In "Introduction to Quantum Mechanics", Griffiths derives the following formulae for counting the number of configurations for N particles.
Distinguishable particles...
$$ N!\prod_{n=1}^\infty \frac {d^{N_n}_n} {N_n !} $$
Fermions...
$$ \prod_{n=1}^\infty \frac {d_n!} {N_n!(d_n-N_n)!}$$...
I had to suffer through a lot of pseudo-science to earn my master's in education. I am suffering through more of it now while I "study" for my administration license. In fact, because anachronistic theories such as VAK (visual, auditory, kinesthetic learners) get floated around so much in the...
The uncertainty principle tells us that there is no state that a particle can be in such that we can predict with certainty both what the result of a position measurement will be and what the result of a momentum measurement will be. This statement is not the same as saying we can't measure the...
Here's the problem and the solution provided online by the author (the problem numbers are different but it's the same question). I think I'm okay up until the last step where he declares the Hamiltonian is (1 1 1 -1). Where did he get those components?
I am working through David Griffiths' "Introduction to Quantum Mechanics". All of the solutions are provided online by Griffiths himself. This is Problem 2.5(e). I understand his solution but I'm confused about one thing. After normalizing Ψ, we find ##A=\frac {1}{\sqrt2}##. Griffiths notes that...
Buffon's needle was presented as a problem in David Griffiths' "Introduction to Quantum Mechanics". In the book, a needle is of length l is dropped randomly on a sheet of ruled paper with the lines of the paper also a distance l apart. It is required to find the probability of the needle...
I always tend to get confused when thinking about non-Euclidean geometry and what straight lines and parallel lines are. If I think of a sphere, I get how two people driving north would almost mysteriously intersect at the North Pole and how the angles of a triangle would not add up to 180...
With automated theorem proving, what is left for mathematicians other than perhaps inputting weird axioms? Or are the machines not as sophisticated yet as I'm assuming they are?
I've been deeply disturbed by recent brain-related experiments that involve driving around insects, rats, and probably other animals like remote-controlled toys. No doubt this will lead somehow to some benefit for humanity. To cut straight to the point, do you feel (because this is a moral...
Many of my foreign language students do not have computers at home, only phones. I have tried unsuccessfully to find ways for translated subtitles to appear in YouTube videos when played on a phone. Does anyone here have a solution? (Specific solutions for Android, iPhone, and Galaxy would be...
You're on Earth. You throw a ball and watch its trajectory. It's curved. That's because the Earth is curving space-time at every point along the trajectory. But the Earth itself is not present along the trajectory - there is no matter along the trajectory (let's ignore the air and any radiation...
Trying to understand the concept of heat. As I understand it, heat is really just kinetic energy. In Newtonian mechanics, it is 1/2mv^2. Here are my questions...
(1) On a microscopic level, are conduction and convection simply atoms bumping into each other and passing along some of their...
##{dx}^2+{dy}^2=3^2+3^2=18##
##{dr}^2+r^2{d\theta}^2=0^2+3^2*(\theta/2)^2\neq18##
I have a feeling that what I'm doing wrong is just plugging numbers into the polar coordinate formula instead of treating it as a curve. For example, I naively plugged in 3 for r even though I know the radius...
I feel a little guilty writing this post because I'm sure there are people here who are tired of answering questions about the twin paradox, hence the FAQ post on the subject, but there's something which is still nagging me. First I have a question about the FAQ post itself. Toward the bottom of...
Reading The Theoretical Minimum by Susskind and Friedman. They state the following...
$$\left|X\right|=\sqrt {\langle X|X \rangle}\\
\left|Y\right|=\sqrt {\langle Y|Y \rangle}\\
\left|X+Y\right|=\sqrt {\left({\left<X\right|+\left<Y\right|}\right)\left({\left|X\right>+\left|Y\right>}\right)}$$...
In "The Theoretical Minimum" (the one on classical mechanics), on page 218, the authors write a Lagrangian
$$L=\frac m 2 (\dot r^2 +r^2\dot \theta ^2)+\frac {GMm} r$$
They then apply the Euler-Lagrange equation ##\frac d {dt}\frac {dL} {d\dot r}=\frac {dL} {dr}## (I know there should be...
I'm reading a book where the author gives the long division solution of ##\frac 1 {1+y^2}## as ##1-y^2+y^4-y^6...##. I'm having trouble duplicating this result and even online calculators such as Symbolab are not helpful. Can anyone explain how to get it?
In a certain derivation, the author begins with
$${g(-t)=}\frac 1 {2\pi}\int_{-\infty}^\infty {G(\omega)}e^{-i\omega t}d\omega$$
and then says he will replace ##t## with ##\omega## and ##\omega## with ##t##. He then writes
$${g(-\omega)=}\frac 1 {2\pi}\int_{-\infty}^\infty {G(t)}e^{-it\omega...
In Mathematical Methods in the Physical Sciences by Mary Boas, the author defines the Laplace transform as...
$${L(f)=}\int_0^\infty{f(t)}e^{-pt}{dt=F(p)}$$
The author then states that "...since we integrate from 0 to ##\infty##, ##{L(f)}## is the same no matter how ##{f(t)}## is defined for...
There is a simple geometric derivation of the area element ## r dr d\theta## in polar coordinates such as in the following link: http://citadel.sjfc.edu/faculty/kgreen/vector/Block3/jacob/node4.html
Is there an algebraic derivation as well beginning with Cartesian coordinates and using ##...
In Paul Nahin's book Inside Interesting Integrals, on pg. 113, he writes the following line (actually he wrote a more complicated function inside the integral where I have simply written f(x))...
## \int_0^\phi \frac {d} {dx} f(x) dx =...
In Microsoft Excel, if I type in the formula =GCD(113,100) then it gives me the correct answer of 1. However, if I type in =GCD(1.13*100,100), which means the same thing, it tells me 4. What's going on and how can I fix it? Thanks
I am reading "Inside Interesting Integrals" by Paul Nahin. Around pg. 59, he goes through a lengthy explanation of how to do the definite integral from 0 to infinity of ∫1/(x4+1)dx. However, he then simply writes down that this integral is equal to ∫x2/(x4+1)dx with the same limits. Now, it's...
I am a middle school teacher who is very concerned about global warming. My students’ memories don’t go back very far and they think it is perfectly normal to see mild weather and little to no snow throughout the winter months in New York City. I would like to compare current temperatures each...
Newton's law of gravitation cannot be compatible with relativity because the gravity from a massive object applies a force to all other masses infinitely fast. General relativity is supposed to correct this flaw by setting a speed limit on how fast the effect of gravity can reach a distant...
Just read the FAQ post "Do photons have mass?" and I'm still confused. The post says that all of the photon's energy is in the pc term of the energy-momentum equation. (1) But isn't p equal to mv, implying there is mass? (2) The post also says there is no inconsistency with E=mc^2 but doesn't...
In this super short video of the derivation of the relativistic kinetic energy, , I'm just stuck on one thing. Around 1:00 minute in, the constants of integration change from 0 to pv when the integration changes from dx to dv. Where does the pv come from? Thanks!
How does r∪(-p∩q∩-r) simplify to r∪(-p∩q) ? The second expression is just the first with the "-r" gone at the end. I'm not seeing how to get from the first expression to the second using any of the basic laws like distribution, de morgan, tautology, etc. What am I missing?
In the image below, why is the third line not \frac {ln(cosx)} {sinx}+c ? Wouldn't dividing by sinx be necessary to cancel out the extra -sinx that you get when taking the derivative of ln(cosx)? Also, wouldn't the negatives cancel?
I know how to prove the quotient rule by using the definition of a derivative using limits (Newton's style). I just saw a proof of the product rule using Leibniz's concept of differentials on Wikipedia. https://en.wikipedia.org/wiki/Product_rule#Discovery
Does anyone know of a Leibniz-style...