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Suppose I have a random variable X with known mean and standard deviation. After n realizations of X, what is the expected maximum of those n realizations? When n is very large, we know the mean of those realizations will be the mean of the distribution. Is the expected maximum simply the...

Interesting, I had wondered if that was okay to do...as the variance of X would then be the sum of the variances of the two normal distributions...and this is, in fact, sqrt(2). Could you explain how these are equivalent pictures? In general, I would like to consider a set of distributions...

Could you elaborate a bit more? I can confirm that the std deviation is indeed sqrt(2), however, I don't understand where the following formula comes from: E(X^2 | E(x) = m) = 1 + E(m^2) From the definition, \sigma_x^2 = E(x^2) - E(x)^2 = E(x^2) - m^2 Presumably, I stick your formula into...

Let X be a random variable with mean \mu and standard deviation 1. Let's add a twist. Suppose \mu is randomly distributed about 0 with standard deviation 1. At each iteration, we select a new \mu according to its distributuion. This mean is then used in the distribution for X. Then we pick...

Here are some thoughts. Please critque them. When speaking informally, we write the microstate as: (for distinct coins) HH HT TH TT However this just shorthand as marcusl implied. In physics, we cannot specify a microstate as above. We are treating the coins as independent. In...

You have implicitly assumed the coins are distinct. If the coins are identical, then there are only 3 microstates. This is exactly what Bose-Einstein statistics is about. From http://en.wikipedia.org/wiki/Satyendra_Nath_Bose: [Broken] "Because photons are indistinguishable from each other...

There is no way that this is correct. The probability of getting a single head when tossing two real coins is always 1/2 and never 1/3. This is _interesting_. Even if I personally am unable to distinguish two coins from one another (thus, they are identical to me and to what I care about)...

...any takers on this? Surely it isn't too difficult.

Maybe this didn't come through clearly. Why are the two coins distinct? I guess I was thinking that distinguishablity was a function of my interest. If I am only interested in the face of the coin, then all coins are identical. Obviously this line of thinking is wrong. I promise, I'm not...

I'm looking for some insight/clarification on the definition of a microstate. Consider two coins. Here are the possible outcomes: T T T H H T H H Now, I have assumed something: The coins are distinct. A better description of each microstate is: { (coin 1, T), (coin 2, T) }...

Wonderful! So, it doesn't appear that there is a closed form solution to this. However, the recurrence relation is quite nice, along with the Euler triangle. Thanks!

Anyone? I know there are smart enough people here.

Suppose I have 4 bosons in a one-dimensional harmonic oscillator potential and that the total energy is E_\text{tot} = 8 \hbar \omega. Recall, E_n = (n+1/2)\hbar\omega. Question: How many quantum states exist? (assume no spin degeneracy) After accounting for the ground state, we have 6...

Thanks. A quick look did not reveal any information on restricting the partitions as I discussed. Any other ideas?

Here is a much better statement of my question: How many integer solutions exist to the following equation: \sum_{i=1}^k n_i = N Let me call this number p(N,k). It is the number of partitions for N such that the partitions are restricted to be of order k or less. Example: N = 5 k = 3...

http://en.wikipedia.org/wiki/Integer_partition The above link should set the context. Given an integer q, the total number of partitions is given by partition function p(q). For example, 4 = 4 = 3+1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1 So, p(4) = 5. In mathematica, one can...

Hi, I'm wondering if someone can point me to "rigorous" developments of the path integral formulation. I've mostly seen arguments based on chopping up a line into a discrete set of points and then taking the limit as the number of points goes to infinity and integrating over all possible values...

Wick's theorem doesn't apply when you have equal-time contractions. I haven't learned this yet, but these equal-time contractions are somehow lumped into the interacting vacuum state. Is this correct?

I don't have pstricks...so I am going to use words. \text{contraction}\{ \overline{\psi}(x_1) \psi(x_1) \} My question: Propogators are usually dealing with different points...but what is the contraction of two quantities evaluated at the same point. Thanks.

...comments? Surely this isn't a difficult question.

First correction: put the natural numbers ellipse completely inside the integers. The attachment has been updated.

Have I correctly classified these sets of numbers? I am trying to diagram algebraic, transcendental, irrational...etc, numbers. Please see the attached picture. Thanks

Then, in principle you should be in support of the death penalty...to be in support of the policy is another issue altogether that shouldn't be confused with the first That's a tough statement for me to accept. There are some _evil_ people out there, and I have no trouble with the idea of...

Well, so long as you admit your decision was not based on sound logic but on emotion, I'm okay with that. The next time I hear a discussion on QM, I'll simply state that I no longer believe in QM. Then, I won't have to worry about those discussions. ;-)

A common argument: Suppose God is all-powerful. Now, let God create a rock so heavy that God cannot lift it. Common conclusions: 1) But if God cannot lift it, then God is not all-powerful. 2) The rock cannot exist since it creates a contradiction. Thus, God is not all-powerful since...

You messed up a couple minus signs. I'm going to copy and paste your code with corrections. \begin{array}{l} \int {\cos \left( {2x} \right) \cdot e^{2x} dx = \frac{1}{2}e^{2x} \cdot \sin \left( {2x} \right) - (-\frac{1}{2}e^{2x} \cdot \cos \left( {2x} \right)} + \int {\cos \left( {2x}...

Could someone spell this out for me? I have convinced myself that there is no pretty way to write a power series for a function of operators (that do not necessarily commute). It seems like you'd have a sum of an infinite product...each term in the product with their own index...so you are...

Hmm...I think that we just take that as the quantization condition. That is, [\phi_r(\vec{x},t),\pi_s(\vec{x}{\,}',t}] = i \delta^3(\vec{x}-\vec{x}{\,}')\delta_{rs} Is this correct?

I am working with a complex scalar field written in terms of two independent real scalar fields and trying to derive the commutator relations. So, \phi = \frac{1}{\sqrt{2}} \left(\phi_1 + i \phi_2) where \phi_1 and \phi_2 are real. When deriving...

So here is what I am actually trying to do. I have: [P^j,\phi_r(x)] = -i \hbar \frac{\partial\phi_r(x)}{\partial x_j} and [P^j,\pi_r(x)] = -i \hbar \frac{\partial\pi_r(x)}{\partial x_j} For a function F\left(\phi_r(x),\pi_r(x)\right), I need to show the following...

Hi, I'm looking for a general power series for a function of F of n operators. As normal, the operators do not necessarily commute. My first guess was: F(x,p) = \sum_{i=0}^\infty \sum_{j=0}^\infty a_{ij} x^i p^j + b_{ij}p^i x^j However, I don't think this is correct as it is possible to...

I think I've found why these are related. In the first question, it is the quanta that are indistinguishable while in the second question, it is the oscillators that are indistinguishable. In other words, it makes sense that the roles of k and n swapped between the questions. Now it also...

I swear I have tested this. This should work. Question: If I have k quanta of energy to divide into n distinguishable harmonic oscillators, what is the total number of possible configurations? Answer: (k+n-1)! / k! / (n-1)! Question: If I have k indistinguishable harmonic oscillators...

Obviously I have butchered this problem. I'm going to have to sort through this and respond again later.

Here is my question stated much more clearly: If I have k quanta of energy to divide into n distinguishable harmonic oscillators, what is the total number of possible configurations? If I have n indistinguishable harmonic oscillators and I do not allow any of them to be excited above the...

Well, someone just posted the solution in the probability forum. I am lost as to why it is the same exact answer as the "typical problem". If I have M units of energy to divide among N particles, then the number of ways to do this is: (M+N-1)!/(N-1)!/M! In my question, I do not have a set...

That's it! And that is a very well-known result too. For some reason, I had convinced myself that it had to be different from that. Thanks.

A typical stat mech. question is the following: If I have 5 bosons and energy E to divide among the bosons, what is the total number of possible configurations? I can't remember this answer, so if someone reading this can post it that would be appreciated. Now, I want to ask a slightly...

Anyone? I am looking for the number of combinations if I make n selections from k objects with replacements. Reptitions are acceptable, permutations are not important. So, if I select from 2 times from the objects {0,1} these are the combinations I care about: 00 01 11 Notice, 10...

Removing all (x,x) does not work in the general case. Below is another example. Suppose I am base 3 and I make three selections with replacement. Then there are 3^3 possible sets. I list them below. 000 001 002 010 011 012 020 021 022 100 101 102 110 111 112 120 121 122...

Suppose I pick two number from {0,1,2} without replacement and suppose I keep track of which one was drawn first. This is a permutation question. There are 3!/(3-2)! = 6 possible permutations: (0,1) (0,2) (1,0) (1,2) (2,0) (2,1) Of course, if I only care which numbers were choosen, then I...

Specifically, he states: if A_i \in \mathcal{F} is a countable sequence of sets then \cup_i A_i \in \mathcal{F} I think this is my answer. Let the sequence consist of only the set \mathcal{F}. Then \mathcal{F} (and hence the empty set as well) is in \mathcal{F}. Correct?

I had a quick question concerning the definition of a \sigma-algebra \mathcal{F} over a set \Omega. Most sources I've seen (e.g. http://en.wikipedia.org/wiki/Sigma-algebra ) require that \Omega or the empty set be an element of \mathcal{F}. Is this necessary? I ask because I am looking at...

Consider the amplitude for a free particle to propogate from \vec{x}_0 to \vec{x}: U(t) = \left\langle \vec{x} | e^{-i H t} | \vec{x}_0 \right\rangle I guess I don't understand what it means for a particle to propogate from one "position" to the next. If the particle is not in a...

Haha! I knew I had seen that sum before! Independence! lol...thanks.

Actually this might not be a Fourier question, but it certainly reminds of Fourier series. Suppose, \sum_{n=0}^\infty a_n \, g_n(x) = 0 Does it necessarily follow that a_n = 0 \: \forall n? If so, please provide a proof. If not, a counterexample would be helpful. If not, can I...

That all sounds good. So let me state this: 1) An electon and I are in the same inertial frame. 2) The electron is in a momentum eigenstate. Thus, 1) The electon is a "rest". 2) The energy of the electron is E = mc^2 3) The energy stored is the fields does not contribute to the...

The energy of an electron at rest is mc^2. 1) Can an electron even be at rest? It seems that the answer is "no" by the uncertainty principle. Thus, it would seem that every electron has energy greater than mc^2. Is this a correct statement? 2) Is this a classical quantity? That is, if I...

...which becomes painfully obvious when I include the effect of the oscillations. It is true that R->infinity makes exp(-R^2 exp(2*I*theta)) very large...but the oscillation causes it to go back and forth between +infinity and -infinity. It seems that no matter which contour I pick (as far...

I am trying to to the Gaussian integral using contour integration. What terrible mistake have I made. I = \int_{-\infty}^\infty \mathrm{e}^{-x^2} \mathrm{d}x I consider the following integral: \int_C \mathrm{e}^{-z^2} \mathrm{d}z where C is the half-circle (of infinite...