This isn't technically a homework problem, but I'm trying to check my understanding of the geometric phase by explicitly calculating the Berry connection for a simple 2x2 Hamiltonian that is not a textbook example of a spin-1/2 particle in a three dimensional magnetic field solved via a Bloch...
The standard reference for open quantum systems is The Theory of Open Quantum Systems by Breuer and Petruccione, which seems well-written but it lacks exercises, as do all the other textbooks I googled. Does someone have a recommendation for a textbook on open quantum systems with problems to...
I'm looking for literature recommendations regarding tensor networks. I never came across singular value decomposition or spectral decomposition in my linear algebra classes, so I need to brush up on the relevant mathematical background as well.
I've joined a group which studies cold atoms a little over two months ago. I was asked to write a Markov Chain Monte Carlo script, which I did, and which turned out really promising. I've been programming in Python for almost a year now so this wasn't too difficult to do. Shortly thereafter, the...
I am looking for a book/notes on the topics mentioned in the title that would be accessible to an undergrad. I have a background in grad quantum and statistical mechanics, but most resources I found on those topics assume a familiarity with QFT, string theory, gauge theory, and general...
For the case when ##B=0## I get: $$Z = \sum_{n_i = 0,1} e^{-\beta H(\{n_i\})} = \sum_{n_i = 0,1} e^{-\beta A \sum_i^N n_i} =\prod_i^N \sum_{n_i = 0,1} e^{-\beta A n_i} = [1+e^{-\beta A}]^N$$
For non-zero ##B## to first order the best I can get is:
$$Z = \sum_{n_i = 0,1}...
I'm considering a hydrogen atom placed in an infinite potential on one side of the nucleus, i.e. ##V(x) = +\infty## for ##x < 0##. I require the wavefunctions to be odd in order to satisfy the boundry condition at ##x=0##. By parity of the spherical harmonics only states with ##l## odd are...
This should be a trivial question. I am trying to compute the spherical tensor ##T_0^{(0)} = \frac{(U_1 V_{-1} + U_{-1} V_1 - U_0 V_0)}{3}## using the general formula (Sakurai 3.11.27), but what I get is:
$$
T_0^{(0)} = \sum_{q_1=-1}^1 \sum_{q_2=-1}^1 \langle 1,1;q_1,q_2|1,1;0,q\rangle...
So I've tried factoring out the eigenkets from the superposed kets in my equation, e.g. ##\cos(\alpha/2)|+\rangle + \sin(\alpha/2)|-\rangle##, and found out that most of the terms cancel. I ended up with the expression ##\cos(\alpha)|\text{singlet}\rangle##. Now I'm only unsure how to...
I'm not sure I understand why I need to do this. Don't the rotation operators act only the corresponding spin states in their Hilbert space, in which case I wouldn't need to find the tensor product? This worked for me when trying to show invariance under rotation about z, unless that was purely...
I have tried doing the obvious thing and multiplied the vectors and matrices, but I don't see a way to rearrange my result to resemble the initial state again:
##(\mathcal{D_{1y}(\alpha)} \otimes \mathcal{D_{2y}(\alpha)} )|\text{singlet}\rangle = \frac{1}{\sqrt{2}}\left[
\begin{pmatrix}...
For ##N = 1 = n_x + n_y + n_z## when you apply the completeness relation you get a sum states in coordinate basis for each ##n_i=1##, for a total of three states, each with an inner product between the coordinate and spherical bases (the bra-kets on the very right in the completeness equation)...
Suppose I have a positive spin-##1/2## eigenstate pointing in the ##z##-direction. If I apply a rotation operator by an angle ##\theta## around the ##z##-axis the state should of course not change. However, if I write it out explicitly, I find something different:
$$R_z(\theta)|\uparrow\rangle =...
What I'm confused about is how the representations were obtained from the closure relation. I understand everything before it. I don't see how to get rid of the inner products after making use of the closure relation.
The assigned textbook for my quantum mechanics class is Sakurai, but I find it too concise and struggle to solve any of the problems. I feel like I understood very little so far because of it and am looking for some alternatives/supplements. Currently I'm torn between Townsend, Ballentine, and...
I'm aware of the row/column vector representation, but I didn't put much attention in thinking about how these change under those operations, so I'll try to look into it. Could you elaborate on what you mentioned regarding eigenvalues. I don't like the language of eigenvalues, bases etc. when...
Thanks, that is very helpful. Does that mean that for a general operator ##A## acting on a bra-ket, I can choose it to act only on the ket or the bra and not on both?
I think that would confuse me more. I'm only asking why I need not consider ##\Delta A \Delta B|\alpha \rangle##. (Or maybe I do and there is something else I don't see.)
So if I do that I do indeed get the required result, provided I only find the eigenvalues of the kets/bras that associate with ##\Delta A##, so I neglect the example I've given before. Now I only need an explanation why I'm allowed to do this.
It's easy to show that ##[\Delta A, \Delta B] = [A,B]##. I'm specifically having issues with evaluating the bra-ket on the RHS of the uncertainty relation:
##\langle \alpha |[A,B]|\alpha\rangle = \langle \alpha |\Delta A \Delta B - \Delta B \Delta A|\alpha\rangle##
The answer is supposed to be...
Is that really the way it was meant to be done? The standard procedure is supposed to be quite messy, and I thought there would be a way to make use of the hint to find the eigenkets. Using the hint to find the eigenvalues is not much quicker than just evaluating the Hamiltonian and solving...
My first most obvious attempt was to use the relation ##<\epsilon> = \frac{3}{5}\epsilon_F## and the formula for kinetic energy, but this doesn't give the right answer and I'm frankly not sure why that's the case. My other idea was to use the Fermi statistic ##f(\epsilon)## which in this case...
Average
Average speed
Fermi
Fermi gas
Fermi-dirac distribution
Fermi-dirac statistics
Gas
Molecules
Quantum statistical mechanics
Speed
Statistical mechanics
I realize the question is quite broad but what research groups working on statistical physics, stochastic processes, and complex systems are generally considered the best? Would like to know about Europe and America alike.
I'm a physics and math major, going into my 3rd year. Suppose I want to do research in theoretical aspects of condensed matter. What would be the mathematics I should be learning as an undergraduate? Here is a rundown of courses I'm considering taking next year:
Abstract Algebra: it seems a...
I'm a student in a UK university and have the option of spending my 3rd year abroad in the USA or Canada. My primary motivation for doing so is the ability to do research during the academic year (which is impossible to do in the UK), which would make me more competitive when applying to...
Update: the examiner has just responded to me, and indeed, he made a mistake; there should be no spatial separation between the charge and the centre of the hemisphere.
He's the lecturer of the course. I tried to inquire about that exact detail at that time but I was still partly in the process of digesting the integration. I have hence tried to get in touch with the examiner again (by email, his lecture course ended) but he did not reply to me yet.
Look, this is THE MODEL SOLUTION that I cosulted with the examiner who made up this question himself. I'm asking for a clarification of one of the steps.
As for the approach you suggest, this is far from simple geometry. I tried to solve it that way for at least 2 weeks, consulting various...
My solution is correct. I should not find the E-field at the point where the charge is, that would be far beyond my abilities. This solution is pretty much the model solution for that question (I dicussed it wih the examiner himself), but now the examiner refuses to respond to my email to...
The following is a past exam question on electrostatics:
And here is my (correct) solution:
I understand the derivation of the E-field at the centre of the flat surface of the hemisphere, but I don't see the justification for using it in the Lorentz force formula. Is one not supposed to use...
I must have misinterpreted your earlier post:
If I were to keep ##r^2## in the integrand, I'd get the right answer.
Well, if ##r## is the distance from the charge, and ##\theta## the angle between ##r## and the ##z##-axis, the E-field due to a small charge is given by ##dE=k\frac{dQ}{r^2}##...
To be frank, I've been only exposed to derivatives of polar coordinates. These are the ones that come up in E&M most of the time. So it's actually easier for me to set up the integration with spherical coordinates in mind. Moreover, it's obvious some offshoot of those need to be used, given the...
I still don't see where this is going. I integrated for ##Q## and got ##Q=2\pi R \rho r^2##, if that's in any way helpful. Regardless of weather this is right or not, this would give me the total charge of the hemisphere, but I don't see how I'm supposed to use it to find the force; we can't...
We can find the force by finding the E-field on the charge first, then applying Lorentz force formula. However, it isn't obvious to me at all how to find the E-field. If the charge were on top of the hemisphere I would be using spherical coordinates, but here I don't know which coordinate system...
Update:
If I were to integrate the acceleration with respect to time I would get:
$$v(x,t)=g(\sin(\theta)-x\frac{\mu_0}{d}\cos(\theta))t$$ Knowing that the maximum occurs when v(x=d)=0, this equation gives \mu_0=\tan(\theta).
Alternatively, the chain rule gives a=\frac{dv}{dx}v, so $$\int a...
Homework Statement
A block of mass m is placed on a rough wedge inclined at an angle α to the horizontal, a distance d up the slope from the bottom of the wedge. The coefficient of kinetic friction between the block and wedge is given by µ_0x/d, where x is the distance down the slope from the...
I get results like mc2(E0-mc2)=2γm2cos2(α)c2, where γ is of the protons after the collision. But again, I should be able to solve the question without introducing any gammas, but I don't see a way to apply conservation of momentum without doing so.
E0 is the total energy of the proton in motion, E0+mc2 is the total energy of the system, which is equally distributed between both protons after the collision.
b). I would refrain from using conservation of momentum since that is specifically mentioned in part c (which I didn't include in the picture), and here it asks to equate the invariant masses.
fHomework Statement
Question b:
Homework Equations
E2=c2p2+m2c4
The Attempt at a Solution
We have c2pinitial2=E02-m2c4, and Ef2=c2p2+m2c4 for each outgoing proton. Combining those equations we get c2p2=Ef2-E02+c2pinitial2. I don't know where to go from here.