Mathematics used in optics and intro to modern physics

[Levi Tate]

Hello.

I am taking intro to modern physics and optics next semester, the intro to modern physics is a 300 level class and the optics course is a 500 level class, at Wayne State University.

I want to come into next semester very well practiced in the math needed for those classes.

Could you folks tell me what mathematics I should concentrate on, the math that will appear in those classes?

Thanks in advance

 

[bossman27]

Generally, the mathematical rigor of courses with similar names can vary a lot from program to program. I say that just as a disclaimer that my courses may be more or less advanced than yours. That said, I did look up your course descriptions and will give some advice based on my experiences (at a top 20 program):

Intro to Modern Physics:
This was the first "upper division" course I could take in my physics program, as it had no pre-reqs beyond basic calc (and intro physics). The math was not very difficult, at least conceptually. Most of the difficulty was in lengthy algebraic manipulations, rather than calculus, etc. The general purpose of the course was to familiarize you with concepts in special relativity and basic quantum phenomenon, not to make you do rigorous calculations. The description of your course sounds to be very similar. My advice is that you don't need to practice any pure math for this class, but I found that playing around with the equations of special relativity before the semester started helped save me a lot time once the class got started, if nothing else. If you'd like to take that route, click http://www.hep.utexas.edu/~kopp/UTexas/phy355/ for some youtube tutorials by my professor.
Also, note that it will help to thoroughly familiarize yourself with using the relativistic "eV units" for energy and rest mass, as opposed to Joules, kilograms etc. It's not too difficult to master, but you'll be one step ahead of your classmates and as I said, you'll have more free time to focus on your more challenging courses.

Optics:
My first "optics" material was part of a course titled "Wave Motion and Optics." Your course description sounds like your optics material may be more advanced than mine was, as it was only a sub-unit in my course curriculum. That said, here are the topics listed in the description (as well as some possible others) and corresponding mathematics you may encounter:

General Wave Equations - partial differential equations

Physical Optics - assuming this is your first introduction, the math will probably be relatively simple, at least initially. Namely, sinusoidal waveforms, constructive/destructive interference, diffraction, etc. If you're not familiar, I believe you can find some basic information on this stuff in the link I gave for modern physics tutorials, or very easily elsewhere on the internet.

Geometrical Optics - mathematically based on short wavelength limit for solutions to hyperbolic partial DEs

Advanced Optics - with another disclaimer that I'm not sure whether it will be covered in your course, you might also encounter Fourier optics. This would involve Fourier Transforms and Green's function.


TL;DR: No need to study math specifically for "Into to Modern Physics"; instead, familiarize yourself with special relativity and using appropriate units. For "optics," focus on partial differential equations, partial differential equations and partial differential equations. If you get bored and want to get a little more optics-specific, try playing with the general 3-D wave equation, exponentially dampened waveforms, the Korteweg–de Vries equation, the Helmholtz equation and maybe some Fourier Transforms just for fun. Honestly though, I think you'd be best served by really mastering partial DE's. All the optics-specific stuff will be that much easier when you learn it in class.

 

[jtbell]

I am taking intro to modern physics and optics next semester, the intro to modern physics is a 300 level class and the optics course is a 500 level class, at Wayne State University.

What are the official prerequisites (courses)? Also, do you know which textbooks you'll be using?

The intro modern course will likely contain an introduction to Schrödinger's equation and some simple solutions of it (infinite square well, etc.). This is a differential equation, but the course may or may not expect you to have taken a DE course beforehand. When I taught that course, we introduced the SE slowly, along with the general concepts of a DE and its solutions. We didn't use the rigorous solution methods that you learn in a DE course, but instead a hand-waving "guess the solution and see if it works" method. Aside from the SE, you probably won't need much calculus, just some basic derivatives and integrals.

The optics course will probably include the differential wave equation, but it might be only an introduction like with Schrödinger's equation in intro modern. You'll need some calculus for that. Otherwise it will probably be mostly algebra. If the course does a lot of geometrical optics, you'll want to review the ray-diagram stuff from your freshman physics course.

 

[Levi Tate]

Wow thanks a lot mates. I'll definitely take your advise Bossman27.

The official prerequisites for the courses are the intro calculus sequence and the intro physics sequence.

There is no book for modern physics at Wayne State this semester, the book for optics is

Principles of PHYSICAL OPTICS by C. A. Bennett, Wiley.

The first thing in the book, from the looking inside a few pages on Amazon, is the wave partial differential equation.

I've taken the calculus sequence and differential equations and done well, but I've always thought even an intro course in physics was harder than any 200 level math class. So I'm a bit scared of this 500 level optics course. I'm going to look into partial differential equations, but I do not know where to begin. In my differential equations class I am taking right now, it seems that none of the equations we use are going to show up in applications, aside from power series solutions, which we are doing now, and the Laplace transform. It seems like an antiquated class that should be way more focused on modeling. I think the MIT course is a lot better than my schools, the MIT open courseware one.

I suppose then when this semester is over I'm going to go over multivariable and vector calculus again and some partial differential equations and read the modern physics part of the intro physics books, get to know the eV units.

I am doing the intro to electromagnetism right now and having a hard time fully understanding what is actually happening in reality. I think a lot of assumptions are made in the course to make the math doable for the level, but I wish I understood it better. That's the main importance to me, to internalize and really understand the physics and the math, that's my primary motivation, to just better understand the world around me. I was taking organic chemistry, on set to be a chemist, and I was like, 'I don't care about these huge molecules at all', and had to switch to the most fundamental of sciences to look for knowledge.

Thanks again so very much for the help, and if you guys have any other ideas about what I should focus on, (I want to study this stuff 8 hours a day in the month between semesters), it will be much appreciated.

Sorry that my comments kind of blurred together, I am new to the site, if I wrote this again, I would answer each of you directly in turn, rather than blending it all together, sorry about that.

 

[jtbell]

I dug up a previous syllabus for the optics course from the WSU web site:

http://physics.clas.wayne.edu/syllabi-w2012/phy5430.pdf

Based on that, i wouldn't freak out about the differential wave equation. It will probably be used mainly to justify the fact that we use sines and cosines to describe electromagnetic waves.

 

[bossman27]

The official prerequisites for the courses are the intro calculus sequence and the intro physics sequence.

If that's the case for optics, don't worry about the more advanced stuff I mentioned. I doubt you'll need to worry about Fourier optics.

I've taken the calculus sequence and differential equations and done well, but I've always thought even an intro course in physics was harder than any 200 level math class. So I'm a bit scared of this 500 level optics course.

That's natural, but if the prereq's were only basic calc and into physics you should be just fine. Especially if you study over break, you'll be one step ahead of everyone else.

I suppose then when this semester is over I'm going to go over multivariable and vector calculus again and some partial differential equations and read the modern physics part of the intro physics books, get to know the eV units.

Sounds like a good plan. Again, don't psyche yourself out about the math. If you just understand the basics of partial differential equations, and do some special relativity problems, you'll be in great shape.

I am doing the intro to electromagnetism right now and having a hard time fully understanding what is actually happening in reality. I think a lot of assumptions are made in the course to make the math doable for the level, but I wish I understood it better. That's the main importance to me, to internalize and really understand the physics and the math, that's my primary motivation, to just better understand the world around me.

Welcome to the club! As I said above, if you aren't getting enough information from your classes, you should absolutely do one of two things:
1) Go to your prof's or TA's office hours. Most of them are more than happy to sit and talk through concepts more thoroughly with you.
2) Don't be afraid to do a little solo research. Luckily we live in the internet age so if you feel like you don't have enough depth of knowledge in a particular area, I guarantee that you can find it online without any trouble.
As far as simplifying assumptions goes, that is true about intro classes, but if you stick with physics it will be a lot more thorough in your intermediate/upper-division E&M and Mechanics Classes.

Thanks again so very much for the help, and if you guys have any other ideas about what I should focus on, (I want to study this stuff 8 hours a day in the month between semesters), it will be much appreciated.

My suggestion is that if you get bored of math, play around with those Modern Physics concepts. There's a lot of fun to be had with special relativity, relativistic collisions, basic quantum phenomenon, etc. and you can never have too good of an intuition about those concepts, so it'll never hurt to play them more. If you think it'd help, I can send you some homework sets from that class for you to figure out.

Sorry that my comments kind of blurred together, I am new to the site, if I wrote this again, I would answer each of you directly in turn, rather than blending it all together, sorry about that.

No apologies necessary amigo, glad to help.

 

[Levi Tate]

jtbell, thanks a lot for finding that. That's my teacher and my book we're using. Huge help to me. I can read the whole book before the semester begins!

Do you think you could find a syllabus for modern physics? I have no idea how you did that.

bossman27, thanks again for the advise, I feel really good going into next semester now knowing things I can do to try to understand what is happening. I think i'll just go through the modern physics part of the intro Serway or Tipler book and read the whole thing.