I am interested in the subsequent measurement of energy, like assume the energy of the electron was measured to be $E_1$ and the wavefunction of the electron is described by a finite superposition of eigenfunction of energy. After the first measurement will the subsequent measurement have the same energy measurement which is $E_1$. I think this question has been asked multiple times, but what I am confused about is that if all the subsequent measurement was of state $E_1$ that means that the uncertainty in energy is zero, which implies that the uncertainty in time measurement has to be infinite.
Just beginning to learn quantum mechanics and my textbook says that all pure eigenstates of the Schrodinger equation ( with a specified potential function) have a zero standard deviation for the expectation values of energy.So every measurement of the energy yields a fixed value. I've also read that pure eigenstates are physically realisable solutions in some situations ( like the infinite potential well). But wouldn't that violate the uncertainty principle? the standard deviation in expectation of energy is zero and hence $$ΔE~Δt=0.$$ Where am I going wrong? Also could someone also give me an insight as to what uncertainty in time even means? I haven't been able to grasp that concept clearly yet.
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Improving Review Queues - Design overview II: Changing review bans and other new features There are 8 crew members, The leading member wants that only a crew of 5 people or more could open a safe he bought, To each member he gave equal amount of keys, and locked the safe with several locks.  What's the minimal number of locks he should put:   at least 5 crew members would be required to open the safe? Any team of 5 crew members could open the safe, but none of the 4 team crew members.   1,2 stands individually of course. What are Virtual Methods? Why would you declare a method as "virtual".   What is the benefit in using virtual? How can I fix a 404 Error when using a PPA or updating my package lists? PPA seems to be constantly offline. Whenever I use sudo apt-get update, this error is shown:  W: Failed to fetch http://ppa.launchpad.net/ 404  Not Found   How do I fix these errors? "Scaled $L^p$ norm" and geometric mean The $L^p$ norm in $\mathbb{R}^n$ is \begin{align} \|x\|_p = \left(\sum_{j=1}^{n} |x_j|^p\right)^{1/p}. \end{align} Playing around with WolframAlpha, I noticed that, if we define the "scaled" $L^p$ norm in $\mathbb{R}^n$ to be  \begin{align} \| x \|_p = \left(\frac{1}{n}\sum_{j=1}^{n} |x_j|^p\right)^{1/p} \end{align} then \begin{align} \lim_{p \to 0} \|x\|_p &amp;= \left( \prod_{j=1}^{n} |x_j| \right)^{1/n}, \end{align} which is the geometric mean of the coordinates' absolute values. This is interesting maybe because the $L^p$ norm doesn't have a nice limit at zero.  My questions:   How do I prove this? Is this definition of "scaled $L^p$ norm" interesting, or known by another name, or used anywhere? Is there any interesting reason to define the $L^0$ norm as the geometric mean, as above? Further reading?   Thanks! proof that $f'$ is continuous at $a$ Let $f:I \rightarrow \mathbb{R}$ derivable in set I. if $\lim x_n = \lim y_n =a \in I$ with $x_n \neq y_n$ and $\lim \frac{f(y_n)-f(x_n)}{y_n - x_n} = f'(a)$ proof that $f'$ is continuous at $a$  my attempt $x_n \neq y_n$ so by MVT $\exists z_n \in I$ like $x_n \geq z_n \geq y_n$ with $\frac{f(y_n)-f(x_n)}{y_n - x_n}=f'(z_n)$ for each $x_n$ and  $y_n$ then $f'(a) = \lim\frac{f(y_n)-f(x_n)}{y_n - x_n}= \lim f'(z_n)$   Using $\lim x_n = \lim y_n = a$ and $x_n \geq z_n \geq y_n \Rightarrow z_n \rightarrow a$  if $ \lim_{x\rightarrow a} f'(x) \neq f'(a)$ then $\exists \epsilon &gt; 0$ such $\forall n\in \mathbb{N}$, $z_n \in [x_n,y_n]$ $0 &lt; |z_n - a| &lt; \frac{1}{n}$ and $|f'(z_n) - f'(a)| \geq \epsilon$. So $z_n \rightarrow a$ but will do not have $\lim f'(z_n) = f'(a)$  so $\lim f'(z_n) = f'(a) = \lim_{x\rightarrow a} f'(x)$ when $z_n \rightarrow a$  and thats implies $f'$ continuous at $a$  my question is, that's correct ? if not, can you help me? thanks Potato Power. Self sustaining medium using a living potato plant possible? equal size numerator and denominator I am trying to size the numerator and denominator of the below equation so that they are equal in size.  I have not had much luck suggestions are appreciated.  \begin{equation} \sum_{i=1}^{n} \frac{\frac{cf_{n}}{(1+i)^n}} {\displaystyle\sum_{i=1}^{n}\frac{cf_n}{(1+i)^n}} , n_{i} \end{equation}