Learning physics is hard. The purpose of this post is to collect a bunch of techniques that can help to make the process of learning (and remembering!) new concepts easier.
Tip #\(1\): Avoid always simplifying formulas as much as possible, group quantities in dimensionally useful combinations.
Explanation: Since elementary school, one is often taught to simplify everything as much as possible. While this is often a good idea if one is doing some calculation, sometimes a final result is better left unsimplified, especially if dimensionally intuitive quantities are grouped together.
Example: The density of states in \(k\)-space of an ideal electron gas, simplified, is:
\[g(k)=\]
However, it is much more preferable to remember it in the unsimplified form:
\[g(k)=\]
Example: The impedance \(Z\) of non-dispersive transverse displacement waves on a string of tension \(T\) and speed \(c\) is, when simplified, \(Z=\sqrt{Tc}\). However, it is preferable to remember it as:
\[Z=\frac{T}{\sqrt{T/c}}\]
since this reinforces that it’s just a “force/velocity”.
Tip #\(2\): Try covering up the microscopic inner workings with a black box.
Explanation: This is basically an engineering mindset. For instance, to use an op-amp, one does not have to understand the detailed transistor networks inside.
Example: A Fabry-Perot interferometer…looks like a diffraction grating viewed as a black box.
Tip #\(3\): Understand systems by isomorphism.
Explanation: Mathematicians love talking about (and finding) isomorphisms between various kinds of spaces and structures, since often one structure \(X\) and easier to understand than another structure \(Y\) but in fact both are really the same, so one can leverage one’s understanding of \(X\) in order to make sense of \(Y\).
Example: Electric circuits are not as intuitive to me as a damped, driven harmonic oscillator. Yet the \(2\) systems are in fact often isomorphic.
Tip #\(4\): Dimensional analysis! And don’t be afraid to package a lot of constants into one’s own custom-defined variables.
Explanation: As a general rule of thumb, use the least number of variable packages that eliminates all numerical factors (a catchy slogan to summarize this?).
Example: The dispersion relation for the \(n^{\text{th}}\) mode of waves propagating along the \(x\)-axis of a \(2\)D waveguide with fixed boundary conditions at \(y=0,\Delta y\) can be written:
\[\omega^2=c^2\left(k_x^2+\frac{n^2\pi^2}{\Delta y^2}\right)\]
This formula can be conceptually simplified by defining the cutoff frequency \(\omega_c:=\pi c/\Delta y\) so that:
\[\omega^2=c^2k_x^2+n\omega_c^2\]
Example: The dispersion relation for electromagnetic waves in a conductor at high frequencies can be written:
(plasma frequency)
Tip #\(5\): Build a “circuit model” of the system.
Explanation:
Example: A transmission line can be viewed as…
Tip #\(6\): Write formulas without any numerical prefactors, focusing on dimensional analysis.
Tip #\(7\): Remember some physical values of quantities in SI units, get a feel for orders of magnitude.
Example: Knowing that Avogadro’s constant is something like \(N_A\sim 10^23\) and Boltzmann’s constant is \(k_B\sim 10^{-23}\) (both when expressed in SI units), it follows that their product should be \(O(1)\) (again in SI units), and in this context one obvious \(O(1)\) constant is the gas constant \(R\approx 8.3\) so this helps to remember that:
\[N_Ak_B=R\]
Tip #\(8\): This tip applies specifically to studying new quantum systems. Basically, the algorithm for understanding is:
\[\mathcal H\to H\to H|\psi\rangle=E|\psi\rangle\]
In other words, first have a clear idea of what the Hilbert state space \(\mathcal H\) of the system is, what the Hamiltonian \(H:\mathcal H\to\mathcal H\) is, and (the hard part that needs to be worked out carefully) the eigenstates \(|\psi\rangle\) and their corresponding eigenenergies \(E\). With that, essentially everything is thus known about the system such as its time evolution, the time evolution of expectation values of observables, and measurement probabilities.
Example: