Questions tagged [matrix-elements]

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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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0answers
25 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
78 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
57 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
45 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
107 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
27 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
72 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
90 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
285 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
182 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
22 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
155 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
163 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
146 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
301 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
197 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
114 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
482 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
134 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
28 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
75 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
307 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
263 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
316 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
243 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
170 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
69 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 ...
0
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1answer
634 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
29 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
422 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 ...
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1answer
85 views

What is norm of matrix element in Fermi Golden Rule

Fermi Golden Rule says: $\Gamma \propto |M_{ij}|^2$ I know how to get $M_{ij}$, but how do I proceed? How do I take a norm of Hermitian matrix? There is no clear (to me) definition in the internet ...
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1answer
108 views

Hermitian Matrix

What is the geometrical significance of the Hermitian matrix? Actually what does the conjugate of the transpose represents. As a determinant if 3 by 3 expresses the volume.
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186 views

Finding a Lens System Matrix for a Planar Concave Thick Lens

How can I go about determining the lens system matrix for a planar concave thick lens? I analyzed that a light ray first enters the lens and refracts, then it goes through the lens, and then it ...
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1answer
97 views

Dimensions/Units and Matrix Inverses

I was going through a homework problem which is essentially a math review in a kinematics frame. This group of problems start as follows, Given $a_x$, constant acceleration, and initial conditions, ...
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2answers
349 views

Which phase shift gate form is correct?

I am trying to figure out the matrix for a gate I'm going to be implementing - a mirror, i.e., a $180^{\circ}$ phase shift. Quantiki gives $$R(\theta)=\begin{bmatrix}1&0\\0&e^{2\pi i\theta}\...
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1answer
865 views

matrix of rotation for quantum states [duplicate]

For the two-level quantum system, we have the bloch sphere representation, and for a rotation we have the exponential operator: $$\text{exp}(\frac{-i \sigma \cdot \hat{n} \phi}{2})$$ where $\sigma = (\...
0
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1answer
362 views

Spin-orbit coupling, magnetic field

The Hamiltonian of spin-orbit coupling in an external magnetic field $\vec{B} = B\vec{e}_z$ is given by $$ H = \beta L\cdot S+\frac{\mu_b}{\hbar}(L_z+2S_z)B.$$ The ladder operators are $$ L_\pm = L_x \...
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1answer
134 views

Representation of the Poincare algebra on the space of smooth functions

The following representation describes how a field $\varphi$ transforms under the Poincaré group $\mathcal{P}$. $$\mathsf{S} : \left\lbrace \begin{aligned} \mathcal{P} \times C^{\infty}(\mathcal{M})...
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2answers
812 views

Matrix representation of rotation operators in QM

The usual matrix form of an operator $X$ is given by matrix components $$\langle a''| X | a' \rangle $$ where $|a' \rangle$ forms a basis for the ket space. In the case where we define matrix of ...
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1answer
187 views

How is simple matrix representation related to quantum probabilities?

I went again through some of my undergraduate books of quantum mechanics to get a new look at it as a futur PhD (not in QM though). I got answers for some old questions that bugged me at the time but ...
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3answers
310 views

truncation before matrix exponential: how to do it right?

I'm trying to compute (numerically) the matrices of some simple quantum optical operations, which in principle are unitary. However, in my case they are unitary in an infinite-dimensional space, so I ...
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0answers
266 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
104 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
883 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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0answers
364 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
77 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
812 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, ...