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The generalised uncertainty principle for any Hermitian operators $\hat{A}$ and $\hat{B}$ can be proven using the fact that the inner product of the following operator is non-negative: $$\hat{O} = \hat{\alpha} + i\lambda\hat{\beta},$$ where $\hat{\alpha} = \hat{A} - \langle\hat{A}\rangle$ and $\hat{\beta} = \hat{B} - \langle\hat{B}\rangle$. These are chosen such that $$(\Delta{A})^2 = \langle(\hat{A} - \langle\hat{A}\rangle)^2\rangle = \langle \hat{\alpha}^2\rangle$$ and $$(\Delta{B})^2 = \langle(\hat{B} - \langle\hat{B}\rangle)^2\rangle = \langle\hat{\beta}^2\rangle.$$ I will write the full proof below for reference.

We construct the operators $\hat{\alpha}$, $\hat{\beta}$, and $\hat{O}$ in order to relate the uncertainties $\Delta A$ and $\Delta B$ of measuring $\hat{A}$ and $\hat{B}$. We use the positive-definite property of the inner product in order to prove the generalised uncertainty principle. This seems like a mathematical trick. My question is: is there any (other) significance of the operator $\hat{O} = \hat{\alpha} + i\lambda\hat{\beta}$. Is there anything deeper to be said about the fact that it is this operator's inner product that gives us the uncertainty principle and not any other's. I understand the proof mathematically, but since the principle itself is so fundamental to nature, I was wondering what the importance of $\hat{O}$ is in nature, if that makes sense.

Full proof: $$\langle\hat{O}\psi|\hat{O}\psi\rangle = (\langle\hat{\alpha}\psi|+i\lambda\langle\hat{\beta}\psi|)\cdot(|\hat{\alpha}\psi\rangle+i\lambda|\hat{\beta}\psi\rangle)$$ $$= \langle\hat{\alpha}^2\rangle+\lambda^2\langle\hat{\beta}^2\rangle + i\lambda\langle[\hat{\alpha},\hat{\beta}]\rangle$$ because $\hat{A}$ and $\hat{B}$ are Hermitian, which implies that $\hat{\alpha}$ and $\hat{\beta}$ are Hermitian.

$$\langle[\hat{\alpha},\hat{\beta}]\rangle = \langle\psi|\hat{\alpha}\hat{\beta}|\psi\rangle - \langle\psi|\hat{\beta}\hat{\alpha}|\psi\rangle\\ = \langle\psi|\hat{\alpha}\hat{\beta}|\psi\rangle-\langle\psi|\hat{\alpha}\hat{\beta}|\psi\rangle^{*} \\ = z - z^{*} \in \mathbb{I}.$$
Define $ib = \langle[\hat{\alpha},\hat{\beta}]\rangle$, where $b \in \mathbb{R}$. Therefore, $$\langle\hat{O}\psi|\hat{O}\psi\rangle = (\Delta{A})^2 + \lambda^2(\Delta{B})^2 -\lambda b \geq 0 ~.$$

This is a quadratic equation in $\lambda$. The positive definite condition of the quaratic implies that its discriminant must be $\leq0$, $$b^2 - 4 {{(\Delta{A})}^{2}} {{(\Delta{B})}^{2}} \leq 0$$ $${{(\Delta{A})}^{2}} {{(\Delta{B})}^{2}} \geq \frac{1}{4}b^2,$$ where $b = |ib| = |\langle[\hat{\alpha},\hat{\beta}]\rangle|$ and it can be shown that $[\hat{\alpha},\hat{\beta}]=[\hat{A},\hat{B}]$. Therefore, $${{(\Delta{A})}^{2}} {{(\Delta{B})}^{2}} \geq \frac{1}{4}|\langle[\hat{A},\hat{B}]\rangle|^2 .$$

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    $\begingroup$ I would caution you to assume that the uncertainty principle "fundamentally" characterizes quantum mechanics. It shows up in any theory of linear systems, including classical ones. I would call it a more or less trivial mathematical statement about certain types of linear operators that just happen to be quite frequent in physics. That quantum mechanics is perfectly linear follows from the assumption that all systems in an ensemble are statistically independent, as far as I know. It's not a physical "fact", at least not any more than the ergodic hypothesis. $\endgroup$ Commented Jun 9, 2023 at 19:35
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    $\begingroup$ It's a fine, elegant, mathematical trick, invented to streamline the flow of logic. Can you explain what more you want from it? As you read in WP, it is moored in the structure of Fourier analysis, and so it obtains in signal processing, and other fields, beyond QM. Even in other formulations of QM it has its formal analogs. The book of nature is written in math, as Galileo observed, and you never need physics dress-up for bad math, as P Morse often warned his students about. $\endgroup$ Commented Jun 9, 2023 at 20:48

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Yes. The eigenstates of the (non-hermitian) operator $$ \hat O=\hat \alpha+i\lambda\hat \beta $$ are the so-called intelligent states. They were introduced in

Aragone, C., Guerri, G., Salamo, S. and Tani, J.L., 1974. Intelligent spin states. Journal of Physics A: Mathematical, Nuclear and General, 7(15), p.L149.

for angular momentum as states that saturate the uncertainly relation, i.e. for those eigenstate $\vert\lambda\rangle$ one has $$ \Delta \alpha^2\Delta \beta^2=\frac{1}{4}\vert\langle \lambda\vert [\hat \alpha,\hat \beta]\vert \lambda\rangle\vert^2\, , $$ where the $=$ sign replaces the $\ge$ sign.

There are several ways of constructing intelligent states for various types of systems. A typical example of paper on the topic is

El Kinani, A.H. and Daoud, M., 2002. Generalized coherent and intelligent states for exact solvable quantum systems. Journal of Mathematical Physics, 43(2), pp.714-733.

but searching for "intelligent states" in GoogleScholar will return a good number of hits.

The coherent states are examples of intelligent states for $\hat x$ and $\hat p$, and indeed the squeezed states for $\hat x$ and $\hat p$ are also "intelligent". In general, since the operator $\hat O$ is not hermitian, intelligent states are not orthogonal, and again the coherent states or squeezed states are examples of non-orthogonal states. Since the operator $\hat O$ is not hermitian, its eigenvalues can be complex.

Intelligent spin states have applications in quantum optics, as discussed for instance in

Wodkiewicz, K. and Eberly, J.H., 1985. Coherent states, squeezed fluctuations, and the SU (2) am SU (1, 1) groups in quantum-optics applications. JOSA B, 2(3), pp.458-466.

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