Mathematics

[DavidW]

David, do you like quantum physics? What position/theory do you hold about the nature below sub-particle level?

Yes but my research area was in gravitational wave physics (specifically with regards to binary black hole coalescence and pulsar detection).  When I teach qm I like to take the more modern approach to integrate it with linear algebra instead of the traditional approach of laboriously working out wave mechanics ad nauseam.  Many people think that quantum mechanics just absolutely destroyed and replaced classical mechanics, which is not true.  They are literally the same equations just re-interpreted.  A big part of that problem is that you learn quantum mechanics very early on, but only learn Hamiltonian mechanics as a grad student (at least in the US and at least when I was in school).  Two things should happen and maybe already are which is that all of classical mechanics needs to be taught earlier.  But also quantum field theory needs to start being taught way earlier, like in undergrad.  You don't need to master all of it, but there are deep ideas that resonate throughout physics that shouldn't wait until second year of grad school.

There are a series of books called The Theoretical Minimum which does a great job of introducing many of the ideas of advanced Physics to the uninitiated with far more rigor than the usual pop sci book, but not as explosively complex and difficult as traditional textbooks are.  Anyone that is not bad at Math, and wants to learn some more Physics without devoting years of study to it should give them a try.

[Karl Henning]

Yes but my research area was in gravitational wave physics (specifically with regards to binary black hole coalescence and pulsar detection).  When I teach qm I like to take the more modern approach to integrate it with linear algebra instead of the traditional approach of laboriously working out wave mechanics ad nauseam.  Many people think that quantum mechanics just absolutely destroyed and replaced classical mechanics, which is not true.  They are literally the same equations just re-interpreted.  A big part of that problem is that you learn quantum mechanics very early on, but only learn Hamiltonian mechanics as a grad student (at least in the US and at least when I was in school).  Two things should happen and maybe already are which is that all of classical mechanics needs to be taught earlier.  But also quantum field theory needs to start being taught way earlier, like in undergrad.  You don't need to master all of it, but there are deep ideas that resonate throughout physics that shouldn't wait until second year of grad school.

There are a series of books called The Theoretical Minimum which does a great job of introducing many of the ideas of advanced Physics to the uninitiated with far more rigor than the usual pop sci book, but not as explosively complex and difficult as traditional textbooks are.  Anyone that is not bad at Math, and wants to learn some more Physics without devoting years of study to it should give them a try.

Interesting, thanks.

[steve ridgway]

1. I didn't much like it but was good at it up to age 18.

2. I did B.Sc. Physics but the maths proved too hard for me in the end, I prefer to know a little about many subjects and the big picture rather than minute detail.

3. I went into IT when I realised the only maths you needed were one and zero.

4. I still find basic maths and mental arithmetic very useful.

[Biffo]

1. I didn't much like it but was good at it up to age 18.

2. I did B.Sc. Physics but the maths proved too hard for me in the end, I prefer to know a little about many subjects and the big picture rather than minute detail.

3. I went into IT when I realised the only maths you needed were one and zero.

4. I still find basic maths and mental arithmetic very useful.

I worked in IT for over 25 years but never mastered binary or hexadecimal. I encountered both on various training courses but had forgotten it all by the time I came to use it - and that was very infrequently. For hexadecimal I used a calculator - this occasionally caused some tut-tutting but I didn't believe in cluttering my brain with things I had no interest in.

[DavidW]

I worked in IT for over 25 years but never mastered binary

Certainly you've mastered turning it off and then on again!

[Karl Henning]

Certainly you've mastered turning it off and then on again!

(* chortle *)

There's a rule that this is always the first suggestion, right?

[Dry Brett Kavanaugh]

Yes but my research area was in gravitational wave physics (specifically with regards to binary black hole coalescence and pulsar detection).  When I teach qm I like to take the more modern approach to integrate it with linear algebra instead of the traditional approach of laboriously working out wave mechanics ad nauseam.  Many people think that quantum mechanics just absolutely destroyed and replaced classical mechanics, which is not true.  They are literally the same equations just re-interpreted.  A big part of that problem is that you learn quantum mechanics very early on, but only learn Hamiltonian mechanics as a grad student (at least in the US and at least when I was in school).  Two things should happen and maybe already are which is that all of classical mechanics needs to be taught earlier.  But also quantum field theory needs to start being taught way earlier, like in undergrad.  You don't need to master all of it, but there are deep ideas that resonate throughout physics that shouldn't wait until second year of grad school.

There are a series of books called The Theoretical Minimum which does a great job of introducing many of the ideas of advanced Physics to the uninitiated with far more rigor than the usual pop sci book, but not as explosively complex and difficult as traditional textbooks are.  Anyone that is not bad at Math, and wants to learn some more Physics without devoting years of study to it should give them a try.

David, thank you for the response. I am interested in Quantum Field Theory too!

[krummholz]

1. Did you like it / were you good at it during elementary / secondary / high school / college years?

2. Are you a graduate of a college where mathematics was among the core courses?

3. If yes, do you use it in your current / former job?

4. If no, do you think, in retrospective, that learning it during elementary / secondary / high school was of any use?

1. Yes.
2. Yes.
3. Yes. (I teach college-level physics, including upper-division courses.)

[Daverz]

1. I thought I was good at it, but I was not a very good student.
2. Yes.  I actually have a BA in Math.
3. I use "advanced" math every day for work, and I also like to read Physics and Math books for pleasure.

[krummholz]

Yes but my research area was in gravitational wave physics (specifically with regards to binary black hole coalescence and pulsar detection).  When I teach qm I like to take the more modern approach to integrate it with linear algebra instead of the traditional approach of laboriously working out wave mechanics ad nauseam.  Many people think that quantum mechanics just absolutely destroyed and replaced classical mechanics, which is not true.  They are literally the same equations just re-interpreted.  A big part of that problem is that you learn quantum mechanics very early on, but only learn Hamiltonian mechanics as a grad student (at least in the US and at least when I was in school).  Two things should happen and maybe already are which is that all of classical mechanics needs to be taught earlier.  But also quantum field theory needs to start being taught way earlier, like in undergrad.  You don't need to master all of it, but there are deep ideas that resonate throughout physics that shouldn't wait until second year of grad school.

Agree on all points (but see below about quantum vs. classical mechanics). One thing though: before you can learn the Hamiltonian formulation, you need to have a solid mastery of the Newtonian formulation in 3 dimensions. Otherwise, Lagrange's equations (on which Hamilton's are based) will seem like a foreign language, and you won't understand why, and when, forces of constraint can be ignored in setting up the equations of motion.

I know this from bitter experience, having been thoroughly turned off of physics as an undergrad at a well-known school that tried to introduce Lagrange's equations on Day One to students who have only seen Newtonian physics in 1-d. Luckily, I returned to physics (after majoring in music as an alternative) and figured everything out with a bit of perseverance.

As far as the equations of QM being those of Hamilton, just reinterpreted... well, there are some major differences between Poisson brackets and commutator brackets... but otherwise, yeah I agree, at least in a formal sense.

There are a series of books called The Theoretical Minimum which does a great job of introducing many of the ideas of advanced Physics to the uninitiated with far more rigor than the usual pop sci book, but not as explosively complex and difficult as traditional textbooks are.  Anyone that is not bad at Math, and wants to learn some more Physics without devoting years of study to it should give them a try.

Great to see, I had never heard of this series. Susskind, a co-author, is of course a well-known figure in string theory. This is on my to-do list to check out at some point.

[DavidW]

I know this from bitter experience, having been thoroughly turned off of physics as an undergrad at a well-known school that tried to introduce Lagrange's equations on Day One to students who have only seen Newtonian physics in 1-d. Luckily, I returned to physics (after majoring in music as an alternative) and figured everything out with a bit of perseverance.

Wow I never heard of that, that is crazy.  I just meant the upper level undergrad course usually only does Lagrangian dynamics and it really should include Hamiltonian dynamics.  That sounds like you went to MIT or Cal Tech!

[krummholz]

Wow I never heard of that, that is crazy.  I just meant the upper level undergrad course usually only does Lagrangian dynamics and it really should include Hamiltonian dynamics.  That sounds like you went to MIT or Cal Tech!

Actually, it was Michigan. Things may be different now though, that was mid-1970s. And you're right: we didn't see Hamilton at all in that course. However, where I teach now, our upper division classical mechanics course does at least show the students Hamilton's equations.

[vandermolen]

No, you can't work out exam percentages just with a calculator. Give a chimpanzee a calculator and ask it to calculate the percentages.  You need to know how to operate the calculator and that means you have understanding of how percentages work. Give credit to yourself for the math you can do! Math is not really about being able to calculate the fifth root of 72 in head*. It is for example about understanding percentages. It is about knowing what someone means by 20 % of 400. If you know it means 80 (thanks to the numbers being easy to calculate in head) you know how to use a calculator to calculate (very accurately) 17.2 % of 386.5 which is nastier to calculate in head. 

* It is not so difficult to make rough estimates in your head: The fifth root of 72 is the number that raised to the fitht power (multiplied 5 times by itself) gives 72. So, could it be about 2? Well 2^5 = 2 * 2 * 2 * 2 * 2 = 4 * 4 * 2 = 16 * 2 = 32. We already know the fifth root of 72 is bigger than 2, but how much bigger? Bigger than 3? Well, 3^5 = 3 * 3 * 3 * 3 * 3 = 9 * 9 * 3 = 81 * 3 = 80 * 3 + 3 = 240 + 3 = 243 which is much bigger  than 72. So,  The fifth root of 72 is in between 2 and 3, but probably closer to 2, because 2^5 is closer to 72 and 3^5. How about something like 2.2 as an rough extimate? If we calculate the fifth root of 72 using a calculator, we get 2.352158045. The error of the rough estimate is 0.152158045 which is less than 7 % of the correct value! If you needed no more than 20 % accuracy, calculating it in head is good enough!

Thank you 71 dB! You made me feel better about my mathematical ineptitude 

[Jo498]

That's interesting. In my German university theoretical physics in the 1990s we started in first semester with Newtonian mechanics, second semester Lagrange and Hamilton, and 4th and 5th semester were QM. So at least in theory everyone starting with QM would be familiar with Hamiltonians and even Poisson brackets (although I think the latter were not given much room in the mechanics class). I think it still depended a bit on the professor and textbook if a more traditional "wave mechanics" or the Feynman/Dirac "bra-kets as soon as possible" approach was taken.
Of course this doesn't smoothly map unto US college because it was more like a 5 years master's degree course with only physics, maths and one small minor subject (like chemistry) to pick, not a Bachelor's with general educational requirements in the first one or two years.  It was not ideally integrated with the necessary maths either, some maths tools one learned "quick and dirty" in physics before (if ever) getting to them in maths (like delta function/distribution). And the silent precondition was that students did have a solid calculus, vector algebra and intro level physics background from before university.

[LKB]

I like cheese.

If you divide one cheddar wedge precisely down the longitudinal axis, you then have two wedges. It's almost like cheesy magic, and the most useful math l know. 

[vandermolen]

I like cheese.

If you divide one cheddar wedge precisely down the longitudinal axis, you then have two wedges. It's almost like cheesy magic, and the most useful math l know. 

In Dickens's 'David Copperfield' the appalling 'Mr Murdstone' makes Davy work out a complex maths problem (for a little boy) involving the purchase of Double Gloucester Cheese.

[LKB]

In Dickens's 'David Copperfield' the appalling 'Mr Murdstone' makes Davy work out a complex maths problem (for a little boy) involving the purchase of Double Gloucester Cheese.

I've heard that name before. One of the most  famous episodes from Rod Serling's The Twilight Zone refers to David Copperfield, and to Mr. Murdstone specifically.

A short while later, the world blows up...

https://pluto.tv/en/on-demand/series/the-twilight-zone/season/1/episode/time-enough-at-last-1959-1-8

[71 dB]

Thank you 71 dB! You made me feel better about my mathematical ineptitude 

You're welcome. Everybody struggles with math. The best Mathematician in the World at the moment, Terrence Tao, struggles with the mysteries of prime numbers for example. At the same time we should appreciate the math we can do and build on that. 

[Biffo]

Certainly you've mastered turning it off and then on again!

Yes but I never worked in IT support. It was the first thing I tried and then I phoned support and got the invariable response.

[71 dB]

David, thank you for the response. I am interested in Quantum Field Theory too!

I was really bad at Quantum Physics in the University. I barely passed the course and that was thanks to relativistic physics being part of the course. Lorentz-transformations are easy enough to learn/understand, but Schrödinger equation? I was so lost, because intuition is how I learn and intuition is useless with Quantum Physics!