Questions tagged [matrix-elements]
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0
votes
1answer
26 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 ...
1
vote
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 ...
1
vote
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 $$\sin(\phi)=-\frac{\epsilon_{ijk}...
1
vote
1answer
57 views
How can quantum operators be expressed as a matrices?
I have just started quantum mechanics with Shankar. In my understanding, quantum operators are linear operators in infinite-dimensional Hilbert spaces.
Shankar has repeatedly treated quantum ...
-3
votes
1answer
45 views
How an operator is converted to a function? [closed]
$$\langle m|F|n\rangle^*=\langle F(n)|m\rangle$$
How does the operator become a function of state $|n\rangle$?
0
votes
3answers
107 views
How to write an operator in matrix form?
Say I have the following operator:
$$\hat { L } =\hbar { \sum_{ \sigma ,l,p } { l } \int_{ 0 }^{ \infty }\!{ \mathrm{d}{ k }_{ 0 }\,\hat { { { a }}}_{ \sigma ,l,p }^{ \dagger } } } \left({ k }_{...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
11
votes
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)...
0
votes
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 ...
0
votes
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 $...
2
votes
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
$$J_1 = i \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 &0 \\ 0 & 0 & 0 ...
3
votes
1answer
163 views
Calculation of Matrix Element in “Old-Fashioned Perturbation Theory”
I would like to better under the manipulations/formalism applied in order to evaluate the following matrix element from Schwartz "Quantum Field Theory and the Standard Model" (Eq. 4.16)
$$\quad V _ {...
1
vote
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 ...
2
votes
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:
$$
\left\langle{\bf r}|{\bf k}\...
2
votes
2answers
197 views
Physical transformation associated with a Pseudo-Orthogonal matrix [closed]
An orthogonal matrix $O$, which belongs to an orthogonal group, is characterized as $O^TO=I$.
Let's take an example of a $2 \times 2$ orthogonal matrix,
$$O = \begin{bmatrix}
\phantom{-} \cos{\...
0
votes
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 ...
0
votes
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. ...
11
votes
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, ...
0
votes
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 ...
0
votes
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 [...
3
votes
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 ...
1
vote
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, ...
2
votes
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:
$$x = \sqrt{\frac{\hbar}{2 m \omega}} \left[
\begin{array}{ccccc}
0 & \sqrt{1} & 0 & 0 & \cdots \\
\sqrt{1} & ...
0
votes
1answer
316 views
Levi-Civita tensor
The Kronecker delta can be represented by a two dimensional matrix:
\begin{gather}
\delta_{ij}=\mathbb{I}=
\begin{bmatrix}
1&0&0\\
0&1&0\\
0&0&1\\
\end{bmatrix}.
\end{gather}
...
0
votes
1answer
243 views
Using Dirac notation to find matrix representation
I am currently reading Sakuria, and I cannot get my head around how one uses the completeness relation to derive the matrix representations of outer products. In the first chapter he states that an ...
0
votes
1answer
170 views
How to derive the Schrödinger Equation from Heisenberg's matrix mechanics and vice-versa?
How do you derive the Schrödinger equation (wave mechanics, time dependent state) from Heisenberg's Matrix Mechanics (matrix based, time dependent operators)
1
vote
1answer
69 views
Proof that elastance matrix is invertible
I was reading this lecture notes from MIT OCW on capacitance .
It says
$V_i =\sum_j P_{ij}Q_{j} $ where the constants $P_{ij}$ are determined
by the geometry of the conductors. This matrix can ...
0
votes
1answer
26 views
using symmetry in the siffness method
Good day All
while trying to solve this exercice
I tried to find a symmetry plan to make my computations easy
and according to my basic understanding
the symmetry must be in term of:
lenght
length
...
0
votes
1answer
634 views
What does the time evolution operator look like in Matrix representation?
In Matrix representation of quantum mechanics, using an energy eigenbasis, we have the state vector:
$$|\psi(t)\rangle=\left(\begin{matrix}
\psi_0(t)\\
\psi_1(t)\\
\psi_2(t)\\
\vdots
\end{matrix}\...
1
vote
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 ...
0
votes
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 ...
-1
votes
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 ...
0
votes
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.
1
vote
0answers
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 ...
1
vote
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, ...
2
votes
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
$$R(\theta)=\begin{bmatrix}1&0\\0&e^{2\pi i\theta}\...
0
votes
1answer
865 views
matrix of rotation for quantum states [duplicate]
For the two-level quantum system, we have the bloch sphere representation, and for a rotation we have the exponential operator: $$\text{exp}(\frac{-i \sigma \cdot \hat{n} \phi}{2})$$ where $\sigma = (\...
0
votes
1answer
362 views
Spin-orbit coupling, magnetic field
The Hamiltonian of spin-orbit coupling in an external magnetic field $\vec{B} = B\vec{e}_z$ is given by $$ H = \beta L\cdot S+\frac{\mu_b}{\hbar}(L_z+2S_z)B.$$ The ladder operators are $$ L_\pm = L_x \...
0
votes
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 $\mathcal{P}$.
$$\mathsf{S} : \left\lbrace \begin{aligned}
\mathcal{P} \times C^{\infty}(\mathcal{M})...
1
vote
2answers
812 views
Matrix representation of rotation operators in QM
The usual matrix form of an operator $X$ is given by matrix components $$\langle a''| X | a' \rangle $$ where $|a' \rangle$ forms a basis for the ket space. In the case where we define matrix of ...
2
votes
1answer
187 views
How is simple matrix representation related to quantum probabilities?
I went again through some of my undergraduate books of quantum mechanics to get a new look at it as a futur PhD (not in QM though). I got answers for some old questions that bugged me at the time but ...
6
votes
3answers
310 views
truncation before matrix exponential: how to do it right?
I'm trying to compute (numerically) the matrices of some simple quantum optical operations, which in principle are unitary.
However, in my case they are unitary in an infinite-dimensional space, so I ...
0
votes
0answers
266 views
Finding the eigenvalues and eigenvectors of this operator
The operator
$$\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),$$
is physically similar to the momentum operator in quantum mechanics. It has the ...
2
votes
1answer
104 views
Singularities of matrix element of composite local operator in QFT
Consider a state $|\psi\rangle$ in a quantum field theory and a local operator $\mathcal{O}(x)$. It's known that the $n$-point function
$\langle \psi | \mathcal{O}(x_1) \cdots \mathcal{O}(x_n) | \...
0
votes
1answer
883 views
Pauli matrices in spherical coordinates
In my work I currently have to work with the following partition function $Z\propto e^{\vec{h}.\vec{\sigma}}$ where $\vec{\sigma}$ are pauli matrices and $h$ is some vector. So far I've been using ...
2
votes
0answers
364 views
Calculation of $b \to s~ l^+ l^-$ penguin diagram
I'd like to calculate the matrix element amplitude for $b \to s~ l^+ l^-$ penguin diagram mediated by Z boson or the photon , like :
These calculations are made of course from many time ago, so if ...
-1
votes
1answer
77 views
Matrix mechanics [closed]
So, I am given a Hamiltonian of a system, represented in the $|e_{i}\rangle$ basis as:
$$H=\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix}$$
where, $|e_{1}\rangle = \begin{...
2
votes
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, ...
