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Let's forget physics for a moment and just talk about the mathematics of waves. The uncertainty principle is a property of waves. Think of a single, narrow pulse traveling along one direction. The pulse is narrow, and so the position of the pulse at any given time is easy to quantify. But this is a single, non-periodic pulse. You can build up such a ...


9

In classical physics you are supposed to be able to measure the coordinates and the velocity (really the momentum) of a mass with infinite precision at the same time. If you try this trick in the lab you notice that that's not the case. Either your position or your momentum measurement or both will always show some non-trivial statistical fluctuations when ...


8

$\newcommand{\d}[1]{\,\mathrm{d}#1}\newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}}\newcommand{\p}{\psi_{100}}\newcommand{\pdvt}[2]{\frac{\partial^2 #1}{\partial #2^2}}$The hydrogen ground state is the following: $$\psi_{100}=Y_{00}\frac{2}{a_0^{3/2}}e^{-r/a_0}$$ The expectation value of the position operator on this state is the following: ...


3

There is a related (and relevant) question with answer called Uncertainty principle and measurement but I want to give a specific answer for this situation and background. You will need top know the difference between precision and accuracy. A precise dart thrower throws darts that land very close to each other, this isn't a deep statement, in fact it is ...


2

What you've fabricated is of course unrealistic in the physical world as CuriousOne stated, but not so in the virtual world of simulation. All the conditions you ask for can be arranged in a simulated universe. If perfectly balanced as its initial condition, the virtual pencil will not fall in this virtual world. It is unstable, but it will not fall until a ...


2

In the universe where this pencil is, there are no outside forces which can affect the pencil, other than gravity. But there are forces afoot inside the pencil. Unless you also chill the pencil to absolute zero and thus stop all molecular activity, the trillions of atoms in the pencil are vibrating in random directions. It won't take long for the ...


1

A highly simplified analogy that attempts to get thinking going in a direction that can sometimes lead to eventual understanding... Think of various photographs taken of a baseball in flight. Different photographs are taken with different exposures. They are part of an attempt to measure both speed and position at the same time. In particular, imagine that ...


1

Which way will the pencil fall, after you let go? Facetious answer: it would fall in the direction to which it was leaning when you let go. Okay, now to justify that: you cannot balance the pencil before letting go. For an object resting on a surface to be balanced, its centre of mass (a single point) must be directly above a point within the area of ...


1

Why don't we observe the infinite violations of conservation of energy The reason we don't see e.g. an atom spontaneously turning into a red giant for a fraction of a second is because of the extremely small timescales that such an energy difference would require - not even light could travel a tiny fraction of a proton radius in that time. The heavy ...


1

$$\Delta E \Delta t \sim \hbar$$ This is a version of the Heisenberg Uncertainty Principle. Instead of using momentum and position,however,the above form uses energy and time ($\Delta$ means change in). How can the uncertainty principle be used to deduce range of a force from properties of the force carrier? Let's take the simplest example: the ...


1

When someone says that spin measured about different axis can't both be known, they mean that whatever state you pick will have variability in at least one of the possible spin measurements you can do. So that is what you will get when measure the spin, you will get variable results. This happens even with entanglement with even just one particle. With ...


1

Let $X$ and $Y$ be random variables with density functions $f$ and $\hat{f}$, where $\hat{f}$ is the Fourier transform of $f$. Then the uncertainty principle is a lower bound on $\sigma_X\cdot \sigma_Y$ (where $\sigma$ is the standard deviation). In particular, $\sigma_X\sigma_Y\ge 1/4\pi$. For example, take a particle moving in one dimension, in a fixed ...



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