This is what I was referring to^

@Mithrandir24601

@MoreAnonymous To start with, I'm talking about the principle of a measurement changing energy, so I'm talking about a single 2-level system in order to understand the concept - if a measurement can change the expectation value of the energy for a 2-level system, then I see no fundamental reason why it can't for a [extremely large number]-level system, which could be e.g. your gas

@Mithrandir24601 Lets say you change the energy by $\Delta E$ my claim is $\langle \Delta E \rangle = 0$

So, for said single 2-level system, you start in your state $\left|\psi\right>$, the expectation value of energy is given by $\left<\psi|H|\psi\right>$. After measuring, it's either $\left<g|H|g\right>$ or $\left<e|H|e\right>$, which is different

@Mithrandir24601 Agreed ... but how many times will you get either $\left<g|H|g\right>$ or $\left<e|H|e\right>$ ?

@MoreAnonymous So you're saying that you have a large number of uncoupled 2-level systems? (which may or may not describe a gas, I don't know)

@Mithrandir24601 Well we can proceed with that example ... And my claim will hold I believe

You expect by the law of large numbers, that, as the number of systems go up, it will tend towards the expectation value

But that doesn't make it impossible that you'd measure everything to be in e or g

@Mithrandir24601 Oh everything will definitely be in e or g ... Its just that this will as you expect it will tend to the expectation value ... I'm sure the measuring device will use more energy than the delta of energy indicated here

@MoreAnonymous Yeah, it's worth being very clear that, in the limit of an infinite number of such systems, you'll get the expectation value of the energy and also that in any real, practical system, thermodynamics/probability (same thing in this case) dictates that you'll be close to the expectation value with some stupidly high probability

It's essentially that thing where the thing that seemingly violates the laws of thermodynamics can technically happen but the probability of it happening is so ridiculously low that you'd be waiting forever and beyond

Yups ... So are we on the same page that I still don't know the answer to my question?

@MoreAnonymous I believe it would depend on whatever correlations were or were not present in the system before measurement as well as things like if you're able to measure faster than the inherent interactions of the system ('can you make the measurements Markovian?'). If you're not able to do the later, then I'm fairly sure that whatever you've just measured will just go back to being in the overall thermal state

Point being, you're essentially measuring an environment, so assuming everything's Markovian, it just forgets whatever

@Mithrandir24601 So basically while I haven't learned to create energy I've managed to destroy it :P Since I can get rid of all the energy in the measuring instrument and have nothing to show :P

Yeah, something like that :P

If you keep track of the numbers I assume you'd be able to use them to calculate the temperature of the system, but that sounds like a very innefficient way to do that :P

Here's something I had realized a while back ... If I use a density matrix and you use the wavefunction to describe the system then $\pho | \psi \rangle $ is an invariant