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A book that I used to learn basic Classical Mechanics, called "No-Nonsense Classical Mechanics" by Jakob Schwichtenberg, defines the probability density in Koopman-Von Neumann Mechanics as $$\rho(x,p,t)=|\Psi(x,p,t)|^2=|c(x,p,t)|^2$$ where $$\Psi(x,p,t)=\int c(x,p,t)e_{x,p} \, dx \, dp$$ where $e_{x,p}$ are the basis vectors for the Hilbert Space.

But, Schwichtenberg says that the above integral vanishes leaving only the constants $c(x,p,t)$, due to the orthonormality of the basis vectors, but does not delve any further. The problem is that I can't seem to find out how it vanishes or why. I have been looking on Quantum Mechanics webpages only to find nothing explaining why this integral should vanish. Furthermore, must this be true in order for $|c(x,p,t)|^2$ to denote the probability of finding the system in the state given by $c(x,p,t)$? If anyone could clarify on this or just help to explain the topic of probability density and its relationship with Wavefunctions and their coefficients in Quantum Mechanics and KvN, that would be incredibly helpful.

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  • $\begingroup$ Are you cool with the KvN formalism and the peculiar orthonormal phase-Hilbert-space basis employed? $\endgroup$ Oct 5, 2020 at 0:20
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    $\begingroup$ The second displayed formula you write is nonsense, since you've already integrated over dummy variables x and p on the right hand side. $\endgroup$ Oct 5, 2020 at 0:28
  • $\begingroup$ Cosmas Zachos I don't know too much about Hilbert space in general, this is my first introduction to it, but I think that the way the Author presented it isn't that great. He gave like 5 pages total to it. $\endgroup$ Oct 5, 2020 at 0:54
  • $\begingroup$ Cosmas Zachos the formulas are straight out of the book. I tried to use this section as a sort of introduction to the way Quantum works, and it has brought me nothing but confusion. $\endgroup$ Oct 5, 2020 at 0:55
  • $\begingroup$ I am using his second book on Quantum as an introduction, should I switch to Shankar instead? $\endgroup$ Oct 5, 2020 at 1:02

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I'll try to eschew the nonsense and write down the corrected second expression leading to the first.

The orthonormal vectors of this peculiar Hilbert space are $|x,p\rangle$, so that $$ \langle x,p | x',p'\rangle= \delta (x-x') \delta (p-p'). $$ Skip the time, since it is an inert parameter in these expressions. I'm using Dirac's bracket notation for basis vectors in Hilbert space. It then follows that $$ |\Psi\rangle= \int \!\! dx dp ~ c(x,p) |x,p\rangle , ~~\leadsto \\ c(x,p)= \langle x,p|\Psi\rangle , $$ by the above orthogonality. Conventionally, you'd write this coefficient $c(x,p)=\Psi(x,p)$, so, in this sense, the integral "vanishes".

The density in phase space is your first equation, $$ \rho(x,p) = |\Psi(x,p)|^2=|c(x,p)|^2. $$

It's not rocket science.

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  • $\begingroup$ Thank you, I feel like I sort of understand now! $\endgroup$ Oct 5, 2020 at 1:31

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