Turns out that the perturbation order that I worked on is purely dissipative and that all that work was pointless lmao.
Update: Turns out that I was just sloppy and forgot to include transition rates in the final step of my calculation. Seems like I can proceed as usual.
Science do be like that :(
I'm sorry for ya, that sucks.
Joke response would be "sounds like a paper!"
I remember working as an undergrad assistant to my math profs for one of their papers and them telling me that math was nice because sometimes you're allowed to publish non results like that.
Well somebody had to find out whether that perturbation order was purely dissipative or not.
Well that'll teach you for learning enough math to master Hamiltonians. They have taken us for absolute fools.
I'm kinda beginning to wish I'd become a scientist in whatever field this is so I could name something new after myself:
Evilsandwichmanians!
Legit though, if I ever discovered something new that people would be applying in their mathematical research for years to come, I'd choose something long and tedious to say but just short enough that you wouldn't abbreviate it and circumvent the intention of being annoying.
It is an interesting discovery, but it‘s not enough for my entire thesis, meaning I basically have to start anew with either a higher perturbation order (more likely option) or a different system entirely. At least I am much better at characterising this shit, so it might not take quite as long.
After years of calculations and fine tuning, it turns out Han Solo couldn't have shot first.
I can try... Generally, in quantum mechanics, you mostly focus on a certain (very small) system and assume that the system's environment has no greater relevant impact on the way your system develops. In physical terms, this means that we assume that our system is closed. However, no system ever truly exists in a vacuum, which is why environmental contributions to your systems have to be factored in, if you want to more accurately describe a system. But because describing both system and environment in full is extremely complicated, we separate our total system into system and environment and only focus on the impact the environment has in interaction with the system without concerning ourselves with the environmental state. This is the basic premise of the theory of open quantum systems.
Quantum thermodynamics is the attempt of reconciliation of quantum mechanics and classical thermodynamics. In classical TD, the microscopic interactions of the system are usually not all that relevant as we are mostly interested in the state of the macroscopic system (if you measure the temperature in a room you don't exactly give a shit about how singular molecules of nitrogen and oxygen bump into each other, for example). For very small systems (those relevant in QM), however, these microscopic interactions are what governs the entire system. Classical concepts such as heat and work don't have solid quantum mechanical definitions either, which is why even the most fundamental theorems of TD are difficult to translate to quantum mechanics. Finding an equivalent of these classical terms for my specific quantum system is basically what I set out to do.
My problem, however, is that when I characterised my quantum system, I (thought that I) found that it is "entirely dissipative in the perturbation order I looked at". These are words only uttered by the completely deranged. Perturbation order means the following: In QM, it is often analytically impossible/extremely complicated to fully describe the dynamics (the way a system behaves in time) of a system. What we can do, however, is to basically decompose our dynamics into terms of ever higher order with less impact in every higher order term (this is not necessarily true for every single order, I think, but as a general trend, this holds). We basically choose an order for which we say "all higher order terms are negligible in our case" and basically ignore the rest. The fact that my system appeared to be entirely dissipative means that the energy of my system itself was completely unaffected by the dynamics in the perturbation order, meaning if I "drive the system", meaning I change characteristics of my environment (basically, I increase the temperature), nothing happens to my system, which would have made any attempt of quantum thermodynamics pointless, as the energy of my system seemed to be unaffected. Luckily, I just turned out to be wrong.
