Questions tagged [matrix-elements]
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231
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How to write a Hamiltomian in matrix form? [closed]
Suppose we have the following Hamiltonian:
$$\begin{aligned} H= & \Delta_a a^{\dagger} a+\Delta_b b^{\dagger} b+\Delta_c c^{\dagger} c+\Omega\left(c^{\dagger}+c\right)+J_1\left(a^{\dagger} c+c^{\...
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Matrix element of third power of $r$ for radial function in $\rm H$ atom [closed]
How to calculate this matrix element in the closed form?
$$\int_{0}^{\infty} r^3 R_{n1}^{*} R_{10} dr$$
$R_{nl}$ denotes radial part of hydrogen atom wavefunction.
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Tensor Index Manipulation
I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem. I have to prove that
$$\partial_{\mu} ...
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Spin 1 Matrix Representation
I want to prove that the spin 1 matrix representations $(x,y,z)$ obey the angular momentum algebra, and i want to do it in another way. Suppose that the $x$-component of a spin 1 matrix is represented ...
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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$$
Here $\Lambda$ are my lorentz transformations such that $\Lambda^T \eta \Lambda=\eta$. $A$ is some matrix.
I know that in terms of ...
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Clebsch-Gordon coefficients for isospin transformations
I'm trying to understand a paper in which the author uses the nuclear matrix elements from beta decay to arrive at the matrix elements for neutral-current neutrino-nucleus excitation. The author ...
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Correct units when calculating matrix elements between bound and continuum states
Consider that the 1D time-independent Schrodinger equation has been solved to obtain both bound states ($\phi_n(x)$, for $n$ discrete) and continuuous states ($\psi_E(x)$, for $E$ continuous). These ...
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Breaking product of three vectors into symmetric and anti-symmetric vectors [closed]
Let's consider we have three arbitrary vectors A, B and C. We have the quantity $A_{\mu}B_{\nu}C_{\rho}$. Is it possible to break the above quantity into sum of symmetric and anti-symmetric vectors in ...
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Matrix Factorization into a Real Times a Unitary
I am working on my Final Degree Project involving entanglement entropy, for which I am reading H. Cassini and M. Huerta's paper Entanglement entropy in free quantum field theory
(J.Phys.A42:504007,...
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How can I demonstrate that, in a degenerate system, if $[H_0,\lambda H_1] = 0$ then $ H_1$ is already diagonalized?
In the context of degenerate perturbation theory, for a perturbed Hamiltonian $H_0 + \lambda H_1$, I've heard of a very useful tool:
"If $[H_0,\lambda H_1] = 0$ holds (both parts of the ...
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Confusion regarding Fermi's golden rule
The Fermi Golden Rule is:
$$\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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About calculation of matrix element of heat kernel component
I'm reading "Introduction to Quantum Effects in Gravity - Mukhanov, Winitzki" and in the Draft version I'm referring to pag. 183, when they state that
$$
\langle x | \hat{K}_1^\Gamma(\tau) | ...
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Representations of continuous operators
I'm currently trying to self-study quantum mechanics from Cohen-Tannoudji, and am struggling to wrap my head around the following claim. In section C-4 he says that
[representations of operators] ...
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Question on Pauli matrices completeness relation
In the derivation on Wikipedia, it says that if
$$2 M_{\alpha \beta} = \delta_{\alpha \beta} M_{\gamma\gamma} + \sum_k \sigma^k_{\alpha \beta} \sigma^k_{\gamma \delta} M_{\delta \gamma}$$
for any ...
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Transpose of a matrix element [closed]
A matrix element is just a number. Now, If I have the following matrix element:
\begin{equation}
\newcommand\bra[1]{\left<{#1}\right|}
\newcommand\ket[1]{\left|{#1}\right>}
A = \bra{B}\bar{b}\...
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What is the matrix representation of a function of an quantum operator?
If we know the matrix representation of a quantum operator, say $J$, will the matrix representation of any function of the operator i.e$f(J)$, same as acting the function on the matrix of the operator?...
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Scattering in Shallow Bound States - Suppression of Matrix Element
In Weinberg's Lectures on Quantum Mechanics, given the S-matrix
$$S_{\beta \alpha}=\delta(\beta-\alpha)-2 \pi i \delta\left(E_\beta-E_\alpha\right) T_{\beta \alpha}\tag{8.8.4}$$
where
$$T_{\beta \...
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A specific question about sakurai modern quantum mechanics
In the kronocker deltas for spin, $\lambda_4$ and $\lambda_3$ places changes. How can the author change their places?
Also, I see both case can appear by calculation so there are two different ...
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Stiffness matrix of an orthotropic transversely isotropic material
I am studying the generalized Hooke's law for an orthotropic transversely isotropic material (the same behaviour along the directions $x_2=y$ and $x_3=z$).
The general elastic law by Hooke states that
...
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Parity operator for a general $N\times N$ Hamiltonian
Assume I've a $N\times N$ tight-binding Hamiltonian that depends on spatial coordinates $k=(k_x, k_y)$; it holds
$$ H(k)|\psi_k\rangle = E_k |\psi_k\rangle \quad .$$
Now I want to find the parity ...
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Indices of the Minkowski metric and Lorentz transformation
I am currently in the process of studying special Relativity and I keep stumbling over a concept I can't make consistent for myself.
It is about the fact which index of a Lorentz transform and the ...
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Issue when trying to represent an operator as a matrix
We say that a ket $|V\rangle$ can be expresed in an orthonormal basis $(|e_1\rangle,|e_2\rangle,...|e_n\rangle)$ as :
$$|V\rangle = \sum_i^n v_i |e_i\rangle$$
where $v_i = \langle i|V\rangle $
for a ...
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Cosmology, transpose of partial derivatives and Riemannian metrics
In general relativity and cosmology, Riemannian metrics play an important role. Here, I have a few related questions about the algebraic manipulation of derivatives and Riemannian metrics in the ...
user252965
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Matrix formulation of the momentum operator
For a quantum state $\Psi=c_{1}\psi_{1}+c_{2}\psi_{2}$ with momentum eigenstates $\psi_{1}$ and $\psi_{2}$, the action of the momentum operator $\hat{p}$ is given by
$$\hat{p}\Psi=p_{1}c_{1}\psi_{1}+...
