Questions tagged [matrix-elements]

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Proving that a matrix consisting of the eigenvectors of a two-state system can simplify the Hamiltonian

I am at the late undergrad level. My professor has asked me to use the matrix $U=\left( {\begin{array}{cc} u & -v \\ v & u \\ \end{array} } \right)$ to simplify the Hamiltonian like ...
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Derivation of equivalence of molecular and atomic orbitals

Intuitively, it seems quite reasonable that n atomic orbitals produce n molecular orbitals. I'm curious about how this can be proven from the generalised eigenvalue equation for the AO coefficients. ...
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Are there ways to find representations of matrices given an algebra?

Given an equation (or a set of equations) involving matrices, is there an algorithm to find possible representations of these matrices? For example, we can consider a matrix $A$ such that $A^2=\begin{...
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Matrix elements of vector current in $\pi^-$ decay

In positive pion decay $\pi^- \rightarrow \mu^- + \overline{\nu}_\mu$, the matrix element contains the following factor: $$<0|\overline{u}(0)\gamma^\mu d(0)|\pi^-(p)>$$ On the book by Nachtmann, ...
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How to diagonalize the following perturbation?

Consider an infinite cubic well: $V(x,y,z)=0$ for $0<x<L, 0<y<L, 0<z<L$ and $V=\infty$ otherwise The eigenstates of this system are given by: $$\Psi(x,y,z)=\bigg(\frac{2}{L}\bigg)^{\...
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Why doesn't the Weinberg-Witten theorem forbid collinear photons?

The Weinberg-Witten theorem tells us that any theory that has an effective graviton, i.e. a massless helicity-2 particle as a state in the free-particle Fock space, cannot have a gauge-invariant and ...
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Special relativity, what is an transformation with just spatial rotation?

Can someone help me understand how to mathematically represent a transformation of a matrix with a rotation of the spatial axes and why this transformation does not change the unit 3x3 matrix? This ...
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1 answer
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Is an Hamiltonian like this non-degenerate? [closed]

If I have a system of only two energy levels $E_0$ and $E_1$ and an Hamiltonian $$H=\begin{pmatrix} E_0 & 0 \\ 0 & E_1 \\ \end{pmatrix}$$ can the Hamiltonian be both degenerate and non-...
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What is the matrix element $\langle \phi_a | e^{- i H t} | \phi_b \rangle$ for a free field theory?

Suppose we have a free scalar field theory where the action is $$ S[\phi] = - \int d^4 x\; \bigg[ \frac{1}{2}(\partial\phi)^2 + \frac{1}{2} m^2 \phi^2 \bigg] \ . $$ Define the position eigenstates of ...
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Matrix elements of the self-energy in the GW approximation

I have been teaching myself the $GW$ approximation (or the $G_0W_0$ version of it). I am able to follow the entire derivation of Hedin's equations, etc. and the $GW$ approximation itself. However, I ...
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Mapping from tensor notation to matrix notation, left right or upper lower to row column?

In the matrix notation $M_{ab}$, the left index goes through the rows ($a = 1,2,3\dots m$ means there are $m$ rows). The right index go through the columns, ($b = 1,2,3\dots n$) means there are $n$ ...
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How does the paring of indices work in matrices?

I am not sure whether the title does convey what I want to ask, but here are two observations from the cited (freely available) articles: Below equation 1 of this article, a matrix $T_{ijkl}$ with ...
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Intrinsic $Z$-$Y$-$Z$ Euler angle sequence

Consider Euler angles $\alpha$, $\beta$, $\gamma$ with the intrinsic $Z$-$Y$-$Z$ convention. The effective rotation $S(\alpha, \beta, \gamma)$ can be written as: \begin{equation} S(\alpha, \beta, \...
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Determining Hyperfine Hamiltonian Term [closed]

I'm having difficulty determining the hyperfine Hamiltonian term for a hydrogen-like atom ($^{87}$Rb & $^{85}$Rb). The hyperfine Hamiltonian is given by: $${\hat{H}}_{hf}=a_{hf}\left(\frac{1}{2}\...
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Question about bra-ket notation for inner products in Quantum Mechanics

I'm trying to understand the notation used to denote inner products in my introductory quantum physics textbook (Introduction to Quantum Mechanics, Griffiths). In section 6.1 where Griffiths ...
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QED vertex as 2 equivalent matrix elements

Consider just right handed fields $e_R,\bar{e}_R$ making up the standard electromagnetic current $A_\mu \bar{e}_R\gamma^\mu e_R$. Consider the matrix elements $$\langle 0|J^\mu|e\bar{e}\rangle, \...
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Is the matrix element symmetric in a cross-section computation?

Let's consider the cross-section of $A+B\rightarrow C+D$. This involves the usage of an "matrix element" : $M$ for the amplitude of the reaction. Is the element matrix of $A+B\rightarrow C+D$...
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Orthogonality of basis functions of irreducible representations

Lets say I have two irreducible representations of a finite group $G$. Let the group have $l$ elements. The elements of the representation shall be linear operators $L$ that act on some function ...
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Twisted Bilayer graphene effective model (Model for the metal-insulator transition in graphene superlattices and beyond)

This question is referred to the article "Model for the metal-insulator transition in graphene superlattices and beyond" The hamiltonian H1 is written using chiral basis. Can someone direct ...
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Differential Decay Rate - What is wrong with my result?

I am trying to calculate the decay rate of pion in a certain BSM model. In this model the pion decays to a photon, and a pair of BSM particles $\chi_1$ and $ \chi_2$; the diagram is a pion decaying to ...
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Relation between two-particle Green's function and density matrix elements

In the article https://doi.org/10.1103/PhysRevA.69.054305, the authors used the following relation between the 2-particle density matrix element and 2-particle Green's function to calculate the ...
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3 votes
3 answers
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Covariant and contravariant for a beginner

I saw that people were representing matrices in two ways. $$\sum_{j=1}^n a_{ij}$$ It is representing a column matrix (vector actually) if we assume $i=1$. $$\begin{bmatrix}a_{11} & a_{12} & ...
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Irreducible Representations: Howard Georgi's book

I am currently studying the book Lie Algebras in Particle Physics by Howard Georgi (2nd edition) and I couldn't understand this part: A representation is completely reducible if it is equivalent to a ...
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Calculation of electromagnetic transition probability

I'm reading From Nucleons to Nucleus by Suhonen, where (page 118) he introduced the electric or magnetic (with an index $\sigma$ such that $\sigma = E$ or $\sigma = M$) by the formula, \begin{align} ...
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What is meant by the "components" of operator?

So far, I have understood that vectors are represented in a basis, and operators are linear maps which map one vector to another in the same space or to a different space. What I don't understand is ...
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3 answers
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How it is possible that a ket precedes a bra in a matrix expression?

Is it possible to rewrite $\langle a| M|b\rangle$ as $|b\rangle \langle a|M$?
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1 answer
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Eigenstate of a Hamiltonain diagonalized using a paraunitary matrx

I have a question regarding the definition of energy eigenstates of Hamiltonian that can be diagonalized using paraunitary matrices. From quantum mechanics, energy eigenstates are defined by, $$ \...
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2 votes
2 answers
286 views

What product is used in Dirac Equation?

I've just learned that the Dirac Hamiltonian in the Dirac Equation is given by $H = c\alpha \cdot \overrightarrow{p} + \beta mc^2$ and $\alpha$ is a 4 X 4 represented as $\begin{bmatrix} 0 & \...
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4 votes
1 answer
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Why is there an $i$ in the definition of hadronic decay constants?

The decay of (for example) a pion can be parameterized by a decay constant $f_\pi$ defined via $$ \langle 0 | \bar d \gamma_\mu \gamma^5 u |\pi^+(p) \rangle = i f_\pi p_\mu $$ $$ f_\pi \approx 131 \...
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2 answers
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Sakurai Quantum Mechanics - Definition of Matrix Elements [closed]

I don't understand how he represents $X$ as a matrix. Any help would be appreciated.
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Can there be singular subtensors in the elasticity tensor and what would it mean in that case?

Can the elastic tensor $C_{ijkl}$ describing the stress strain relationship as $$\sigma_{ij} = C_{ijkl} \, \varepsilon_{kl}$$ contain singular subtensors? In that case, what would it mean mechanically?...
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Relation between $\delta$-system

I'm reading Pavel Grinfeld's book "Introduction to tensor analysis and the calculus of moving surfaces". I've reached the section where the author talks about $\delta$-systems and the ...
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3 answers
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Proof of form of 4D rotation matrices

I am considering rotations in 4D space. We use $x, y, z, w$ as coordinates in a Cartesian basis. I have found sources that give a parameterization of the rotation matrices as \begin{align} &R_{...
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1 answer
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Photon-Gluon annihilation in QCD

I am starting to learn about QCD, and I wanted to calculate the squared matrix elements for photon-gluon annihilation into a quark and an anti-quark. However, I am having trouble writing down the ...
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1 answer
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Interpretation of indices in Matrix representation of operators [duplicate]

Say, an operator is being expressed in a matrix form, using its orthonormal eigenstates as a basis. We have $ A|n\rangle = k|n\rangle$. To get the matrix representation, what we do is : $ A_{mn} = \...
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1 vote
1 answer
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Matrix Elements of the raising and lowering operators for angular momentum

I was taught in our chemistry spectroscopy class (NMR module) that the matrix elements of $J_+$ and $J_-$ operators are respectively $\sqrt{(s-m)(s+m+1)}\hbar$ and $\sqrt{(s+m)(s-m+1)}\hbar$, but the ...
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2 answers
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Kronecker Delta to Matrix form of operator

I've recently seen the expression of the creation and annihilation operator in the harmonic oscillator in terms of matrices. What I've understood is that: $$ A_{ij} = \langle i | A | j\rangle. $$ ...
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2 answers
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Derivative of eigenvalues with respect to parameter

By trying to find precise ways to calculate the derivative of numerical Hermitian matrices, I've recently stumbled upon this post in Math Stack Exchange. From the first answer on that post we get an ...
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1 vote
1 answer
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Ensemble average of the interaction energy in second quantization

I have a problem with the calculation of the ensemble average for the second quantization interaction energy. It is definded as: $$E_{int} = \left < \frac{1}{2} \sum_{i,j,k,l}w_{i,j,k,l} \hat{c}_{i}...
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2 votes
1 answer
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SVD of $2\times 2$ matrix where entries have different units

I have the following matrix: $$ M = \begin{pmatrix} 1 + xy & y \\ x & 1 \end{pmatrix} $$ where $x$ has the unit $m^{-1}$ (per meter) and $y$ has the unit $m$ (meter). This matrix acts on real ...
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2 votes
1 answer
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Matrix components defined as a trace of other matrix involving pauli matrices [closed]

For a given 2 $\times$ 2, unit determinant, complex matrix $S$, define the matrix $\Lambda$ by the following. $$\Lambda_{\mu \nu}=\frac{1}{2}tr(\sigma_{\mu} S \sigma_{\nu} S^{\dagger})$$ Show that $\...
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1 vote
1 answer
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Bhabha scattering in the spinor-helicity formalism

I am trying to calculate the square amplitude for Bhabha scattering $e^-(p_1)e^+(p_2)\rightarrow e^-(p_3)e^+(p_4)$ using the spinor-helicity formalism but one of the interference terms just will not ...
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1 vote
2 answers
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Matrix Representation of Lorentz Group Generators

Let $\Lambda^{\alpha}{}_{\beta}$ denote a generic Lorentz transformation. Then, an infinitesimal transformation can be written like $$\Lambda^{\mu}{}_{\nu} = \delta^{\mu}{}_{\nu} + \omega^{\mu}{}_{\...
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2 answers
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How is the spontaneous decay rate of an atomic level calculated?

I am trying to understand the transition rate $\Gamma$ between two atomic state levels when there are multiple decay channels involved. Let's say I am considering the spontaneous decay from an upper $...
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Where do the traces come from in Casimir's trick?

I am following the derivation for electron-muon scattering amplitude in Griffiths textbook and got to the section where they use Casimir's trick. I can't see where the traces come from. Equation 7.123 ...
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Defining the inverse of a tensor via the adjugate tensor

My professor definied the adjugate of a tensor $\mathbf{t}\in T^{1}_{1}(E)$ (E is just a vector space of dimension n) by defining its components as $adj(\mathbf{t})^{a}_{b}=\frac{1}{(n-1)!}\...
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Definitions of Determinants and Permanents in QFT

I have been recently reading a QFT book called: "QFT For the Gifted Amateur". It states in footnote 4 on p. 40 that the determinants and permanents of matrices can be defined as follows: $$ \...
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Summing of neutrino spins in Muon decay

This might be a silly question, but I am just a beginner so please bear with me. When we average the matrix element over spins, do we also have to sum over neutrino spins since neutrinos are only Left-...
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4 answers
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Matrix algebra of rotations

I have heard that the basis vectors and the components of the vector transform differently keeping the vector same. Normally the Matrix notation for expressing rotation is: $ \left(x' \quad y'\quad z'...
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4 answers
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Matrix element of powers of position operator for quantum harmonic oscillator

A similar question has been asked here before, but that did not contain the particular solution I am after and is now closed. I was wondering if there is a compact analytical formula for matrix ...
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