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1. ### I About the expectation value

Hi there, Thanks for the reply. Your comment give me some hints on how to view the problem in different aspect. But I found that there is an issue in the number I gave since it will cause the expectation value (without redraw) to over 1, which means the player have advantage to win in long...
2. ### I About the expectation value

Hi all, 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...
3. ### I About binomial statistics

Hi all, 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...
4. ### I Finding the average time with given probability

Hi all, 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...
5. ### I Statistical problem of drawing colored balls from boxes

##E_n## is obtained from ##D## in the way ## E_n = \frac{n}{1- D\sum q_i} ## Well, I just find that there is a bug in my program, I corrected it and my simulation shows the same result (3.034) now. So I think the above calculation should be correct. Here is what I did in my code Setup a...
6. ### I Statistical problem of drawing colored balls from boxes

Sorry, I didn't complete the math. I typed all but after I preview that some part were gone, don't know why. Anyway, let me rephrase it here. One thing is very important I missed in the question, in second box game, place the drawn ball back to the box, so the probability of drawing...
7. ### I Statistical problem of drawing colored balls from boxes

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...
8. ### I About control random variables

Hi all, 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...
9. ### I About standard deviation

Thank you so much for your explanation. I think I get some points from here. So from the first comment you made, when you say population mean and population SD, do you mean the real mean and SD that computed from every single entities in the sampling space. But in actual case, it is not possible...
10. ### I About standard deviation

Hi all, 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...
11. ### I How to understand diffusive and ballistic transport?

Thanks for reference. It looks like a very good article to read. When I first think of the problem, I am thinking of the transport in gas.
12. ### I How to understand diffusive and ballistic transport?

Hi there, 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...
13. ### B About creating a product state

Thanks for the reply. I didn't confuse those two things. I though I clarify that at the beginning by saying that they are two questions but obviously I didn't make it clear enough. Anyway, my last question is about how to measure ##\langle a|b\rangle## (nothing to do with the product state...
14. ### B About creating a product state

Thanks. I think I understand that now. But I am still looking for the answer for the second question. Let say I have two states ##|a\rangle## and ##|b\rangle##, what is the significance of this ##\langle a | b \rangle##, by reading some examples found in the text, can I say that it stands for...
15. ### B About creating a product state

Thanks for that information. I understand that in math. But how actually people do that in experiment?
16. ### B About creating a product state

Hi all, 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...
17. ### I About creating standing wave with laser

Thanks DrClaude. Let say the laser beam is of the form $$y(x) = A\cos(kx + \omega t)$$ So by adding two counter-propagating waves, we have $$A\cos(kx + \omega t) + A\cos(-kx + \omega t) = 2\cos(\omega t)\cos(kx)$$ If the laser frequency is ##\omega=2\pi c/ \lambda##, it will have the...
18. ### I About creating standing wave with laser

Hi all, 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...
19. ### I About normalization of periodic wave function

Thanks for your reply. I am still reading your reply but I am still confusing on some parts. Since you mention the momentum space, I wonder if the following is physically possible or not. Taking crystal as example, in the text they always start the discussion with periodic lattice in position...
20. ### I About normalization of periodic wave function

Hi all, 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...
21. ### I Odd/even function in integral

I am still looking for a way to prove hat those two integral gives zero. Following andrewkirk's comment, I am trying to break my proof into 4 integral by taking care of 4 different ##m## such that the following identities can be used I1: ##m=4t+1, \sin[m(x+\pi/2)] = \sin(mx+\pi/2) = \cos(mx)...
22. ### I Odd/even function in integral

Thanks. Do you mean ##\sin[m(x+\pi/2)] \cdot \cos[(m+1)(x+\pi/2)]## will be odd function if ##m## is odd, the cosine part will be even and the sine part will be even too so their multiplication is odd; when ##m## is even, the cosine part is odd but the sine part is still odd, so their...
23. ### I Odd/even function in integral

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...
24. ### I Integral of a special trigonometic functions

Hi all, 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...
25. ### I About magnetic momentum in quantum mechanics

Hi all, 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...
26. ### I Why the energy inversely propotional to n^2 in Bohr model

Thanks for clarifying it. I appreciate your help.
27. ### I Why the energy inversely propotional to n^2 in Bohr model

Thanks a lot. I think I misunderstood some context in the text. I always think the ##\Delta E=\hbar\omega## is universal for all quantum system. Your reply help me to recall the harmonic oscillator. So like what you said, the explicit form of energy for a quantum system is really depending on...
28. ### I Why the energy inversely propotional to n^2 in Bohr model

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 =...
29. ### I Does the +/- 1 term in bosonic and fermionic statistics matter

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} =...
30. ### I About classical and quantum-mechanical statistics

Hi all, 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...
31. ### I About precessing directon of angular motion in quantum mechanics

Thanks a lot. I think I get some idea now.
32. ### I About precessing directon of angular motion in quantum mechanics

Thanks. I am still digesting this information. But as I can understand form above so far, since the angular momentum does not have prefer direction, so you could always call a specific direction as z so you could project whatever angular momentum onto that z and the associated x and y, is that...
33. ### I About precessing directon of angular motion in quantum mechanics

Hi all, 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...
34. ### I Why is the s state spherically symmetric?

Thanks a lot.
35. ### I Why is the s state spherically symmetric?

Sorry that I give the wrong reference. I just get it corrected. And in the document given by the link, in page 3, note below the caption of FIGURE 7.3, it is said that "for l=0 we have a completely flattened ellipse that goes through the nucleus, this will be later called an s state."
36. ### I Why is the s state spherically symmetric?

Hi there, 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...
37. ### I Solving a Second-Order Wave Equation

Thanks all above. Yes, by looking at the force, it makes sense to have 2nd order PDE.
38. ### I Solving a Second-Order Wave Equation

Thanks a lot for your explanation. I agree the reason to have a second order PDE for the wave equation in terms of mathematics. But what I want to know more is why physically it should be 2nd order and if it is any thing to do with the time reversibility. In physical system, there is 1st order...
39. ### I Solving a Second-Order Wave Equation

Yes, it is a typo there. I just get it fixed. Thanks for that.
40. ### I Solving a Second-Order Wave Equation

Hi there, 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...

thanks
42. ### Phase velocity is confusing

Oh, so to briefly summarize it, can I say phase speed is the speed of each frequency component composing the wave. Phase speed could be same or different for each component. If phase speed is different for each components, there will be dispersion.
43. ### Phase velocity is confusing

Hi all, 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...
44. ### About current from a battery going through our body

thanks for your explanation. So if for regular battery, C=0.1, you mean the maximum current is just 200mA by assuming the load is not zero ohm, is that right? I am curious where do we find the parameter of the battery for that C and mAh.
45. ### About current from a battery going through our body

Hi all, 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...
46. ### I About the time evolution of a wave packet

Hi Twigg, I will first appreciate your detail explanation on this. It clears more doubts from me. I also get some clue from where and what I should read further to get more information. It really makes sense (at least to me) to understand why Gaussian in terms of minimal uncertainty. In your...
47. ### I About the time evolution of a wave packet

I think those statements are more understandable. I am actually trying to find a reason convincing myself Gaussian has its advantage to be the choice. Thanks.
48. ### I About the time evolution of a wave packet

Ok. I understand vanhees71's explanation from math's point of view. Now if it is said that Gaussian waveform is not limited to free particle and it is confusing me again why Gaussian will be that typical so many physical system will have such waveform. I understand that it is square integrable...
49. ### I About the time evolution of a wave packet

Thanks for that. So does it mean the Gaussian solution is only eligible for free particle?
50. ### I About the time evolution of a wave packet

By the way, I know that in quantum mechanics, we have ##E=\hbar\omega##. In some text, I saw they use ##E(k)## instead of ##\omega## in the expression of time evolution. I am quite confusing. Is it always true to do that no matter what explicit form of ##E(k)## will be? My last question is...