What are some less famous equations that you must know as an undergraduate physicist?

Join the discussion
Ask a follow-up here, or get your own question answered by working scientists, mathematicians and engineers — people, not an autocomplete.
Real named experts · corrections over time · the nuance an AI answer skips
Sign up Log in
21 replies · 6K views
pines-demon
Science Advisor
Gold Member
Messages
1,101
Reaction score
950
I was thinking of writing a motivational essay for physics undergrads where I provide a list of equations which have the property of telling you when you mastered (at undergraduate/early graduate level) a field of physics. Here are some examples:
  • Electromagnetism: ##{\partial}^2 A^{\sigma} = \mu_{0} \, J^{\sigma}##, it shows that you master all of Maxwell's equations, potentials and you have written them in relativistic formulation
  • Mechanics: ##\{f, g\} = \sum_{i=1}^{N} \left( \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i}\right).##, it shows that you have gone beyond Newtonian mechanics, into Lagrangian, and past it into Hamiltonian mechanics. Alternative it could be Hamilton-Jacobi equations.
I am looking for equivalents for thermodynamics/statistical mechanics (maybe the grand canonical partition function?), special relativity (a relativistic field Hamiltonian? stress-energy tensor?), non-relativistic quantum mechanics (maybe Bell's theorem?, von Neumann's equation?). I am just trying to find iconic but not so common that you could find it in popular science videos on famous equations. Any suggestions are welcome.

Extra points if you have any other idea like these for other advanced topics condensed matter, general relativity, quantum field theory and so on (or for a subdivision of the topics like Newtonian mechanics or thermodynamics). The idea is that when you reach one of those equations you know you mastered and it's time to go into something different or more advanced.
 
Last edited:
Reply
  • Like
Likes   Reactions: mathisrad and PhDeezNutz
Physics news on Phys.org
ergospherical said:
to be cynical, why does knowing an equation make you a master?

you will make a more insightful article by explaining just 1 equation, rather than 50 equations...
(we have wikipedia for that)
It is more like a checklist, it is about making a list of goals. Listing topics is already widely used, I was thinking about proposing something different.
 
Reply
  • Like
Likes   Reactions: PhDeezNutz
This does beg the question: when do you really understand an equation?

But I'll toss in some of votes:
- Schrödinger, of course
- E2=(pc)2+(mc2)2 plus all the ## \gamma ## stuff.
- Navier-Stokes
- Fourier & Laplace transforms.
- Maybe Stokes' & Green's theorems?
- Something from basic statistics, probably definitions. Like χ2?
- KVL & KCL. Yes, they are included in Maxwell's, but they are simple and everyone knows and uses them at some point.

Yes, some are really just math, but every physicists uses those.
 
DaveE said:
This does beg the question: when do you really understand an equation?

But I'll toss in some of votes:
- Schrödinger, of course
- E2=(pc)2+(mc2)2 plus all the ## \gamma ## stuff.
- Navier-Stokes
- Fourier & Laplace transforms.
- Maybe Stokes' & Green's theorems?
- Something from basic statistics, probably definitions. Like χ2?
- KVL & KCL. Yes, they are included in Maxwell's, but they are simple and everyone knows and uses them at some point.

Yes, some are really just math, but every physicists uses those.
Again the idea is to use less well known equations, Schrödinger equation can be found in t-shirts, same with Einstein's energy-mass relation. I want end-of-course equations not the main equation. Fourier and Laplace transforms and Green's theorem are kind of that but not specific to any physics course.
 
Reply
  • Like
Likes   Reactions: MatinSAR
Oops, sorry. I missed the part about your equations being obscure. I'm not sure why you want that, it seems like an arbitrary requirement for an undergraduate physics knowledge metric. But, good luck. Please share your list with us when you're done.
 
DaveE said:
Oops, sorry. I missed the part about your equations being obscure. I'm not sure why you want that, it seems like an arbitrary requirement for an undergraduate physics knowledge metric. But, good luck. Please share your list with us when you're done.
I would not qualify them as obscure, any modern physicist will recognize these equations. These equations indicate the final topics that you encounter before graduation.
 
Reply
  • Like
Likes   Reactions: DaveE
Included in the list should be the Planck blackbody function ## L(\lambda, T)=\frac{2 hc^2}{\lambda^5(\exp(\frac{hc}{\lambda k_b T})- 1)} ## along with ## \int\limits_0^{+\infty} L(\lambda, T) \, d \lambda=\frac{\sigma T^4}{\pi} ##.

You could even include ## \sigma=(\pi^2/60)(k_b^4/(\hbar^3 c^2) ##.
 
Reply
  • Like
Likes   Reactions: Astronuc and dextercioby
Charles Link said:
Included in the list should be the Planck blackbody function ## L(\lambda, T)=\frac{2 hc^2}{\lambda^5(\exp{\frac{hc}{\lambda k_b T}}- 1)} ## along with ## \int\limits_0^{+\infty} L(\lambda, T) \, d \lambda=\frac{\sigma T^4}{\pi} ##.

You could even include ## \sigma=(\pi^2/60)(k_b^4/(\hbar^3 c^2) ##.
That's a formula that you learn at the beginning of quantum mechanics not at the end.
 
pines-demon said:
That's a formula that you learn at the beginning of quantum mechanics not at the end.
Not sure if you are giving it a thumbs up or a thumbs down, but IMO the 3 of these are really very useful, especially for someone at the advanced undergraduate level.
 
Reply
  • Like
Likes   Reactions: dextercioby
Charles Link said:
Not sure if you are giving it a thumbs up or a thumbs down, but IMO the 3 of these are really very useful, especially for someone at the advanced undergraduate level.
I was looking for end course equations.
 
pines-demon said:
I was looking for end course equations.
I don't think that I saw the second and third formulas until I was a graduate student. I don't think today's students are any more advanced than my generation was 40+ years ago=at least in the Statistical Physics realm. The three formulas kind of go together, but they weren't introduced to us that way=we all saw the Planck formula first, but very sparingly.
 
Reply
  • Like
Likes   Reactions: Astronuc
Charles Link said:
I don't think that I saw the second and third formulas until I was a graduate student. I don't think today's students are any more advanced than my generation was 40+ years ago=at least in the Statistical Physics realm.
Some do. Anyway, Planck's law is at the beginning of any modern physics course.
 
Let me try one more: Linear response theory
## V_{out}(t)=\int\limits_{-\infty}^{t} m(t-t')V_{in}(t') \, dt' ##
with ## \tilde{V}_{out}(\omega)=\tilde{m}(\omega) \tilde{V}_{in}(\omega) ##.

Edit: With Fourier transform ## \tilde{F}(\omega)=\int\limits_{-\infty}^{+\infty} F(t) \exp(-i \omega t) \, dt ##
and inverse transform ## F(t)=\frac{1}{2 \pi} \int\limits_{-\infty}^{+\infty} \tilde{F}(\omega) \exp(i \omega t) \, d \omega ##.
 
Last edited:
Reply
  • Like
Likes   Reactions: Astronuc and WWGD
Dirac Delta function ?
 
Reply
  • Like
Likes   Reactions: Charles Link
pines-demon said:
That's a formula that you learn at the beginning of quantum mechanics not at the end.
Is there an end to quantum mechanics?