Questions tagged [matrix-elements]

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How to determine correct spinor in Feynman diagram

Setup Consider Moller scattering, that is $$e^-(\vec p_1, \alpha)+e^-(\vec p_2, \beta) \quad\longrightarrow\quad e^-(\vec q_1, \gamma)+e^-(\vec q_2, \delta),$$ where the $\vec{p}_i,\vec q_i$ label the ...
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2answers
117 views

Physical meaning of the rows and columns matrix representation of operators in Quantum Mechanics

When any operator is written in its matrix form, do the individual rows and columns of the matrices have any physical meaning to them? For example, the rows and columns of momentum and position ...
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1answer
46 views

Does a $U(1)$ matrix, as in QED, have a single entry? A $1×1$ matrix ( degree 1)? [closed]

This is a very simple question, but somehow, I am getting confused.... Some examples of Q.E.D U(1) matrices I see gave four (2×2) elements, like the Weak SU(2) matrices....
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1answer
72 views

When does a Hermitian operator have real matrix elements?

I will use braket-notation, but my question is not specific to quantum mechanics. Instead, I would be interested in a general answer for operators in some Hilbert space. Let $H$ be a Hermitian ...
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2answers
74 views

Matrix elements of powers of the position operator for a SHO [closed]

I'm asked to compute the matrix elements $\left( x^i \right)_{nm}$ where $i=2,3,4$ and $x$ represents the position operator for the quantum harmonic oscillator. I know that for $i=1$: \begin{equation} ...
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1answer
40 views

Taking derivatives of traces over matrix products

I started with evaluating the following derivative with respect to a general element of an $n\times n$ matrix, $$\frac{\partial}{\partial X_{ab}}\left(\mathrm{Tr}{(XX)}\right)$$ I wrote out the ...
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1answer
54 views

Spinor transformation matrix derivation

The spinor in the Dirac equation should transform via a 4x4 matrix $S$ that depends on the specific Lorentz boost/rotation: $\psi '(x')=S(\Lambda )\psi(x)\tag1$ Where S satisfies: $S^{-1}\gamma ^{\...
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1answer
39 views

Relation between Wigner $D$-matrix and its complex conjugate

I have an expression with two Wigner $D$-matrices $$\mathcal{D}^{j_1}_{m'_1,m_1}(\phi,\theta,-\phi)\mathcal{D}^{*j_2}_{m'_2,m_2}(\phi,\theta,-\phi)$$ and I would like to write the second matrix in ...
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2answers
81 views

Diagonalizing Hamiltonian in Second Quantization

I have a fermionic system with states 1,2. They are coupled by a harmonic oscillator. The Hamiltonian of the system should then be $$ H=\left[\gamma(a^\dagger+a)-\delta\right]\left( c^\dagger_1 c_2+c^...
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2answers
57 views

Representation of operator as block matrix

I've two operators that commute: $A=\begin{pmatrix} 2 & 0 & i\\ 0 & 1 & 0\\ -i & 0 & 2 \end{pmatrix}$ and $B=\dfrac{1}{2}\begin{pmatrix} 3 & -i\sqrt{2} & i\\ i\sqrt{2} &...
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1answer
37 views

Hamiltonian matrix elements involving ladder operators for spin-1 state

I am reading the Doctorate thesis 'Zero-Field Anisotropic Spin Hamiltonians in First-Row Transition Metal Complexes: Theory, Models and Applications' (link: https://tel.archives-ouvertes.fr/tel-...
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37 views

Three Gell-Mann Matrices Identity

I have an expression with three Gell-Mann matrices and I have to calculate the color factor. I know that for two of them there is an expression in terms of delta functions: $$\frac{1}{4}\sum_{a}\...
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1answer
81 views

Why does this amplitude not factorise into subamplitudes?

Consider the process $Xq\rightarrow Yq$ at tree level via exchange of a photon. It is depicted in the following Feynman diagram. In various literature it is said that for a nearly on-shell photon this ...
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1answer
29 views

Significance of sign of perturbation matrix

I understand that the off-diagonal elements of a Hamiltonian denote the interaction between different states. The magnitude of off-diagonal elements therefore tells us how strong the interaction is. ...
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1answer
39 views

What transformation gives a Weyl-like representation by flipping $\gamma^0$ and $\gamma^5$?

The usual Weyl representation of the Dirac matrices is defined like this: $$\tag{1}\gamma_W^a = T_W \, \gamma^a \, T_W^{-1},$$ where \begin{align}\tag{2} T_W &= \frac{1}{\sqrt{2}} (1 + \gamma^5 \, ...
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1answer
51 views

Show that the set of Lorentz boosts in the $x$ direction, represented by the matrices $Λ$, form a group under successive application [closed]

So the following question has been causing me problems. Show that the set of Lorentz boosts in the $x$ direction, represented by the matrices $Λ$, form a group under successive application $$ \left[\...
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2answers
80 views

Matrix representation of the operator $\hat{S}_x$ in the standard basis

I have recently been introduced to the idea of spectral decomposition of spin angular momentum operators in Quantum-Mechanics. Out of curiosity I was wondering if the the spin angular momentum ...
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2answers
150 views

Meaning of matrix elements

What does $⟨ϕ|Q|ψ⟩$ physically mean, where $|ψ⟩$ and $|ϕ⟩$ are states and $Q$ is a linear operator? I know what its mathematical meaning is but I am looking for an interpretation: What does $Q$ do to $...
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1answer
112 views

What exactly this means in Dirac notation?

I am pretty new to QM, I get the basics of Dirac notation. I understand this is the expectation value of an observable in a state $|\psi\rangle$ is given by: $$\langle\hat A\rangle = \int\limits_{-\...
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1answer
22 views

Momentum matrix elements for two-photon absorption in semiconductors

I am trying to follow the paper "Two-photon absorption with exciton effect for degenerate valence bands" (to be found here: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.9.3502). It gives the ...
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2answers
91 views

Need help understanding matrix representation of a linear operator

I'm struggling with linear algebra. Specifically, understanding the following: $\newcommand{\ket}[1]{|#1\rangle}$ Suppose $A:V \rightarrow W$ is a linear operator between vector spaces $V$ and $W$. ...
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25 views

Matrix element for the neutron decay using the exact $SU(3)$-symmetric limit

I want to calculate the electron spectrum in the decay process $n\rightarrow p~e^-~\bar\nu_e$. The matrix element should be written in the exact $SU(3)$-symmetric limit. Could you recommend any ...
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0answers
52 views

How to demonstrate that Fierz-like identity for 2-components Weyl spinors? [duplicate]

Consider the 2-components Weyl spinors with the following scalar product \begin{equation}\tag{1} \langle \, \phi, \, \psi \, \rangle = \phi^{\top} \, \sigma_y \: \phi, \end{equation} where $\sigma_y$ ...
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1answer
46 views

Simultaneous diagonalization of Cartan generators of $SO(6)$

This question is naive but for some reason I'm not getting the expected result. The generators of $SO(6)$ can be written in this way: $$(J_{ab})_{cd}=i(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc}),...
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33 views

Matrix representation of position operator in terms of Hamiltionian eigenfunctions

Consider the normalized eigenfunctions $\psi_1(x),\dots,\psi_N(x)$ of the hermitian Hamiltonian $H(x,p_x)$. I want to find the matrix representation of $x$ in the basis of the eigenfunctions of $H$. ...
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1answer
40 views

$r$-representation of Operator

I am watching this video https://www.youtube.com/watch?v=sYgX5pdncG8 at 14:30, it has $\langle r|H|r'\rangle = H(r) \delta(r-r') $ Can you help me to understand why it is so? I thought it should ...
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2answers
49 views

Why is a sum remaining here?

Substituting eqn (5.9) into the time-dependent equation gives $$ i\hbar \sum_n \dot c_n(t) |u_n\rangle e^{-iE_nt/\hbar} = \sum_n V(t) |u_n\rangle e^{-iE_nt/\hbar}c_n(t) \tag{5.10} $$ Now take the ...
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27 views

doubt about Bogoliubov for diagonalize matrix

I have the following equations: $$\begin{pmatrix} \dfrac{d}{dt}C \\ \dfrac{d}{dt}C^{*} \end{pmatrix}= - \dfrac{1}{i} \begin{pmatrix} A& B \\ -B^{*} & -A^{*} \end{pmatrix} \begin{...
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1answer
78 views

Why coefficient of states for non-positive Hamiltonian matrix are all non-negative?

For a Hamiltonian $H$, if the all elements of matrix is non-positive under a set of basis $\{|\phi\rangle\}$:$$\langle\phi|H|\phi'\rangle\leq0$$ then the ground state of $H$ will be the linear ...
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1answer
59 views

Understanding completeness relation and writing Hamiltonian in matrix form

A three level system hamiltonian I found where it is written as: $$\frac{H}{\hbar}= \Omega_1|e\rangle \langle g_1| + \Omega_1^* |g_1\rangle \langle e| + \Omega_2|e\rangle \langle g_2|+ \Omega_2^* |...
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31 views

How to incorporate refractive index in transfer matrix method

I need to determine the TE and TM reflections at the interface between a uniaxial crystal and air, using a matrix-based method, such as the TMM. I can do so using the Fresnel equation with ease, but ...
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1answer
404 views

Heisenberg Hamiltonian 2-Spin Terms in Matrix Representation

I am stuck on the interpretation/derivation of the 2-spin terms of the quantum Heisenberg model Hamiltonian. In this model, our electrons, with spin up or down, are confined to sites on a lattice. ...
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1answer
31 views

Simplified computation of matrices for normal modes?

In normal modes, we often refer the total potential energy of the system to be: $$V = q^T B q$$ where $V$ is the total potential energy, $q$ is the coordinates of the system and $B$ is just some ...
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2answers
147 views

Momentum matrix elements in a crystal

I am trying to follow along a derivation (E. I. Blount, Solid State Phys. 13, 305 (1962)) in which he derives the matrix elements of the true momentum $p_{n,n'}(k,k')$ (not the crystal momentum). He ...
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0answers
42 views

For degenerate perturbation theory, how do we interpret the eigenvectors and eigenvalues of $\hat V$?

For the eigenvectors that are unmixed by the matrix $\hat V$, the eigenvalues are the energy corrections of this eigenbasis. However, the eigenbasis tends to always be (as far as I'm aware) a linear ...
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1answer
173 views

Rotation matrix - levi-civita symbol

I'm trying to solve the following problem: Given a rotation matrix $R_{ij}$, show that $$n_k=\frac{-R_{ij}\epsilon_{ijk}}{\sqrt{(3-tr(R))(1+tr(R))}}$$ and that $$\sin(\phi)=-\frac{\epsilon_{ijk}...
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1answer
107 views

How can quantum operators be expressed as a matrices?

I have just started quantum mechanics with Shankar. In my understanding, quantum operators are linear operators in infinite-dimensional Hilbert spaces. Shankar has repeatedly treated quantum ...
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1answer
49 views

How an operator is converted to a function? [closed]

$$\langle m|F|n\rangle^*=\langle F(n)|m\rangle$$ How does the operator become a function of state $|n\rangle$?
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3answers
566 views

How to write an operator in matrix form?

Say I have the following operator: $$\hat { L } =\hbar { \sum_{ \sigma ,l,p } { l } \int_{ 0 }^{ \infty }\!{ \mathrm{d}{ k }_{ 0 }\,\hat { { { a }}}_{ \sigma ,l,p }^{ \dagger } } } \left({ k }_{...
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1answer
33 views

How to construct a 2-partite matrix

Let's assume we have an internal hamiltonian $H_0 = \mid 1\rangle \langle 1\mid$. Now let's assume we have two systems with identical Hamiltonians $H^1_0$,$H^2_0$ and I want to compute the joint ...
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1answer
98 views

Constructing a Hamiltonian for $N$-qubits

Let us assume we have a qubit with an internal Hamiltonian $H_0 = \sum_i \varepsilon_i |i\rangle\langle i|$. Now let's assume we have 2 such qubits. How would their joint Hamiltonian look like? I ...
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2answers
129 views

Rank of a density matix

I was just trying to understand the meaning of rank of a density matrix. I came across the following post, which says that the rank of density matrix is the number of non-zero eigenvalues. And for a ...
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2answers
397 views

Matrix elements of the free particle Hamiltonian

The Hamiltonian of a free particle is $\hat H = \frac{\hat p^2}{2m}$, in position representation $$ \hat H = -\frac{\hbar^2}{2m} \Delta \;. $$ Now consider two wave functions $\psi_1(x)$ and $\psi_2(x)...
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1answer
346 views

Operators in Dirac notation and matrix representation (intuition)

So, I'm taking a QM 1 course, and we have reached a point where we used Dirac notation to solve two-level systems more efficiently, but our professor never really bothered to explain it further (he ...
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1answer
28 views

Relation of vector and scalar matrix elements for hadronic transitions

Consider a matrix element $$ F_{\mu}(p_{h'}, p_{h}, \dots) = \langle h'(p_{h'})|\bar{q}_{i}\gamma_{\mu}q_{j}|p_{h}\rangle, $$ describing transition of some initial hadron $h$ that contains a quark $...
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1answer
231 views

Explicit construction of $(\frac{1}{2}, \frac{1}{2})$ representation of Lorentz group

For the vector representation of the Lorentz group (actually the algebra), the $J^1$ generator is $$J_1 = i \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 &0 \\ 0 & 0 & 0 ...
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1answer
304 views

Calculation of Matrix Element in “Old-Fashioned Perturbation Theory”

I would like to better under the manipulations/formalism applied in order to evaluate the following matrix element from Schwartz "Quantum Field Theory and the Standard Model" (Eq. 4.16) $$\quad V _ {...
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1answer
231 views

Tensor Product Explanation

I'm currently doing a research project involving 3 particle spins and have developed a simple function for the Hamiltonian: I understand how to code my work but the physics behind it is unfamiliar ...
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2answers
378 views

How to find the coordinate representation of the kinetic operator?

From my professor's notes on statistical mechanics. $\left|\bf{k}\right\rangle$ is eigenstate of the hamiltonian of the free particle with periodic boundary conditions: $$ \left\langle{\bf r}|{\bf k}\...
3
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2answers
309 views

Physical transformation associated with a Pseudo-Orthogonal matrix [closed]

An orthogonal matrix $O$, which belongs to an orthogonal group, is characterized as $O^TO=I$. Let's take an example of a $2 \times 2$ orthogonal matrix, $$O = \begin{bmatrix} \phantom{-} \cos{\...