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I can show how: \begin{equation} ΔqΔp \geq \frac{\hbar}{2} \end{equation} In a general way. You just need to consider the commutation properties of 2 generic operators to do so.

Both $p$ and $q$ satisfy this relation.

On the other hand, I can show how $h^{f}$ should be the area of the phase hypersurface, where $f$ is the dimension of the phase space. This is used when you want to integrate like: \begin{equation} \Gamma=\frac{∫ dp dq}{h^{f}} \end{equation}

Now i need to understand how to link both this infos. I've been taught that the cell in the hypersurface is the smaller cell you can find to be certain to have a particle detected in the phase space. It means that this cell is equal to the least uncertainty of $p$ and $q$. This makes sense, but it's false, according to the first equation.

How can I explain it?

Qmechanic
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Matteo
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    How you can show the statement about $h^f$? – warlock Sep 30 '22 at 08:43
  • Related: https://physics.stackexchange.com/questions/63318 – AlmostClueless Sep 30 '22 at 09:12
  • You have a box of side L. You want to enumerate the number of states. Let us consider the wave function $u(\vec{x})=e^{i\frac{2\pi} {L}\sum_{i}x_{i}n_{i}}$ so that it's periodic. We are fixing the boundaries, but when we let $L \rightarrow\infty$ they are not to be considered. Then we apply $p_{i}=-i\hbar\partial_{x}$. So $p|u(x)\rangle=\frac{2\pi n\hbar} {L} |u(x)\rangle=\frac{hn} {L}|u(x)\rangle$. Then the number of state would be: $\frac{L^{N}}{h^{N}}\sum_{p_{1}}...\sum_{p{N}}=\frac{\int d^{N}qd^{N}p} {h^{N}}$ – Matteo Sep 30 '22 at 20:56

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