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

[pines-demon]

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.

 

[ergospherical]

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)

 

[pines-demon]

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.

 

[DaveE]

This does beg the question: when do you really understand an equation?

But I'll toss in some of votes:
- Schrodinger, 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]

Or just steal it from the web. There's lots of potential victims...
https://physics.info/equations/

 

[pines-demon]

This does beg the question: when do you really understand an equation?

But I'll toss in some of votes:
- Schrodinger, 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.

 

[pines-demon]

Or just steal it from the web. There's lots of potential victims...
https://physics.info/equations/

The Physics Hypertexbook does not even cite my examples.

 

[DaveE]

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.

 

[pines-demon]

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.

 

[Charles Link]

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 $\sigma=(\pi^2/60)(k_b^4/(\hbar^3 c^2)$.

 

[pines-demon]

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.

 

[Charles Link]

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.

 

[pines-demon]

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.

 

[Charles Link]

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.

 

[pines-demon]

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.

 

[Charles Link]

I tried. Maybe someone else can give you something that you might be looking for.

 

[jasonRF]

Boltzmann transport equation
Fluctuation-dissipation theorem

 

[Charles Link]

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 $F(t)=\frac{1}{2 \pi} \int\limits_{-\infty}^{+\infty} \tilde{F}(\omega) \exp(i \omega t) \, d \omega$.