Thank you for your answers. If I understand correctly, the second method is to use the already established formulas such as the ones given here:
https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)#Second-order_and_higher-order_corrections
e.g.:
$$E_{n}(\epsilon) =...
Hello,
I am looking for a reference which describe perturbation theory with two parameters instead of one. So far, I did not find anything on the topic. It might have a specific name and I am using the wrong keywords. Any help is appreciated.
To be clear, I mean I have ##H =...
I tried by hand at first just to see if there was some kind of simplification due to the fact that the matrix is Hermitian, but it seemed still to be complicated. Thank you for confirming it.
I tried with Mathematica as well but I am barely using it, so I was not confident that I had...
Hello,
I am looking for a worked out solution to diagonalize a general 4x4 Hermitian matrix. Is there any book or course where the calculation is performed? If not, does this exist for the particular case of a traceless matrix? Thank you!
Hello,
I am looking for a comprehensive theoretical book or article related to the hole-phonon or exciton-phonon interaction in semiconductors. To be more precise, what I am looking for is:
1- Second quantization for the phonons (especially acoustic phonons)
2- Derivation of the Hamiltonian...
Thank you for the document. In equation (10), they get rid of ##V_{d}## by defining the density of states. But what if it was defined wihout dividing by this volume? And then assume I have a real system with a finite volume ##V = L_{x}L_{y}L_{z}##. Can I relate ##V_{d}## to V? E.g...
Hello,
I want to convert a summation in reciprocal space and I am unsure about the integration volume. I have started with the formula:
$$\sum_{\vec{k}} \rightarrow \frac{V_{k}}{(2\pi)^{3}}\int\int\int \mathrm{d}V_{k}$$
where:
$$\mathrm{d}V_{k} = k^{2}\mathrm{d}k...
Thank you for your answer. I think there are not many symmetry groups below C2 anyway. If you have nice explanations that do not make use of group theory, I would like to hear them. But I think I will also just study group theory a lot now, this self-isolation period might be a good moment to...
Hi, if I have to define the field, I would say semiconductor nanostructures although I do not know if this is the official name! The document where I have read this is not anything published that I could find in a book so far, but I can provide an example of an analysis that I cannot understand...
If we are dealing with a wave function that has a spin degree of freedom, I do not think it is possible to just omit it. Normally, there is an equation telling us that the application of a unitary operator on a function is equivalent to the application of the inverse transformation on the...
So the point is to show that ##S\psi## is an eigenvector of ##C_2## if ##\psi## is an eigenvector of ##C_2##? If this is the case, I think I understood your reasoning.
But in this case, how can I apply it to a function with a spin degree of freedom? I think we have ##C_2\psi(r,\theta,z)\phi =...
Thank you for your answer but I don't get it. What should I collapse?
For example, if I want to know the symmetry properties of a wave function in a system which possesses the C2 symmetry, I thought I could start by finding the eigenvalues and eigenvectors of the rotation operator for spin 3/2...
Hello,
My question is simple. I have read that isotropic biaxial strain does not lower C2 symmetry, but no proof whatsoever was provided. I would like to know if it is actually true and have a solid proof. If someone can provide it, that would be wonderful. But also explaining me how to start...
Hello,
I have two questions into one. First I would like to know what books are considered the best to introduce the theory of quantum dots, so for example with the k.p method, tight-binding, empirical pseudopotentials, and other techniques, analytical derivations, optical properties, band...
Ok, so assuming the parity does not act on spin 1/2, I assume my first equation is correct? I could say, calling ##R_{y}## my reflection operator:
$$R_{y}|\frac{3}{2},\frac{3}{2}\rangle = e^{-\frac{i}{\hbar}\hat{J}_{y}\pi}\Pi|\frac{3}{2},\frac{3}{2}\rangle \\...
Thank you for your answer but I am not sure to understand. You are saying I could write my reflection as $$R=QR_{0},$$ but it implies writing another reflection while my question is about how to write such a reflection, either directly or as a composition of other operators which are not...
In case it is unclear, I will take an example that is of interest to me. I wanted to ask if I can write an operator for reflection along y as:
$$\Pi e^{-\frac{i}{\hbar}\hat{L}_{y}\pi}\otimes e^{-\frac{i}{\hbar}\hat{S}_{y}\pi} = \Pi e^{-\frac{i}{\hbar}\hat{J}_{y}\pi}$$
when I want to perform a...
Hello,
I know we have the parity operator for inversion in quantum mechanics and for rotations we have the exponentials of the angular momentum/spin operators. But what if I want to write the operator that represent a reflection for example just switching y to -y, the matrix in real space...
Thanks, your answer is very clear for this example. But I would like to remove one of the wavefunction by taking the inner product on only half of the subspaces with two functions depending on ##\vec{x}##. Is it possible to separate the integration in this case?
Hello,
Thank you for your answer. I thought about the tensor product but it is unclear to me because of the inner product.
\langle\psi | \chi \rangle = \langle\psi | \psi \rangle |\phi \rangle = \left(\int dx \psi^{*}(x) \psi(x)\right) |\phi \rangle should be true since we have separate...
Hello,
I would like to write a product of two wave functions with a single ket. Although it looks simple, I do not remember seeing this in any textbook on quantum mechanics. Assume we have the following:
##\chi(x) = \psi(x)\phi(x) = \langle x | \psi \rangle \langle x | \phi \rangle##
I would...
Thank you for all the references, this is exactly what I am interested in. In fact, I followed a short introduction on group theory and I can see the symmetry elements in a crystal or a molecule. But it is less clear when I have to consider the magnetic field, the spin matrices, a photon, a...
Hello, thank you for your answer. Do you know any book or good pdf that would have exercises with detailed answers on this topic? (find a hamiltonian from group theory)
Hello,
I am currently struggling to understand how one can write a Hamiltonian using group theory and change its form according to the symmetry of the system that is considered. The main issue is of course that I have no real experience in using group theory.
So to make my question a bit less...
So in the book, they say: the amplitude ##\langle x(t_{x})|y(t_{y}\rangle## is known as a propagator. (...) Propagators for single particles have a neat mathematical property: they are the Green's function of the equation of motion of the particle. Then they define the general equation for...
Hello, thank you for your answers.
What I want to calculate is very general. It is simply about going from the wave function to the Green's function picture, so just solving the equation in the first message. In the book, this is almost the first definition of the Green's function, just after...
Here I could also say:
##G^{+}(x,t_{x},y,t_{y}) = \langle x(t_{x}-t_{y})|y(0)\rangle##
So a particle started at position y at time 0 and ends at position x at time ##t_{x}-t_{y}##. But this time can then be negative in the wavefunction representation or do we need the Heaviside function...