegel's proof of a functional equation for the Dedekind eta function (the second one they list.) This is a critical step in the calculation of Hardy and Ramanujan's asymptotic formula for the number of partitions of n. Maybe I'll post that when I understand all the details.