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Questions tagged [matrix-elements]

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303 views

Finding the eigenvalues and eigenvectors of this operator

The operator $$\mathcal{\hat{G}} = (\xi - 1) \sum_{j=1}^N \int dk_j \; k_j \hat{a}^\dagger_j(k_j)\hat{a}_j(k_j),$$ is physically similar to the momentum operator in quantum mechanics. It has the ...
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1answer
122 views

Singularities of matrix element of composite local operator in QFT

Consider a state $|\psi\rangle$ in a quantum field theory and a local operator $\mathcal{O}(x)$. It's known that the $n$-point function $\langle \psi | \mathcal{O}(x_1) \cdots \mathcal{O}(x_n) | \...
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1answer
1k views

Pauli matrices in spherical coordinates

In my work I currently have to work with the following partition function $Z\propto e^{\vec{h}.\vec{\sigma}}$ where $\vec{\sigma}$ are pauli matrices and $h$ is some vector. So far I've been using ...
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396 views

Calculation of $b \to s~ l^+ l^-$ penguin diagram

I'd like to calculate the matrix element amplitude for $b \to s~ l^+ l^-$ penguin diagram mediated by Z boson or the photon , like : These calculations are made of course from many time ago, so if ...
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1answer
80 views

Matrix mechanics [closed]

So, I am given a Hamiltonian of a system, represented in the $|e_{i}\rangle$ basis as: $$H=\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix}$$ where, $|e_{1}\rangle = \begin{...
2
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1answer
898 views

Partial completeness relation for Dirac spinors

in studying trace techniques to obtain matrix elements, I came across a problem when we treat scattering of neutrinos on protons. Indeed, since those neutrinos are supposedly created in a weak decay, ...
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1answer
468 views

What is the point of degenerate perturbation theory in quantum mechanics?

What is the point of degenerate perturbation theory in quantum mechanics? Let's disregard for a moment the issue of constructing the perturbed wave functions and assume that the 1st order correction ...
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1answer
937 views

What type of mathematical object is the “Pauli vector”?

The three pauli matrices $\sigma_x$, $\sigma_y$, $\sigma_z$ are sometimes combined in the "Pauli vector", usually denoted $\boldsymbol{\sigma} = \sigma_{x} \boldsymbol{e_x} + \sigma_y \boldsymbol{e_y} ...
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3answers
5k views

How do one show that the Pauli Matrices together with the Unit matrix form a basis in the space of complex 2 x 2 matrices?

In other words, show that a complex 2 x 2 Matrix can in a unique way be written as $$ M = \lambda _ 0 I+\lambda _1 \sigma _ x + \lambda _2 \sigma _y + \lambda _ 3 \sigma_z $$ If$$M = \Big(\begin{...
2
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1answer
113 views

PDFs expressed through matrix elements of bi-local operators

Extracted from 'At the frontier of ParticlePhysics, handbook of QCD, volume 2', '...in the physical Bjorken $x$-space formulation, an equivalent definition of PDFs can be given in terms of matrix ...
2
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1answer
207 views

Is there a list of hydrogenic transition matrix elements online?

Calculating transition matrix elements can be difficult, and I have found myself needing to use hydrogenic electric dipole transition matrix elements a fair bit. $$\mathbf{r}_{nlm}^{n'l'm'}=\langle\...
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1k views

Raising and Lowering Indices using the Metric Tensor

Given the next tensor: $X^{\mu \nu}= \left(\begin{array}{cccc} 2 & 0 & 1 & -1 \\ -1 & 0 & 3 & 2 \\ -1 & 1 & 0 & 0 \\ -2 & 1 & 1 & -2 \\ \end{array}\...
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2answers
354 views

Index notation matrix calculation (Intro to Relativity)

Consider the next matrix: $$M_{ab} = \left(\begin{array}{cccc} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\...
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1answer
598 views

A question about metric tensor and their minors and cofactors in general relativity

In Einstein's book- 'the meaning of relativity', he says- The equation 55 mentioned is this one- I don't understand what the equation (62) means or how it can be proved. I know that the metric tensor ...
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1answer
2k views

Selection rules for electric quadrupole radiation

The selection rules for electric quadrupole radiation in a Hydrogen-like atom are: $$ \begin{aligned} \Delta l &= 0,\pm2 \hspace{1cm}(l=0\leftrightarrow l'=0 \textrm{ is forbidden}) \\ \Delta m &...
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2answers
301 views

Individual terms in a Hamiltonian matrix

Reference to Problem 2, Chapter 2 in Modern Quantum Mechanics by JJ Sakurai, Consider the following Hamiltonian of a two state system $$ H=H_{11}|1\rangle\langle1|+H_{22}|2\rangle\langle2|+H_{12}|1\...
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0answers
764 views

How to construct the matrix of Hamiltonians for a hexagonal lattice

For part of a project I need to solve the time-independent Schrödinger equation, $\mathbf H\Psi = E\Psi$ (where $\mathbf H$ is the matrix with elements $\langle\Psi_i|H|\Psi_j\rangle$, and $\mathbf S$ ...
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3answers
166 views

Physical meaning of $\left \langle \psi_{nml} | x | \psi_{n'm'l'} \right \rangle$ for hydrogen atom?

I know how to calculate the matrix element $\left\langle \psi_{nml} | x | \psi_{n'm'l'} \right\rangle$, but what is the physical meaning of it? In general, what does the following mean: $$ \left \...
2
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2answers
343 views

Expectation value of an imaginary operator acting on a real function

In a video (http://youtu.be/r_gBQ_qhg8U?t=9m58s) it's stated that a matrix element of an imaginary operator acting on a real wave function is zero, i.e. $$\langle\text{real}|\text{imaginary}|\text{...
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1answer
2k views

Determining the three Euler angles from the acceleration

I want to know, given the measurement of an accelerometer at rest (so not really an acceleration but a force per unit of mass) the inclination of this accelerometer, along the three axis. For this, I ...
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2answers
4k views

3D frame stiffness matrix local to global

I am working on a simple script to be able to solve frame structure using direct stiffness method. I am having following stiffness matrix for 2 node frame element: What is the correct way of ...
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1answer
357 views

Colour decomposition in QCD

I am looking to compute the matrix element for the process gg -> u ubar at leading order. It is straightforward to calculate the non-colour part of the usual s, t and u channels. I will call these ...
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1answer
225 views

Scattering from a potential, matrix elements of momentum eigenstates, and the Fourier transform [closed]

I am working on my last quantum homework and don't know where to begin with part (i) in this question 4. Do I need to use a product rule in the FT and use convolution? Not sure how to go about the ...
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0answers
128 views

matrix elements of the electronic molecular Hamiltonian between a hartree product and a Slater determinant

This may belong in Chemistry, but I thought I might try my luck here first. In Szabo's book, an exercise requires a proof that = (N!)^(1/2) * given that |K(HP)> is the Hartree product wave ...
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1answer
2k views

Matrix elements of the operator $\hat{x} \hat{p}$ in position and momentum basis

I want to calculate the matrix elements of the operator $\hat{x} \hat{p}$ in momentum and position basis, that is the two quantities ($p,q$ - momenta, $x,y$ - positions): $$\langle p|\hat{x} \hat{p}|...
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1answer
697 views

Perturbation of an operator - Meaning of matrix element [closed]

Let be $B$ an operator and $\left|\Psi\right>$, $\left|\Phi\right>$ two states (not necessarily equals). What is the meaning of a matrix element $\left<\Psi\right| B \left|\Phi\right>\...
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1answer
1k views

How are matrices used to represent quantities, and what is the meaning of a matrix?

So I'm reading this text on Quantum Mechanics, and it goes through a few chapters that I understand fairly well including probability. But then it says that all quantities, like position and energy ...
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1answer
257 views

Derivation of matrix element

I have tried to understand paragraph 10.7 (Kallen-Lehmann Representation) in Weinberg's Quantum theory of fields (vol.1). He calculated matrix element $$\langle0|\Phi(0)|p\rangle =(2\pi)^{-3/2}\left(...
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1answer
158 views

Bath Hamiltonian in second quantization

I'm trying to write down the bath Hamiltonian for a system of dimers and trimers. Imagine each of the monomers in the excited state can interact with several phonons with given frequencies. The bath ...
3
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1answer
1k views

1D Ising Model (NN and NNN interactions) with 2 transfer matrices

I've tried an alternative solution for finding the partition function of this model. So is what I've done correct? If it isn't then please prove and explain why not. (I'm pretty sure I made a ...
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1answer
162 views

Matrix elements of a one-fermion operator (first and second quantizations)

I'm currently struggling with the expression of operators in second quantization. I did an exercise in which I had to consider a fermion in a central potential $V(\vec{r})$ and show that the matrix ...
3
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1answer
730 views

How to show that basis of space of Dirac gamma-matrices is given by following matrices?

How to show that 16 matrices $$ \mathbf E , \quad \gamma^{\mu}, \quad \gamma^{5} = \frac{i}{4}\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}, \quad \eta^{\mu \nu} = -\frac{1}{4}\left(\gamma^{\mu}\gamma^{\nu}...
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3answers
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When Eigenfunctions/Wavefunctions are real?

When the Hamiltonian is Hermitian(i,e. beyond the effective mass approximation), generally under which conditions the eigenfunctions/wavefunctions are real? What happens in 1D case like the finite ...
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1answer
95 views

Couple Masses - Change in Basis

I'm having trouble with the linear algebra used to solved a coupled mass problem. $\ddot{x}_1 = -(2k/m)x_1 + (k/m)x_2$ and $\ddot{x}_2 = (k/m)x_1 - (2k/m)x_2$ Shankar then sets the equation up in ...
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2answers
516 views

What is the correct arrangement of the elements of Pauli matrices?

I'm dealing with angular momentum, or particularly spin, on my quantum mechanics course; I guess the Pauli matrices thing is a more general one, but I'd like to illustrate my doubt with them (maybe ...
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1answer
89 views

Diagonal matrix in k-space

I'm having some trouble with an integration I hope you guys can help me with. I have that: ${{\mathbf{v}}_{i}}\left( \mathbf{k} \right)=\frac{\hbar {{\mathbf{k}}_{i}}}{m}$ and ${{\mathbf{v}}_{j}}\...
3
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2answers
269 views

Element of area in 4-dimensional space-time

How would you proof that $$ \mathrm {Tr} (\mathbf{S\cdot \bar S })=0$$ where $\mathbf S$ is an element of area delimited for the 4-vectors $\mathbf u$ and $\mathbf v$ given by $$S^{\alpha \beta}\equiv ...
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4answers
2k views

Is the momentum operator well-defined in the basis of standing waves?

Suppose I want to describe an arbitrary state of a quantum particle in a box of side $L$. The relevant eigenmodes are those of standing waves, namely $$ \left<x|n\right>=\sqrt{\frac{2}{L}}\cdot ...
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2answers
554 views

How to express continuous values as a matrix

Usually a quantity of a matrix is defined as the eigenvalues of the matrix. If so, how can anyone express continuous values, as in Schrodinger picture, into a matrix?
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2answers
3k views

Matrix Representations of Quantum States and Hamiltonians

I am a high school student trying to teach himself quantum mechanics just for fun, and I am a bit confused. As a fun test of my programming/quantum mechanics skill, I decided to create a computer ...
4
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1answer
381 views

Creation and Annialation Operators and Kinetic Energy Matrix Elements

I'd like to write equations for $c_{ij}(t)$, With a hamiltonian of the form $$H=\sum_{kn}a^{\dagger}_k t_{kn}a_n + \frac{1}{2}\sum_{klmn}a^{\dagger}_k a^{\dagger}_l v_{klmn}a_m a_n$$ with $t_{kn}$ ...
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2answers
423 views

If the S-matrix has symmetry group G, must the fields be representations of G?

If the fields in QFT are representations of the Poincare group (or generally speaking the symmetry group of interest), then I think it's a straight forward consequence that the matrix elements and ...
3
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3answers
969 views

Vector space of $\mathbb{C}^4$ and its basis, the Pauli matrices

How do I write an arbitrary $2\times 2$ matrix as a linear combination of the three Pauli Matrices and the $2\times 2$ unit matrix? Any example for the same might help ?
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0answers
175 views

Decay Amplitudes Notation

This question is mostly about how to interpret notation used in Particle Physics. I am given that at lowest order the rate of $b\rightarrow s\gamma$ is proportional to $\langle B_p|b^\dagger b|B_p\...