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        "quote": "\n\n> [!theorem] 定理（海森堡不确定性）\n>\n> $$\n> \\frac {\\langle \\psi | Q ^ {2} | \\psi \\rangle}{\\langle \\psi | \\psi \\rangle} \\frac {\\langle \\psi | P ^ {2} | \\psi \\rangle}{\\langle \\psi | \\psi \\rangle} \\geq \\frac {1}{4}\n> $$\n\n> [!proof]- 证明\n> 对于任意实数 $λ$，有\n>\n> $$\n> \\langle (Q + i \\lambda P) \\psi | (Q + i \\lambda P) \\psi \\rangle \\geq 0\n> $$\n>\n> 但是，利用$Q$和$P ,$的自伴性以及关系$[ Q , P ] = i$，有\n>\n> $$\n> \\begin{aligned}\n> \\langle (Q + i \\lambda P) \\psi | (Q + i \\lambda P) \\psi \\rangle &= \\lambda^ {2} \\langle \\psi | P ^ {2} \\psi \\rangle + i \\lambda \\langle \\psi | Q P \\psi \\rangle - i \\lambda \\langle \\psi | P Q \\psi \\rangle + \\langle \\psi | Q ^ {2} \\psi \\rangle \\\\[0.45em]\n> &= \\lambda^ {2} \\langle \\psi | P ^ {2} \\psi \\rangle + \\lambda (- \\langle \\psi | \\psi \\rangle) + \\langle \\psi | Q ^ {2} \\psi \\rangle\n> \\end{aligned}\n> $$\n>\n> 对于所有 $λ$，这将是非负的，如果\n>\n> $$\n> \\langle \\psi | \\psi \\rangle^ {2} \\leq 4 \\langle \\psi | P ^ {2} \\psi \\rangle \\langle \\psi | Q ^ {2} \\psi \\rangle\n> $$\n",
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        "quote": "\n\n> [!theorem] Theorem (Heisenberg uncertainty)\n>\n> $$\n> \\frac {\\langle \\psi | Q ^ {2} | \\psi \\rangle}{\\langle \\psi | \\psi \\rangle} \\frac {\\langle \\psi | P ^ {2} | \\psi \\rangle}{\\langle \\psi | \\psi \\rangle} \\geq \\frac {1}{4}\n> $$\n\n> [!proof]- Proof\n> For any real $λ$ one has\n>\n> $$\n> \\langle (Q + i \\lambda P) \\psi | (Q + i \\lambda P) \\psi \\rangle \\geq 0\n> $$\n>\n> but, using self-adjointness of $Q$ and $P ,$ as well as the relation $[ Q , P ] = i$ one has\n>\n> $$\n> \\begin{aligned}\n> \\langle (Q + i \\lambda P) \\psi | (Q + i \\lambda P) \\psi \\rangle &= \\lambda^ {2} \\langle \\psi | P ^ {2} \\psi \\rangle + i \\lambda \\langle \\psi | Q P \\psi \\rangle - i \\lambda \\langle \\psi | P Q \\psi \\rangle + \\langle \\psi | Q ^ {2} \\psi \\rangle \\\\[0.45em]\n> &= \\lambda^ {2} \\langle \\psi | P ^ {2} \\psi \\rangle + \\lambda (- \\langle \\psi | \\psi \\rangle) + \\langle \\psi | Q ^ {2} \\psi \\rangle\n> \\end{aligned}\n> $$\n>\n> This will be non-negative for all $λ$ if\n>\n> $$\n> \\langle \\psi | \\psi \\rangle^ {2} \\leq 4 \\langle \\psi | P ^ {2} \\psi \\rangle \\langle \\psi | Q ^ {2} \\psi \\rangle\n> $$\n",
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