Interesting! I just found prof. Tonti's book and am reading through it now. It seems like the methods he's looking at apply to a whole bunch of different branches/subjects; I'm surprised the Youtube videos don't have more views than they do. How did you find these, anyway?
Yeah, Feynman has some good examples that show that Hamilton's Principle does apply to some problems in E&M (and implicitly uses the Euler-Lagrange equation to solve it by introducing a variation, doing the difference between the varied answer and the the minimum answer, and setting the...
Hello all, so I’ve been reading Jennifer Coopersmith’s The Lazy Universe: An Introduction to the Principle of Least Action, and on page 72 it says:
If I understand it right, she’s saying that in our Euler-Lagrange equation ## \frac {\partial L} {\partial q} - \frac {d} {dt} \frac {\partial L}...
Interesting! The video was pretty useful, though I'm afraid I don't have the background in variational calculus / the Hamiltonian reformulation of classical mechanics to quite follow Schrödinger's thought process yet. I've started reading Robert Weinstock's Calculus of Variations with...
Hahah, yeah, I noticed it produces a negative energy (or a sign-change flip-flop) that can only be resolved by changing all the ##i##'s to ##-i##'s in the operators. Whoops!
Huh, interesting. Does that carry any deeper meaning? Does it imply that the rules for quantum mechanics would produce...
I was working through some examples and it seems like, if you were trying to solve Schrodinger's equation in some "alternate universe" where the phasors rotated counter-clockwise instead of clockwise, you could do it: you'd just have to set up the left side of Schrodinger's equation to read...
Okay, I'm with you so far, but one thing is bothering me: it seems, then, that the direction we chose our phasors to rotate in was somewhat arbitrary. If all you can measure is the relative difference in phase between different energy eigenstates, then it seems like all that matters is that you...
Hello all,
So I've been working through the solutions to some simple introductory problems for the Schrodinger Equation like the infinite square well, and I'm trying to make sense of how to think about the phase component. For simplicity's sake, let's start off by assuming we've measured an...
Great! Thanks for all your help. One last question: if we built a sort of "black box" around the atom and detector #2, so that we have the photon gun firing photons into the box and detector #1 detecting photons that exit the box on the opposite side, can we use the energy distribution at...
Ah okay, that makes a lot more sense. I think my main problem was in assuming that just because the photons from the photon gun didn’t have a well-defined energy (i.e. energy eigenvalue), that meant that the photons didn’t carry any information at all about their energy. But if, like you said...
Okay, let me see if I can articulate a little better where my confusion is coming from:
We assume that photons, as they’re fired from the photon gun on their way towards the atom, have no predetermined energy value: there’s no “nametag” or “stamp” on the photons to indicate what energy value...
Imagine for a second that you do the calibration test with no atom present; you observe the calibration energy distribution as a result. Now imagine you add the atom and measure the energy distribution of the photons which pass through it and don't get absorbed; this time, you get a lower energy...
And to clarify, I mean "get shifted down" from the energy distribution we'd expect, not "get shifted down" from the energy they had previous to passing through the atom, since we've already established that we can't talk about the energy of anyone photon before it's measured
Hmm, okay. It makes sense that the combined data looks the same as the calibration data, and it makes sense that the data from detector #2 is consistent with the energy distribution for the atomic transition. But can you explain how the photons at detector #1 wind up with a lower energy...
Hi all, thanks for the insights. This has helped to clarify things, though now it's introduced a new question to me that I can't make heads or tails of: it would seem that the photons know, as they leave the photon gun, whether or not they'll encounter an atom at some point along their way...
Hmm, okay. So let me see if I understand:
Let's say we start off with a photon gun configured to fire photons with, on average, an energy of 10.199 eV. To confirm that the gun is working as expected, we set up a detector to measure these photons' energy. Running this test many times, we see a...
Okay, in that case, let's modify step 1 of the procedure so that we do measure the energy of the incident photon each time we run the experiment, then we continue steps 2 and 3 as normal - so each time we run the experiment, we measure both the energy of the photon fired at the atom, and the...
Thanks for the reply! Okay, I think that makes sense. So let's say for a second that the following is our procedure:
1.) We've set up a "photon gun" to fire photons at the atom. It's designed to fire photons at 10.199 eV, but on any given run of the experiment, the actual energy of the photon...
Sorry to necro an old thread, but did anyone ever figure this one out? Does a hydrogenic electron in its first excited state emit a photon with (on average) an expectation value of 10.2 eV when it decays back into its ground state, or is the expectation value of the emitted photon instead equal...
Ah okay, that makes sense. But does this mean that there's a way in which we can use the uncertainty of the excited state's energy level to "cheat" conservation of energy laws? For example, if the expected transition energy for the electron is 10.2 eV but we start firing photons at it with an...
Quick question: let's say we have an atomic electron in the ground state which requires, say, one "unit" of energy* to jump up to the next orbital energy state. If a photon arrives with a bit more or less than this, say 1.00003 or 0.99997 units of energy, is there some finite, non-zero...