I'm trying to relate an analogy from Brian Greene about entropy microstates/macrostates to the real world. In the analogy, you have 100 coins that you flip. The microstate is which particular coins landed heads up. The macrostate is the total number of coins that are heads up. So a low entropy...
Given a wave function \Psi which is an eigenstate of a Hermitian operator \hat{Q}, we can measure a definite value of the observable corresponding to \hat{Q}, and the value of this observable is the eigenvalue Q of the eigenstate
$$
\hat{Q}\Psi = Q\Psi
$$
My question is whether it's a postulate...
The formula for the expected value of a continuous random variable is E(x) = \int_{-\infty}^{\infty} x\cdot f(x). This leads me to believe that the expected value of a function g(x) is E(x) = \int_{-\infty}^{\infty} g(x)\cdot f(g(x)). However, the correct formula is E(x) =...
So I've done some problems where a sphere intersects with a cylinder and I needed to find the volume of the intersected region using triple integrals. For example, if I needed to find the domain of integration for the intersection of the sphere $$x^2+y^2+z^2=a^2$$ and the cylinder...
Thanks, but I'm still not sure what to do from here. I'm not sure how to deal with the sum, I guess.
EDIT: I guess I kind of get it now. For the summand, if we keep changing i then we're finding the number of ways to choose 2k elements from n-1 elements by choosing them from each sized set. So...
Homework Statement
Prove that for any positive integer n, k,
$$\binom{n}{2k+1}=\sum_{i=1}^n{\binom{i-1}{k}\binom{n-i}{k}}$$
Homework Equations
$$\binom{n}{k}=\frac{n!}{(n-k)!k!}$$
The Attempt at a Solution
I'm looking for a starting point. I've been given a hint that says,
But I don't know...
I recently learned how to calculate the field from a cylinder (inside and outside the cylinder) using Gauss's law. I was wondering how I would be able to derive the same formula without using Gauss's law (just for practice). My idea is that you would need to integrate the electric field from...
Say you have the diff eq. x'=2x(x-13); x(0)=20. After separating and integrating we get,
ln|(x-13)/x|=26t+C
From here, we raise e to the power of each side to get rid of the natural log. Does this get rid of the absolute value signs? If so, why? If not, why is there only one solution (as...
Just like there's LaTeX for writing math papers and LaTeX editors to parse it, is there anything for writing papers in programming? Specifically a nice way to automatically format code.
I thought division might be involved, resulting in non-integers.
What's the difference? I thought if something was random, the probability for each result would be the same (uniform).
I'll try it out, see if I can come up with anything.
If anyone else has any other ideas, please share them...
But I recently saw someone attach a wristband to their hand which was connected to a conducting pad which was connected to ground. After doing this he said it was safe to work with an electronic component sitting on the pad. So if only the electronic component needed to be grounded, he could...
Well the relationship is that after being given this random number from 0-3, you need to perform mathematical operations (functions such as floor and ceiling are allowed) to arrive at a new number from 0-6. Now if you performed these same steps for the random input number many times, you would...
This is actually for a program, but the challenging part is really the math. So you're given a number 0, 1, 2, or 3 and the probability of getting each is the same. Now given this random number, I need to output a number 0, 1, 2, 3, 4, 5, or 6 but the probability of getting each number needs to...
Let's saying you're working with something that has electricity running through it. If you ground yourself by touch something metal, aren't you making it easier for charge to flow through you through the metal, so a greater charge would flow through you? And if you didn't touch something metal...