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Questions tagged [matrix-elements]

Matrix elements are the components, or entries, of a matrix, typically considered in a certain basis.

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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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Rotations of coordinate for differently scaled x and y axes

I have a primitive coordinate system where x and y axis are the basis vectors that are orthogonal. Now the lattice constant along x and y is differnt let us say $a_x$ and $a_y$. I have a vector $\vec{...
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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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Matrix element including weak currents

If a $\overline{B^0}$ meson decays into $D^+$, $e^-$ and $\overline{\nu}_e$ we have a matrix element $$\langle{D^+ \, e^- \, \overline{\nu}_e |\, (\overline{c_L} \gamma^{\mu} b_L) \, (\overline{e_L} \...
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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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How to compute the hopping matrix considering the change of multiple states?

My question is quite simple, but I couldn't find the answer anywhere. Hubbard Models usually have a hopping term as follow: $$H_{hop} = -t \sum_{<i,j>} \left( c^{\dagger}_{j} c_{i} + c^{\dagger}...
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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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How do I find the matrix for the Su-Schrieffer-Heeger model's Hamiltonian?

I will not bother to write down the tensor product in the joint basis of A and B here in this post, where A and B are atoms/electrons and m denotes a unit cell. The Hamiltonian for this model is given ...
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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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$T$ Matrix elements in Scattering Theory (Sakurai 2nd edition)

I am currently unable to see how the $T$ Matrix elements discussed in 6.1 of Sakurai's Modern Quantum mechanics 2$^\mathtt{nd}$ edition can be expressed as they are in equation (6.1.26) (see below). ...
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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 rotate a 3D matrix around a line by a certain degree? [closed]

Physics Ch 67.1 Advanced E&M: Review Vectors (15 of 55) Coordinate Transformation in 3-D: Ex. 2 I was watching the following video where the goal was to find the Coordinate Transformation matrix ...
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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 is a cross section, really? [closed]

Upon looking at different resources, there is a common definition of a cross section (in the context of QFT) to be the probability that some scattering process occurs. For example, here is a ...
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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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Query on Expectation value of Spin-X operator being Invariant with transformation of BASIS

Please correct my calculations below. Where did I go wrong? What did I assume wrong? Thanks & Regards, Sarath Chandra.
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Understanding matrix element calculation in Schwartz (4.18) -- (4.21)

After a number of years out of grad school and in industry I'm trying to brush up on my fundamentals. Going through Schwartz's "Quantum Field Theory and the Standard Model", I'm having ...
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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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Matrix element of two nucleon interaction

I have a couple of questions regarding the following problem: Let the interaction between two nucleons given by: $\hat{V}_{1,2}=\boldsymbol{\sigma}^{(1)}\boldsymbol{\sigma}^{(2)}$. What I need help ...
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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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How to get metric tensor components?

I was wondering how do we get the components of the metric tensor? Why, in euclidian 3D space, the metric tensor is represented like this : $$ g_{\mu\nu}=\left[\begin{matrix}1&0&0\\0&1&...
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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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How to determine where the matrix elements go for the general angular momentum matrix in quantum mechanics?

Hello I have question on how to determine the matrix elements for the general angular momentum operator. So I know that $\langle j',m' \vert J^2 \vert j,m \rangle = (\hbar)j(j+1)\delta_{(j′j)}δ_{(m′m)}...
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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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Matrix representation of squeezing operator

In quantum mechanics we know that every operator can be represented by a matrix.Being a beginner of quantum optics, my question is does there exist a matrix for squeezing operator also? If does, can ...
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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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Expectation values of fermionic operators

I'm trying to compute the matrix elements of an Hamiltonian expressed in terms of fermionic operators. The system is an Agassi model, N interacting fermions on 2 levels separated by energy $\epsilon$, ...
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Matrix representation of Lorentz boost acting on a scalar

For the unitary translation operator $\hat{X}_a$ in 1 dimension it is easy to show that its matrix elements can be expressed purely with a delta distribution or equivalently as a combination of a ...
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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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Matrix elements of the commutator of the delta function with $p^2$ [closed]

Suppose we have potential $V(x) = \delta(x)$. I want to evaluate $\langle x' \vert [p^2, \delta(x)] \vert x'' \rangle$. Unprimed quantities are operators while primed quantities are eigenvalues/...
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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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Strange manipulation of Hamiltonian operator and gradient

I'm reading M. V. Berry's Quantal Phase Factors Accompanying Adiabatic Changes and came across an unfamiliar identity in eq. (8), namely $\langle m | \nabla _Rn \rangle = \frac{\langle m | \nabla_R \...
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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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Matrix elements in second quantisation formalism

In a system with two orbitals $c$ and $d$ (each with two spin degrees of freedom), consider the Hamiltonian $$H=V(d^{\dagger}_{\uparrow} c_{\uparrow} + c^{\dagger}_{\uparrow}d_{\uparrow}+d^{\dagger}_{\...
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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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How to convert the following Matrix equation to tensor notation?

Consider the following equation : $$\Lambda^{-1}\Lambda^T \Lambda=A$$ Here $\Lambda$ are my lorentz transformations such that $\Lambda^T \eta \Lambda=\eta$. $A$ is some matrix. I know that in terms of ...
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Breaking product of three vectors into symmetric and anti-symmetric vectors [closed]

Let's consider we have three arbitrary vectors A, B and C. We have the quantity $A_{\mu}B_{\nu}C_{\rho}$. Is it possible to break the above quantity into sum of symmetric and anti-symmetric vectors in ...
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How can I demonstrate that, in a degenerate system, if $[H_0,\lambda H_1] = 0$ then $ H_1$ is already diagonalized?

In the context of degenerate perturbation theory, for a perturbed Hamiltonian $H_0 + \lambda H_1$, I've heard of a very useful tool: "If $[H_0,\lambda H_1] = 0$ holds (both parts of the ...
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Confusion regarding Fermi's golden rule

The Fermi Golden Rule is: $$\Gamma_{i\to f}=\frac{2\pi}{\hbar}|\langle f|H'|i\rangle|^2\rho(E_f)$$ In this equation, $|\langle f|H'|i\rangle|$ is giving information about the coupling. However, $f$ ...
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Why is this rearrangement of factors in the Compton Scattering matrix element valid?

I've been trying to follow through this derivation of the total squared matrix element for Compton scattering. We have two first-order diagrams: Using Feynman rules, the matrix elements for each ...
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Simplify calculation for matrix elements in quantum electrodynamics

So I am learning quantum field theory. At the moment I have a look at the interactions between electrons/positrons and photons, which is quantum electrodynamics. I want to calculate matrix elements of ...
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Transforming the field strength tensor - Why do we need to use $\Lambda^T$, instead of a row times row multiplication?

The transformation law is \begin{align} F'^{\mu\nu} = {\Lambda^{\mu}}_{\alpha} {\Lambda^{\nu}}_{\beta} F^{\alpha \beta} = {\Lambda^{\mu}}_{\alpha} F^{\alpha \beta} {\Lambda^{\nu}}_{\beta} \end{align} ...
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Why can we use $Tr(AB) = Tr(A)Tr(B)$ when finding matrix element squared? [closed]

Here is a $t$-channel Feynman diagram I'm working on: Using Feynman's rules, we can find the matrix element as $$ M_t = -ig^2\frac{\bar u^{s_3}(p_3)u^{s_2}(p_2)\bar u^{s_4}(p_4)u^{s_1}(p_1)}{(p_2-p_3)...
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The Physical Meaning of Variance of Random Matrix Entries

I am trying to make some physical sense of the Hamiltonian described on pages 1, 2 here. The part I don't get is in the image attached below. I understand what the variance of each entry term tells me ...
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How to reduce matrix element of a spherical tensor operator?

I am studying about Wigner-Eckart theorem, and I have a question about the reduced matrix element. Wigner-Eckart theorem: (I follow the terms as in Wikipedia, https://en.wikipedia.org/wiki/Wigner%E2%...
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Hamiltonian as a matrix and its elements [closed]

Let us consider an electron in an infinitely deep one-dimensional potential well of thickness L with zero potential energy at the bottom of the well. The normalised eigenfunction solutions to this can ...
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Numerically approximating eigenstates and energies of a particle subject to some potential. Question about dirac orthonormality and infinite terms

Context I am attempting to approximate the eigenstates and energies of a particle over an interval $[-a,a]$ subject to some potential. The goal is then to approximate the wavefunction $\Psi$ with ...
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Dipole matrix elements for Bloch wavefunctions?

Suppose we have a one-dimensional periodic system with lattice constant $a_0$. From Bloch's theorem, we can express the wavefunction for an electron in band $m$ with crystal momentum $k$ $\left\langle ...
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Matrix elements of angular momentum operator

I was reading about the matrix elements of the generalized angular momentum operator in my QM textbook, when I came upon the following passage: The matrix representation of $J_+$ follows. It is \...
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What is the value of $e^{i \frac{\pi}{2} \frac{1}{\sqrt{3}}}$? [closed]

Context: There is a Unitary Matrix given: $\bf{U}=e^{i\pi\frac{\bf{H}}{2}}$ where $\bf{H}$ is a Hermitian matrix. And $\bf{H}=\sqrt{3}\begin{bmatrix}\frac{1}{3}&0&\frac{\sqrt{2}}{3}\\0&\...
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