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I am creating a game for fun, which need some math skill to work out the chance of winning and the way to keep the banker never lose. The configuration of the game is like this: five boxes marked no.1, no.2, no.3, no.4 and no.5; there are many balls in different color in each box. For...
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I am solving a practical math problem. There is a sale in one of the shopping mall in my town. The mall gives 10 coupons to a new customer. The customer could use one coupon at a time and when it is used, one could spin a fortune wheel to win more 10 more coupons. If one doesn't win...
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I am thinking a problem of drawing a ball in a sealed box. Assuming there is a box, contains plenty red and white balls, the number of red and white balls are unknown but let's assume there will be ##p## chance to draw a red ball and ##q=1-p## chance to get a white one. Those...
Last week I went to a state fair which I saw a game of lucky draw. There is two sealed boxes, contains bunch of 4 different color balls: red, blue, green and white. Here is the game rule. Players make an initial draw on box one, if players get a white ball, lose the game; if getting a red one...
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I am developing a very simple computer game to randomly move a point to on a bound region and check how many steps it takes to have the point landing to a certain place. To make it simple, I assume it is a 1D problem, the point could start on origin or any location on positive x axis...
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I learn some statistics some times ago. It has been while but I still remember some characteristics and property of the normal distribution. One of them is the standard deviation could be used to estimated the probability of finding the entity around the mean in the range ##[-n\sigma...
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I am always confusing in the difference between diffusive and ballistic transport. My understanding on the diffusive transport is from it's name, particles diffuse from the high density region into the low density region. I think the diffusion happens towards all direction, is it why...
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I am reading a book about fundamental quantum mechanics, in which there mentioned many time about the product state ##|a\rangle|b\rangle## of two states ##|a\rangle## and ##|b\rangle## . To my understanding, product state means to combine two small systems to get a bigger one. So I am...
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I remember the standing wave is introduced in a chapter of mechanical wave in my undergraduate physics times ago. It is said that two waves of the same frequency propagating the opposite directions will form a standing wave in space. I wonder if it is possible to produce the standing...
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I am reading something on wave function in quantum mechanics. I am thinking a situation if we have particles distributed over a periodic potential such that the wave function is periodic as well. For example, it could be a superposition of a series of equal-amplitude plane waves with...
I am reviewing some basic calculus with basic trigonometric functions. I remember for periodic function, one can use the feature of odd/even function to help computing some integral. I got two integrals from a book some times ago (I can't recall which book are they from). I expect those...
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I am working on the following integral
##
\int_0^{2\pi}\frac{1+\cos[\alpha x]}{1+\sin x}dx
##
where ##\alpha## is odd integer. Unless I set the ##\alpha## to a number then I can find the integral with mathematica easily. For general case with symbolic ##\alpha##, I cannot find the...
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I have been reading lots of materials regarding the classical and quantum mechanics. The first subject I read is Bohr model, in which it is assumed the electron is in circular motion around the nucleus on the so-called orbital. I think it is semi-classical. With this assumption, we...
After reading some materials on Bohr model, I understand the model is more or less incorrect, especially in terms of "orbital". I just wonder if the energy expression is also wrong or not.
In my text for general quantum theory, the energy about two neighboring level is given as ##\Delta E =...
I am reading an articles introducing the Nobel Price on Bose-Einstein condensates from where I have further reading on Bosonic and Fermionic statistics on some texts. I know one of the mathematical difference is the +/- 1 term in the denominator of the distribution function as below
##f_{BE} =...
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I am reading an introduction on classical and quantum-mechanical statistics. The material considers a 4-particle system with discrete energy level 0E, 1E, 2E, 3E, 4E, 5E and 6E. It is said that the classical particle is indistinguishable but you can identify the different particle by...
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I got a feeling that learning quantum mechanics is easy and hard. Most of time, it is easy to "accept" all the concepts given in the book by simply looking at their mathematical interpretation. But it is hard if you really take it serious to try to understand everything from the...
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I am reading something about quantum numbers, there the author introduce the quantum number by solving Schrodinger equation for Hydrogen atom. It gives me an example when the principal quantum number n=4, there are four different sub-level ##s, p, d, f##. It also depicts the sublevel...
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I am solving a second-order wave equation of the following form:
##v^2\dfrac{\partial^2 y}{\partial x^2} = \dfrac{\partial^2y}{\partial t^2}##
where ##t## is time; ##y## is the wave function. I learn the wave function by assuming the system is linear and has a linear solution. The...
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I have a difficulty to understand the concept of phase velocity of a wave packet. I read some materials online and my text, I didn't find a very straightforward explanation on this concept except the math definition. I also read some threads from this forum, but still have some...
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I am reading some materials on electrical parameters of a AA battery which give 1.5V and about 2000mAh. As my understanding, the 2000mAh means it will produce 2A if we keep taking current from the battery for an hour. 2A is huge current for human being. I wonder why it won't hurt the...
In some texts of fundamental quantum mechanics, it introduces the wave packet by Fourier transformation of a momentum wave into a spatial version. This is easy to understand because, analogy to the optical wave, a typical beam could compose waves of more than one frequencies. The general form is...
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I start to read some popular science on quantum physics. The term "entanglement" appears everywhere in quantum communication and other related fields. I am trying to understand this concept based on my undergraduate knowledge on physics. From some online resource, I have a feeling that...
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I took the course of quantum mechanics long time ago. From there I learn how to describe an atom with wave function. For example, Hydrogen has the wave function in (spherical coordinates) space. In the book they consider a reduced mass for the nucleus and the only external electron...
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I am reading a book regarding fundamental atomic physics, in which it introduces one kind of electronic scattering called Kapitsa–Dirac effect. I read the some introduction in wiki https://en.wikipedia.org/wiki/Kapitsa%E2%80%93Dirac_effect, and it states that the effect was first...
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I am reading some introducing materials on quantum information and quantum walk. In some materials, the author mentions to implement the related system with cold atoms and they mention the internal states. I learned the quantum mechanics some times ago but I didn't see any chapter in...
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I am reading a math book with a lot of examples on mechanical physics. I saw a math term about second moment. In wiki, it is said that second moment is just moment of inertia in physics and has definition as below
##\int(x-\mu)^2f(x)dx##
here ##\mu## is the average and ##f(x)## is...
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I saw some figures about potential (contour) plot in some articles that has some beautiful gradient lines. The shape is quite weird but you can see clearly there are some strong concentration of potential at some places. For example, if you have 300x300 grids, there shows strong...
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I am reading an article about uncertainty principle. If we consider a Gaussian wave packet which standard deviation of momentum ##\sigma_p##. The uncertainty principle states that the multiplication of variance of x and variance of p is larger or equal to half ##\hbar##
##\Delta...
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The Gaussian wave packet is widely discussed in the text. I got the following expression for wave packet in momentum space
##\psi(p, 0) = A \exp\left[-(p-p_0)^2/ (2\sigma_p^2)\right]##
with ##A=\sqrt{2\sigma_p/\sqrt{2\pi}}##
As my understanding, the corresponding wave packet in...