Questions tagged [matrix-elements]

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Beta decay transitions

Why do we say that the matrix element for beta(-) decay is 1 for the Fermi's transitions while is 3 for the Gamow's transitions?
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2answers
105 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
74 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_{-\infty}^{\...
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7 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
64 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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24 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
34 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
44 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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29 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
38 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
47 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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26 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
64 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
54 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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23 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
210 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
29 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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1answer
69 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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38 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
139 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
91 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
48 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
409 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
92 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
119 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
354 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
263 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
26 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
209 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
254 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
196 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
342 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}\...
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2answers
267 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{\...
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1answer
151 views

Heisenberg's derivation of Schrödinger's Equation [duplicate]

In Heisenberg's book "The Physical Principles of the Quantum Theory", he presents the following derivation of the Schrödinger Equation from his own, Matrix-based, Quantum Mechanics. A matrix $x$ has ...
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1answer
783 views

What is the matrix element?

Can someone give me an Eli5 description of what the matrix element is, particularly in regards to Fermi's Golden Rule? Fermi's golden rule describes the likelihood of a transition per unit time. ...
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3answers
2k views

Definition of an operator in quantum mechanics

In J.J. Sakurai's Modern Quantum Mechanics, the same operator $X$ acts on both, elements of the ket space and the bra space to produce elements of the ket and bra space, respectively. Mathematically, ...
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1answer
189 views

covariant and contravariant form of a matrix

I'm following a paper to solve this equation: $y_{j}=y_{o}$ + A$\eta^{T}$ (Eq. 2) My question is about the term $\eta^{T}$. In the paper says: "With symbol $\eta$, we denoted a 1 × 6 ...
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1answer
36 views

Property of nonadiabatic vector coupling matrix

I just tried to derive the "dressed" kinetic energy operator (for the Hamiltonian $\mathbf{H} = \frac{1}{2M}\left(\mathbf{P} -\mathrm{i}\hbar \mathbf{F} \right)^2 +\mathbf{V}$) in the adiabatic basis [...
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1answer
82 views

$U(2)$ or $SU(2)$? Interferometers and Jones matrices

Recently I've been trying to understand why the scattering matrices that describe an interferometer should be $SU(2)$ matrices rather than $U(2)$. The condition of unitarity is undiscussed as it ...
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1answer
393 views

Computing reduced matrix element without Wigner-Eckart theorem

Lets have a problem: suppose we need to calculate reduced matrix element of some transition of a particle from some higher-order spin(or rather total angular momentum state, it does not really matter, ...
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3answers
400 views

How does one calculate the position eigenvalues of the matrix corresponding to the position operator?

The matrix representation corresponding to the position operator is: $$x = \sqrt{\frac{\hbar}{2 m \omega}} \left[ \begin{array}{ccccc} 0 & \sqrt{1} & 0 & 0 & \cdots \\ \sqrt{1} & ...
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1answer
510 views

Levi-Civita tensor

The Kronecker delta can be represented by a two dimensional matrix: \begin{gather} \delta_{ij}=\mathbb{I}= \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ \end{bmatrix}. \end{gather} ...
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1answer
344 views

Using Dirac notation to find matrix representation

I am currently reading Sakuria, and I cannot get my head around how one uses the completeness relation to derive the matrix representations of outer products. In the first chapter he states that an ...
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1answer
240 views

How to derive the Schrödinger Equation from Heisenberg's matrix mechanics and vice-versa?

How do you derive the Schrödinger equation (wave mechanics, time dependent state) from Heisenberg's Matrix Mechanics (matrix based, time dependent operators)
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1answer
97 views

Proof that elastance matrix is invertible

I was reading this lecture notes from MIT OCW on capacitance . It says $V_i =\sum_j P_{ij}Q_{j} $ where the constants $P_{ij}$ are determined by the geometry of the conductors. This matrix can ...
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1answer
26 views

using symmetry in the siffness method

Good day All while trying to solve this exercice I tried to find a symmetry plan to make my computations easy and according to my basic understanding the symmetry must be in term of: lenght length ...
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1answer
999 views

What does the time evolution operator look like in Matrix representation?

In Matrix representation of quantum mechanics, using an energy eigenbasis, we have the state vector: $$|\psi(t)\rangle=\left(\begin{matrix} \psi_0(t)\\ \psi_1(t)\\ \psi_2(t)\\ \vdots \end{matrix}\...
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0answers
30 views

Stiffness matrix issue

Good day All, while trying to solve this exercice I was puzzeld by the solution approach indeed, they use the symmetry of the structure, they have made a cut on the hinge where the force F is applied ...
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2answers
581 views

Sign issue with the matrix of rotated elements (stiffness matrix)

Good day All I have a doubt regarding the derivation of the following matrix according to my basic understanding we want to go from the basis u1, v1, u2, v2, to the basis u'1, v'1,u'2 ,v'2, and for ...