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Physics processes are independent on the measurement units that you use. Your question is not about physics but about math. If you square a number less than one it will result in a smaller number, if you square a number larger than one you will get a larger number. But changing units will not change anything. For instance, suppose you have a square that is ...

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I'm not sure about your rain comment and how you were taught rain formed, but I can give you a bit of info on units. It doesn't matter what length you choose to be the base unit, as long as you are consistent. Right now our definition of the meter is the length light travels in $1/c$ seconds, where $c$ is the defined speed of light. You have to ...

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There is no reason why you can't measure the rate frequently. However, in order to estimate the half life, you need to see a change in the rate of decay. How long you need to measure for, and how far apart you need to change your measurements, depends on the number of decays per second that you observe as well as the required accuracy. For example, if you ...

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you should resolve the differential equation of N. If you do that you'll get that $$A = A_0 e^{\lambda (t-t_0)},$$ where $A_0$ is the activity at time $t_0$. From there you can obtain the value of $\lambda$ from two measurements of the activity whatever the time interval.

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Firstly, the activity formula is in fact: $$-\frac{dN}{dt}=A=λN,$$ because $\frac{dN}{dt}<0$. [...] is there any particular reason why our time interval for measuring the number of remaining Radionuclides should be close to the half-life of the substance? No and that's not how it's done in practice. $\lambda$ and the half-life are determined by ...

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The state is "measured" in the sense you are imagining - that is, it becomes definite - at whatever time it becomes possible in principle to infer its having a particular measurement outcome. In your case, if the probe provides unambiguous information about the measurement result, the time of measurement will be found to have been delta-t back in time. This ...

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The sky at night (no clouds) typically measures -48c to -21c at my location (Seattle), currently -44c. The clouds measure -5c to -1c right now. The trees across the road are at 2c. The reason I can measure the clouds is that the atmosphere is largely transparent to IR, so my thermometer "sees" the radiation from the clouds, but not the atmosphere - the ...

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The expectation value is a single number, it is the sum of all the possible values with a weight based on how often you get the result. So $\langle S_x\rangle=P(+\frac \hbar 2)\frac{\hbar}{2}+P(-\frac \hbar 2)\frac{-\hbar}{2}$ And you can get the probabilities by projecting the original spinor onto the eigenspaces of the operator and comparing the $L^2$ ...

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The mathematical formalism starts from a C*-algebra $A$, that of observables, and a set of physical states. For any observable $a\in A$, i.e. a self-adjoint element ($a^*=a$) and a state $\omega\in A^*$, the expectation value of $a$ on the state $\omega$ is simply $$\omega(a)$$ If you consider the C*-algebra generated by $a$ you can then apply the ...

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Probabilities of independent outcomes of the same measurement are always additive, which means that the expression $$P(U) = \sum_{n\in J}P(a_n),$$ is perfectly correct. If you want something that looks more formal, you can express $P(U)$ in as the expectation value of an appropriate operator, the projector  \Pi_U=\sum_{n\in ...

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In order to measure an object's speed, you need at least two measurements of its position at different times. This is not the case. The radar guns used by police to determine if you are exceeding the speed limit do not use position measurements. They instead measure the frequency difference between the outgoing and reflected signals. No position ...

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The uncertainty principle doesn't say anything about simultaneous measurements of a particle - that's just a myth which originated from Heisenberg's interpretation of it. Let us first describe the basis of quantum physics and let's start with the most innocent looking object: the quantum state. We can see a quantum state as a prescription to prepare a ...

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The uncertainty principle says something a deeper than "it is impossible to measure both position and momentum to arbitrary accuracy". It says 1) The accuracy is precisely limited by $\Delta x \Delta p > \hbar/2$. 2) In fact, this is not a limit of our measuring procedure, but a limit of reality. If something has well-defined position, it does not have ...

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I think the question is correct. While i was reading the bra ket algebra over and over again i wonder where the hell these are just the properties of quantum mechanics. Following the derivation of uncertanity relation i have found that all the arguments can be applied to classical mechanics equally well. Except for one postulate that after the measurement ...

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The work function of the photocathode is such that it can eject no more than one electron for each photon. Well, it's possible that very energetic photons can eject two, but that process is rare, and can be suppressed with filters. So that's the sense that it's said that PMTs detect single photons. But only 30% to 70% of photons in the proper energy ...

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Unfortunately PMTs don't do that, at least not cleanly. One of the crucial performance characteristics for photon counting is the peak-to-valley ratio of the PMT, which you get by measuring a pulse-height distribution. PMTs produce dark noise, which means they indicate "photons" that aren't there and they can't cleanly tell the difference between one and two ...

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