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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.

Andrew
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Daniel Waters
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  • Are you cool with the KvN formalism and the peculiar orthonormal phase-Hilbert-space basis employed? – Cosmas Zachos Oct 05 '20 at 00:20
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    The second displayed formula you write is nonsense, since you've already integrated over dummy variables x and p on the right hand side. – Cosmas Zachos Oct 05 '20 at 00:28
  • 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. – Daniel Waters Oct 05 '20 at 00:54
  • 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. – Daniel Waters Oct 05 '20 at 00:55
  • I am using his second book on Quantum as an introduction, should I switch to Shankar instead? – Daniel Waters Oct 05 '20 at 01:02
  • I am not familiar with his book, but you may rest assured Shankar's book is sound and tasteful. My guess is that it is superior. Lip-smacking nonsense is not a nice thing to throw at students. – Cosmas Zachos Oct 05 '20 at 01:04
  • Thanks! I got his books because they claim to explain everything simply, and he did a good job with Noether's Theorem and Lagrangian/Hamiltonian Mechanics, but when it came to this, he fell short. I will probably use his quantum book, and double check everything with Shankar or Griffiths, just to be safe. It seems like when he has more time to get to the heart of something he explains it well, just not a complex topic like KvN given that he tried to do it in 7 pages. Thanks again for your help! – Daniel Waters Oct 05 '20 at 01:09
  • It is conceivable by "vanishes" he just means "collapses". I'll write a placeholder answer of how I would go from your corrected second equation to your first. – Cosmas Zachos Oct 05 '20 at 01:15

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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.

Cosmas Zachos
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