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We can assume WLOG that $\bar x=\bar p=0$ and $\hbar =1$. We don't assume that the wave-functions are normalised. Let $$\sigma_x\equiv \frac{\int \mathrm dx\; |x|\;|\psi(x)|^2}{\int\mathrm dx\; |\psi(x)|^2}$$ and $$\sigma_p\equiv \frac{\int \mathrm dp\; |p|\;|\tilde \psi(p)|^2}{\int\mathrm dx\; |\psi(x)|^2}$$ Using $$\int\mathrm dp\ |p|\;\mathrm ... 6 The temperature limit for laser cooling is not related to gravity but to the always-present momentum kick during absoprtion/emission of photons. Ultracold atom experiments typically use laser cooling at an initial stage and afterwards evaporative cooling is used to reach the lowest temperatures. In evaporative cooling the most energetic atoms are discarded ... 4 I went back to the derivation of the Heisenberg uncertainty principle and tried to modify it. Not sure if what I've come up with is worth anything, but you'll be the judge: The original derivation Let \hat{A} = \hat{x} - \bar{x} and \hat{B} = \hat{p} - \bar{p}. Then the inner product of the state | \phi\rangle = \left(\hat{A} + i \lambda ... 4 This is a great example of how hard it is to popularize quantum mechanics. Greene's example is not quite right, because classically, the butterfly does have a definite position and momentum, at all times. We can also measure these values simultaneously to arbitrary accuracy, as your friend says. (As for your concern about exposure time, we could decrease ... 4 Summary Using the entropic uncertainty principle, one can show that μ_qμ_p≥\frac{π}{4e}, where μ is the mean deviation. This corresponds to F≥\frac{π^2}{4e}=0.9077 using the notations of AccidentalFourierTransform’s answer. I don’t think this bound is optimal, but didn’t manage to find a better proof. To simplify the expressions, I’ll assume ℏ=1, ... 3 It cannot be proven, because "wave-particle duality" is not a mathematical statement. It most definitely is not "logically true". Can you try to make it mathematical? A mathematical framework The "complementarity principle" was introduced in order to better understand some features of quantum mechanics in the early days. The problem is that if you consider ... 2 I) In this answer we will consider the microscopic description of classical E&M only. The Lorentz force reads$$ \tag{1} {\bf F}~:=~q({\bf E}+{\bf v}\times {\bf B})~=~\frac{\mathrm d}{\mathrm dt}\frac{\partial U}{\partial {\bf v}}- \frac{\partial U}{\partial {\bf r}}~=~-q\frac{\mathrm d{\bf A}}{\mathrm dt} - \frac{\partial U}{\partial {\bf r}},  ...

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Models based on string theory have consistent quantization of gravity. Within these , there is theoretical work carried out for generalizing the uncertainty principle for quantum gravity. One example : It generalizes the usual space momentum uncertainty,(formula 15) and another This one examines an uncertainty in space time, formula 2.2. It is ...

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The point dipole is an approximation from classical physics - note that it also involves an infinite field strength in its center, where the field amplitude is not differentiable. I think such a source is not compatible with the common approach to quantum mechanics. If you take such a very small, subwavelength source, it is true that the evanescent near ...

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The uncertainty principle never said that nothing can be measured simultaneously with accuracy. Uncertainty principle states that it is not possible to measure two canonically conjugate quantities at the same time with accuracy. Like you cannot measure the x component of momentum $p_x$ and the x coordinate position simultaneously with accuracy. But the x ...

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There is yet another solution (maybe more elementary)$^1$, with some components of the answers from Qmechanic and JoshPhysics (Currently I'm taking my first QM course and I don't quite understand the solution of Qmechanic, and this answers complement JoshPhysics's answer) the solution uses the Heisenberg Equations: The time evolution of an operator ...

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