Questions tagged [matrix-elements]
Matrix elements are the components, or entries, of a matrix, typically considered in a certain basis.
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Matrix element of order $n$ [closed]
Does anyone know an explicit formula to calculate the following matrix element? It is in the context of quantum optics,
$$\langle\alpha | a^n \mathcal{D}(\beta) | m\rangle,$$
where $|\alpha\rangle$ is ...
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What is the correct term for this quantity?
I am dealing with direct and exchange Coulomb matrix elements in periodic systems.
When computing these, there are terms within that look like this:
$$\rho_{ik} (\mathbf{r}) = \phi_{i}(\mathbf{r}) \...
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Hermicity not preserved in Hamiltonian [closed]
My question is in the context of a matrix element in a diatomic molecule. I will rephrase it as well as possible to remove any unnecessary complexity.
I denote the spherical harmonic as $Y_m^l = |m,l\...
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Matrix Element of angular Momentum and Vector Operator
I am reading the book The Theory of Atomic Spectra by E. U. Condon and G. H. Shortley. On page 60, they claim that the matrix element of the operator
$$\sum_{j=1}^3\langle j'm'|J_i J_j T_j|jm\rangle$$
...
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Can the projection in the completness operator be applied multiple times in the integral
I'm trying to rewrite the matrix element $\langle k | V |k' \rangle$ of the potential V in terms of position space using the completeness relation.
I know that the completeness relation in position ...
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Possible errata Landau and Lifshitz in $\S29$ Matrix elements of vectors in Quantum Mechanics Third Edition [closed]
Context
L&L write,
Let $\mathbf{A}$ be some vector physical quantity characterizing a closed system... In the particular case where $\mathbf{A}$ is the radius vector of the particle... we find ...
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Meaning of Fourier expansions in Heisenberg's matrix mechanics [closed]
I am trying to figure out the passages Heisenberg followed in developing matrix mechanics as presented in his 1925 Umdeutung paper. In developing the virtual oscillators model, Heisenberg uses the ...
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Confused in David Bohm's *Quantum Theory*
In discussing matrix mechanics Bohm says in chapter 16 of Quantum Theory that the $rm$ element of operator $A$, $a_{rm}$, is given by
$$a_{rm} = \int dx\ \psi^*_r(x)A\psi_m(x) \hspace{1in} (Eqn. ...
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What is the advantage of using spherical tensor over cartesian tensor?
I am trying to train a machine-learning model to forecast the polarizability of atoms within a molecule. Typically, the tensor is characterised as a Cartesian rank-2 tensor, like this:
$$\alpha= \...
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Connect gaussian beam waist to ABDC matrix coefficient in bow-tie cavity
As part of my master thesis, my task is to find the optimal parameters to set-up a bow-tie optical parametric oscillator (OPO) for squeezed states generation. I'm currently looking at the possible ...
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Component notation and matrix notation for gradient of vector
I'm trying to understand vector and tensor notation, but I'm coming across some difficulties. Say I have vector $\vec{u}$ and I compute its gradient $\nabla \vec{u}$. Then I get a tensor $\frac{\...
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Non-perturbative matrix element calculation
Following Peskin & Schroeder's Sec.7's notation, I would like to compute the matrix element
$$
\left<\lambda_\vec{p}| \phi(x)^2 |\Omega\right>\tag{1}
$$
where $\langle\lambda_{\vec{p}}|$ is ...
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Adjoint and index notation in Weyl field context
In the answer to a question I previously asked, the following manipulation was done but I don’t understand it$.$
$$ (U_{jm}\psi_m)^\dagger=\psi_m^\dagger U_{mj}^\dagger $$
aside from the context from ...
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(L&L vol. 4, sec. 59) Matrix Elements of the Electromagnetic Interaction Operator
In Sec. 59 ('The Scattering Tensor') of the fourth Landau and Lifshitz gives the matrix elements of the electromagnetic interaction operator $\hat{V}=-\hat{\boldsymbol{d}} \cdot \hat{\boldsymbol{E}}$ ...
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How to efficiently calculate the inverse of the overlap matrix?
Now, we consider a non-orthonormal basis:
$$\mathcal{S}_N=\{|\alpha\rangle,a^\dagger|\alpha\rangle,a^{\dagger 2}|\alpha\rangle,\ldots,a^{\dagger N}|\alpha\rangle\},$$
where $|\alpha\rangle$ is the ...
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What does the matrix mean in matrix models?
I'm learning what a matrix model means, for example, in Yang–Mills matrix models, IKKT matrix model and BFSS matrix model. I have consulted a large amount of information but still not sure what the ...
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How can i write the matrix representation of the following Hatano - Nelson model Hamiltonian?
I have a $1$D and one band lattice model with hopping constants $J_R $ (to the right) and $J_L$ (to the left) and under open boundary condition. It has the following Hamiltonian :
$$H = \sum_{n} (J_R ...
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Tensor Index Manipulation
I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem. I have to prove that
$$\partial_{\mu} ...
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Physical meaning of each component of the metric tensor in GR
I am searching, without success, what is the meaning of each component of the metric tensor in the context of General Relativity.
$$
g_{\mu\nu}=\left[\begin{matrix}g_{00}&g_{01}&g_{02}&g_{...
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Second-order tensor contractions and matrix multiplication
The fluid dynamics book I'm reading lists the possible contractions of $A_{ij}B_{kl}$ where $\mathbf{A}$ and $\mathbf{B}$ are second order tensors. Since I'm dealing with fluid dynamics, assume $ 1 \...
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Calculation of the decay rate of the $W$ boson
I am trying to calculate the decay rate of the $W^-$ boson to a charged lepton and the corresponding antineutrino.
I denote the four momentum of the $W$ boson with $q = (M_W, \vec{0})$. The momenta of ...
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Is the propagator the same as the matrix elements of the time evolution operator?
So Sakurai in their QM book defines the propagator in wave mechanics as:
$$K(x'',t;x',t_0)=\sum_{a'}\langle x''\vert a'\rangle \langle a'\vert x'\rangle \exp\left[\dfrac{-iE_{a'}(t-t_0)}{\hbar}\right]....
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Why Spherical tensors are more convenient to use in quantum mechanics than Cartesian tensors?
I know that spherical tensors (more appropriately, tensors in spherical basis) are irreducible representations of the rotation group unlike the Cartesian tensors (more appropriately, tensors in ...
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Non-linear sigma model quantization
Given the following lagrangian for the non-linear sigma model:
$$
\mathcal{L}=\frac{1}{2}\sum_{a,b}\partial_\mu\phi^a\partial^\mu\phi^b f_{ab}(\phi)
$$
where $f_{ab}(\phi)$ is a matrix function.
My ...
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Eigenvalue Decomposition of Operators
I have a question about the eigenvalue decomposition of an operator, more specifically about the matrix with the eigenvectors as columns.
If i have an operator that i decompose as follows:
$$
\hat{A} =...
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