it captures the idea that, as you shrink an interval of with $\Delta x$ to zero, the probability of finding the particle there will shrink as well. but it'll shrink at a rate proportional to $\Delta x$
the point is that this is not an especially "quantum" thing
this is exactly how probabilities have to work for any continuous distribution
the way you quantify the mass density at some point in an object is to extract a sample of said object that includes said point, and is "small enough" that the mass density is approximately constant in that sample
then the mass density at that point is just sample mass / sample volume
or $\left(\int_{\delta V} \rho(x)\,dx\right)/\delta V$
it's the same idea for probability density: you pick some small interval in which to look for the particle, and divide the probability of such by the size of the interval
@Semiclassical Well, to be honest I don't really know what to do right now. I think I will have some break for now. But I'm very thankful for your effort in explaining me. Thanks so much! This chat was more helpful than my instructor's notes.
the mass $M=\int \rho(\vec{r})dV(\vec{r}),$ the center of mass $\vec{r}_{CM} =M^{-1}\int \vec{r} \rho(\vec{r})\, dV(\vec{r})$, and the moment of inertia $I=\int r^2 \rho(\vec{r})\,dV(\vec{r})$
the claim i want to make is that those are not actually so different than the expectation values $\langle 1\rangle$, $\langle x\rangle$, $\langle x^2\rangle$
with the middle case being essentially a one-to-one correspondence
Of course it doesn't end up doing that but they are very useful to check, apart from qft the amount of updates L&L need for what they do is relatively small
e.g. they do quantum stat mech from first principles at the beginning (they set up the classical version first but you can literally ignore that the way they wrote it, after the beginning they do both simultaneously)
They are still based off his teaching notes, one of the biggest differences is they set up qft for fields without starting-from/setting-up the $[\hat{\phi}(x),\hat{\pi}(x')] = ...$ type commutation relations, which is actually an unbelievable virtue
@Slereah I've skimmed/read it (their intro one) at least a hundred times and I'll have to look at it again and it was worth it :p
@DIRAC1930 they derive the grand canonical distribution in two ways, section 35 and 36, the latter can be done after skimming section 4/6 in a few lines
I missed that section when looking through that book
So it turns out average number of particles in a subsystem is conserved
Does average number of particles in a subsystem mean that total number of particles is conserved as a consequence?
My reasons are that if the environment and the subsystem have 2N particles in total and they are both the same size etc, the subsystem should on average have <N>=N particles
if this average rises to say <N>=N+1
then that would imply that the exact number of particles in the whole system is 2N+2
sure, but usually you should think of the reservoir as just containing infinitely many particles - the system can't "drain" the reservoir, it's an idealization of the notion of environment
In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the particle density is low enough to render quantum effects negligible.
The expected number of particles with energy
ε
i
{\displaystyle \varepsilon _{i}}
for Maxwell–Boltzmann statistics is
⟨
N
i...
Lagrange multipliers are only in the microcanonical section there
I think I remember the rationalization I came up with before, if you are studying these kinds of relativistic systems, they are not really thermodynamical in that the effects of changing particle number are instantaneous and local, but as long as the average particle number change is zero you can apply thermodynamical thinking to them, they basically do this as part of setting up black body radiation
As it says there, most books just use $N$, which I'm used to... I guess it makes sense and you can view it as an extension of using $N$ and we now have $[\hat{Q},\hat{N}] = 0$, and $Q =0$ for photons so it obviously agrees with the black body case, otherwise it looks pretty much the same right... that's a bit of a shock
though, come to think of it, one of the exercises my students had to do this semester was show that $[H,N]=0$ still held when doing second quantization
I'd say this simple change from $N$ to $Q$ is all you need to do and you can then just repeat the rest of a stat mech book, I can't believe I didn't think of that well done
They start waffling at the start of the black body chapter to rationalize this, but what they say seems correct
i remember at least one source arguing that writing stuff like $|JM\rangle = \sum c^{JM}_{j_1 m_1}|j_1m_1\rangle$ was wrong b/c it has the wrong number of quantum numbers on both sides
For photon production, is it equally likely for 2 photons to be produced with an energy each of E as it is for one photon to be produced with energy 2E?
Three big stat mech books don't mention this issue about charge vs particle number being conserved, if you find one that does make this distinction like those notes do let me know
So that's why the black body thing is going to be the same if you replace $Q$ by $N$, but if you use $N$ you need to invent a rationalization for why $\mu =0$, whether it's really valid to even use the $N$ version is a question now
I think maybe it doesn't matter because wouldn't you usually calculate the correlation functions in a perturbative expansion with the coupling constant
In statistical mechanics, a grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium (thermal and chemical) with a reservoir. The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible...
They say $N$ is what we called $Q$ when you allow anti-particles, so yeah...
In theoretical physics, anti-de Sitter/condensed matter theory correspondence is the program to apply string theory to condensed matter theory using the AdS/CFT correspondence.
== Overview ==
Over the decades, experimental condensed matter physicists have discovered a number of exotic states of matter, including superconductors and superfluids. These states are described using the formalism of quantum field theory, but some phenomena are difficult to explain using standard field theoretic techniques. Some condensed matter theorists including Subir Sachdev hope that the AdS/CFT correspondence will...