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...
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.
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...
My book stated the following theorem: If the functions P(x) and Q(x) are continuous on the open interval I containing the point x0, then the initial value problem dy/dx + P(x)y = Q(x), y(x0)=y0 has a unique solution y(x) on I, given by the formula y=1/I(x)\intI(x)Q(x)dx where I(x) is the...
Homework Statement
An electron and positron have mass me and separation R and are initially at rest. Find the time T it takes for them to collide.
Homework Equations
F=kq1q2/R2
The Attempt at a Solution
My problem is that the force is not constant. Do I integrate the force then...
Homework Statement
I had a question in my physics class today that was relatively easy but what I was confused about was a misplaced negative sign. "An electron is rotating around a proton at a distance R. The mass of the electron is me and the charge is -e. Find the period."
Homework...
I was on a plane the other day and it was bright outside so I put on my sunglasses and looked outside and saw a spectrum of colors. Just looking through the window or just looking through the glasses didn't produce the colors. Any idea what's going on?
After looking at the graphs of single slit and double slit interference of intensity vs. angle, it seems that the intensity drops off very steeply with single slit interference but doesn't drop off at all with double slit interference patterns. Why is that?
The question is: Which species has exactly five pairs of electrons around
the central atom?
(A) ClF5 (B) SF4 (C) SF5– (D) XeF4
The answer is B. Can someone explain why? I'm not too good with these geometries that use the d-orbital.
The question is: Which substance has the highest melting point?
(A) CO (B) CO2 (C) SiO2 (D) P2O5
The answer is C but I don't know why (you obviously can't look up the values). I need to be able to figure these types of questions out regardless of the...
Could someone please explain the math behind the Second Ace Paradox? Here's the problem: http://www.futilitycloset.com/2009/02/06/the-paradox-of-the-second-ace/ [Broken]
I want to know how the poster got those probabilities... I can't figure it out on my own.
I'm trying to understand this: http://en.wikipedia.org/wiki/Equilibrium_constant#Derivation_from_Gibbs_Free_Energy
And there are few steps that I don't get. My first question is probably a stupid one, but where does the \Sigmav_{j}A_{j}=0 come from? What exactly is it summing up, the number of...
My teacher explained that when a large nucleus has too few neutrons, the binding energy is not enough to hold the nucleus together. But by adding neutrons the binding energy increases without increasing the repulsion between nucleons, and thus the nucleus can stay together. But atoms that have...
Why is there a logarithm in the entropy formula? Why is it S=kln(N) where k is the Boltzmann constant and N is the number of microstates? Why isn't it S=N?
So we can use the Taylor's theorem to come up with a Taylor series represent certain functions. This series is a power series. So far (I'm in my second year of calc, senior in high school), I've never seen a power series that wasn't a Taylor series. So are all power series taylor series? Whether...
For a body to undergo uniform circular motion, a centripetal force which is perpendicular to the velocity at all times must be applied to the body. For the moon and the earth, the gravitational force exerted on the moon by the earth causes the moon to move in a circle instead of continue on a...
Imagine a situation where an object is placed infinitely far away from a convex, converging lens such that the light rays that come from the object are parallel to the principal axis. The real image forms at the focal point. In this situation, the magnification is 0, so if a screen was placed at...
"What is the next number?" question
I saw this question online but an answer wasn't provided. I've thought about it for a long time but I can't find an answer. Can anyone solve it?
What is the next number?
10, 13.33, 20, 26.66, 40, 46.66, 60, 66.66, 80, 100
So Godel's Incompleteness Theorem states that we cannot prove all facts about the natural numbers. Are we still able to prove that those proofs are impossible? Or are we stuck not knowing whether it is possible to prove something?
The second part of the theorem says that any system that can...
I looked up the derivation for the fact that the integral of 1/x from 1 to infinity is ln(x), but I couldn't find anything. If you have one I would love to see it.
A lot of people were saying stuff like "It's a definition. You can't derive it". Is that correct? If so how could anyone know...
I never really understood why using the Root Mean Square of negative data is the preferred method of finding the average of the data as opposed to taking the absolute value of the data and taking the average (arithmetic mean) of that. The example that recently made me wonder about this is...
I want to obtain a degree in physics. Here is the list of colleges I'm applying to:
University of Michigan
University of Chicago
Cornell
California Institute of Technology
Princeton
Stanford
Harvard
MIT
Are there any other colleges I should consider? Any schools in between Michigan...