I must confess that I, too, was attracted because of the “bra” notation. Perhaps I was expecting some sort of 3-dimensional normal distribution. I was too deep into it before I realized Dirac had hooked me with a misspelling. One of my statistics professors was an FRS who got his PhD from the University of Bristol, but I was never brave enough to ask him whether he knew Dirac.
I was just a pure math schmuck.
I never knew Dirac published in physics, let alone Quantum Physics. I did a Graph Theory course where PAM Dirac’s name was on several theorems. There was nothing about bras, but I did attend a few seminars with rather racy counterexample graphs.
I hope you like it. The funny thing here is that, from my perspective, Dirac was always just a theoretical quantum physicist. Which, of course, requires some very serious math skills. I've been working through his book "The Principles of Quantum Mechanics" but it's a hard slog — he's even more math-y than Penrose!
Here’s an interesting math tidbit. I don’t know if you’ve got to it yet. The “Dirac delta function” is a function he introduces that has the seemingly magical ability to give a finite (non-zero) answer to an integral at a point. Every first-year calculus student knows that’s supposed to be impossible (with Riemann’s integral theory anyway). I was reminded of the Heaviside step function; there were other foreshadowings in the 19th century.
It turns out Dirac was giving an example of a “generalized function,” though he only ever used it multiplied by another function before integrating, so it was a legitimate use).
A French mathematician in the 1940s called it “distributions,” but there was a much-loved but obscure Russian mathematician in the early Stalin era who published on generalized functions. See Sergei Sobolev and Laurent Schwartz.
(Since the Russian publications appeared in Russian-language journals, it was unknown in the West. I believe Sobolev tells the story of the early Russian. Sometimes Schwartz is seen as extending Sobolev’s work.)
Yes, returns zero everywhere on the real number line except precisely 0.0 where it returns ∞. My math skills aren't up to it, but AIUI, it integrates over the real number line to 1.0. It pops up a lot in QM as does the somewhat similar Kronecker delta, δ(i,j), which is 1 when i=j but 0 otherwise.
It bothered me when I first saw it, so I created a sequence which converges to 0 in the limit. I then multiplied this sequence by the target function to get a sequence of integrals. That seemed to do the trick, but it left the concept of generalized functions up in the air as an unnecessary adjunct.
Dirac obviously decided not to do it that way, so I must have missed something in what he was trying to say with the physics. I suppose it doesn’t matter because Dirac didn’t really violate any rules in the way he used his delta function. It was just weird in isolation.
From a pedagogical point of view, I still don’t understand why you wouldn’t teach it with the limit concept so as to avoid the math objections. The only objection I see is that this workaround is a one time “solution” to a math paradox. I didn’t think that objection bothered physicists all that much.
We're beyond my math here, but I *think* the physical application is to represent a collapsed wavefunction or one with a precise location or momentum. For an instant, the wavefunction should vanish everywhere except at one real number coordinate.
The Wiki page for the function says it can be defined as a Gaussian in the limit as width vanishes. In which case, it may well be a valid solution to the Schrödinger equation, which would cause its width to instantly begin to expand.
But I'm guessing off a scant number of data points.
I’ve always been dubious of my ability to understand physics. When I was in college, I had an interesting conversation with my physics professor. He said that after doing his undergraduate degree (math-physics, I think), he had a hard time deciding between math or physics for grad school.
He finally decided on physics because, in physics, he always had an intuition about what the next question was. In math, not so much. I’m kind of the reverse of my former physics professor.
One of the most amazing things about him was his ability to explain undergraduate math, particularly anything to do with calculus (the applied stuff — he stayed away from Analysis). My classmates said they had a weekly seminar in his office on the latest differential equations homework. I don’t know why the math department didn’t have him teach the course.
A friend of mine (in math) told me she really enjoyed his course on Special Relativity, but it was tough. She got her usual A anyway. Another physicist in the department was an experimentalist who specialized in x-ray crystallography, and the senior professor went into administration (registrar and college computer services director, I think.) Bright guys, these physicists.
The best anecdotes came from a guy with a Masters in Engineering and Statistics. The stories came from his time in Bell Research labs. I’ll tell you about “hit it with a Laplace” sometime. (And why my former physics professor may have been responsible for Canada winning the Canada-Russia hockey series of the early 1970s.)
Interesting. I suppose I'm far more on the physics side than the maths side, but I'm largely self-taught on both and make no claims to expertise. I'm a lot more comfortable with physics concepts than math. The latter sometimes makes my eyes cross and my brain despair of ever getting it, but the former is more accessible to me.
Special Relativity, for instance, I felt I had such a good handle on it (at a pop sci level) that I presumed to write a whole series of posts explaining it on my old blog. [And I still insist that the Twins Paradox is solvable entirely in the framework of SR, no GR required. I had a very long debate with a guy about that once.]
General Relativity, though, is so math-y that I doubt I'll ever try to go there. I'd be happy if I could learn to solve the differential equations in the Schrödinger equation, but from what I hear, that's some pretty high-level maths. My eventual end goal is creating some animations of quantum "particle" behaviors.
I’m sure you can master the Schrödinger differential equation easily. (The Heisenberg matrix/operator stuff requires a lot more preparation. I came at it from the Hilbert development, which is [unnecessarily?] math-heavy.)
I find the notation on the General Relativity stuff very dense. I even had math professors who harshly didn’t like it and steered me away.
I accidentally bought a used book on Tensor Analysis years after I finished college. I found I didn’t mind the stuff; I liked the take on geometry more than the Relativity stuff. But I only got my head around the Relativity after I read a full development of the Schwarzkopf solution to the Einstein field equations.
I must confess that I, too, was attracted because of the “bra” notation. Perhaps I was expecting some sort of 3-dimensional normal distribution. I was too deep into it before I realized Dirac had hooked me with a misspelling. One of my statistics professors was an FRS who got his PhD from the University of Bristol, but I was never brave enough to ask him whether he knew Dirac.
I was just a pure math schmuck.
I never knew Dirac published in physics, let alone Quantum Physics. I did a Graph Theory course where PAM Dirac’s name was on several theorems. There was nothing about bras, but I did attend a few seminars with rather racy counterexample graphs.
I look forward to reading your article.
I hope you like it. The funny thing here is that, from my perspective, Dirac was always just a theoretical quantum physicist. Which, of course, requires some very serious math skills. I've been working through his book "The Principles of Quantum Mechanics" but it's a hard slog — he's even more math-y than Penrose!
Here’s an interesting math tidbit. I don’t know if you’ve got to it yet. The “Dirac delta function” is a function he introduces that has the seemingly magical ability to give a finite (non-zero) answer to an integral at a point. Every first-year calculus student knows that’s supposed to be impossible (with Riemann’s integral theory anyway). I was reminded of the Heaviside step function; there were other foreshadowings in the 19th century.
It turns out Dirac was giving an example of a “generalized function,” though he only ever used it multiplied by another function before integrating, so it was a legitimate use).
A French mathematician in the 1940s called it “distributions,” but there was a much-loved but obscure Russian mathematician in the early Stalin era who published on generalized functions. See Sergei Sobolev and Laurent Schwartz.
(Since the Russian publications appeared in Russian-language journals, it was unknown in the West. I believe Sobolev tells the story of the early Russian. Sometimes Schwartz is seen as extending Sobolev’s work.)
Yes, returns zero everywhere on the real number line except precisely 0.0 where it returns ∞. My math skills aren't up to it, but AIUI, it integrates over the real number line to 1.0. It pops up a lot in QM as does the somewhat similar Kronecker delta, δ(i,j), which is 1 when i=j but 0 otherwise.
It bothered me when I first saw it, so I created a sequence which converges to 0 in the limit. I then multiplied this sequence by the target function to get a sequence of integrals. That seemed to do the trick, but it left the concept of generalized functions up in the air as an unnecessary adjunct.
Dirac obviously decided not to do it that way, so I must have missed something in what he was trying to say with the physics. I suppose it doesn’t matter because Dirac didn’t really violate any rules in the way he used his delta function. It was just weird in isolation.
From a pedagogical point of view, I still don’t understand why you wouldn’t teach it with the limit concept so as to avoid the math objections. The only objection I see is that this workaround is a one time “solution” to a math paradox. I didn’t think that objection bothered physicists all that much.
We're beyond my math here, but I *think* the physical application is to represent a collapsed wavefunction or one with a precise location or momentum. For an instant, the wavefunction should vanish everywhere except at one real number coordinate.
The Wiki page for the function says it can be defined as a Gaussian in the limit as width vanishes. In which case, it may well be a valid solution to the Schrödinger equation, which would cause its width to instantly begin to expand.
But I'm guessing off a scant number of data points.
I’ve always been dubious of my ability to understand physics. When I was in college, I had an interesting conversation with my physics professor. He said that after doing his undergraduate degree (math-physics, I think), he had a hard time deciding between math or physics for grad school.
He finally decided on physics because, in physics, he always had an intuition about what the next question was. In math, not so much. I’m kind of the reverse of my former physics professor.
One of the most amazing things about him was his ability to explain undergraduate math, particularly anything to do with calculus (the applied stuff — he stayed away from Analysis). My classmates said they had a weekly seminar in his office on the latest differential equations homework. I don’t know why the math department didn’t have him teach the course.
A friend of mine (in math) told me she really enjoyed his course on Special Relativity, but it was tough. She got her usual A anyway. Another physicist in the department was an experimentalist who specialized in x-ray crystallography, and the senior professor went into administration (registrar and college computer services director, I think.) Bright guys, these physicists.
The best anecdotes came from a guy with a Masters in Engineering and Statistics. The stories came from his time in Bell Research labs. I’ll tell you about “hit it with a Laplace” sometime. (And why my former physics professor may have been responsible for Canada winning the Canada-Russia hockey series of the early 1970s.)
Interesting. I suppose I'm far more on the physics side than the maths side, but I'm largely self-taught on both and make no claims to expertise. I'm a lot more comfortable with physics concepts than math. The latter sometimes makes my eyes cross and my brain despair of ever getting it, but the former is more accessible to me.
Special Relativity, for instance, I felt I had such a good handle on it (at a pop sci level) that I presumed to write a whole series of posts explaining it on my old blog. [And I still insist that the Twins Paradox is solvable entirely in the framework of SR, no GR required. I had a very long debate with a guy about that once.]
General Relativity, though, is so math-y that I doubt I'll ever try to go there. I'd be happy if I could learn to solve the differential equations in the Schrödinger equation, but from what I hear, that's some pretty high-level maths. My eventual end goal is creating some animations of quantum "particle" behaviors.
I’m sure you can master the Schrödinger differential equation easily. (The Heisenberg matrix/operator stuff requires a lot more preparation. I came at it from the Hilbert development, which is [unnecessarily?] math-heavy.)
I find the notation on the General Relativity stuff very dense. I even had math professors who harshly didn’t like it and steered me away.
I accidentally bought a used book on Tensor Analysis years after I finished college. I found I didn’t mind the stuff; I liked the take on geometry more than the Relativity stuff. But I only got my head around the Relativity after I read a full development of the Schwarzkopf solution to the Einstein field equations.
Thanks for the Subscribe.