Questions tagged [matrix-elements]
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92
questions
0
votes
1answer
43 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 ...
0
votes
1answer
44 views
Understanding completeness relation and writing Hamiltonian in matrix form
A three level system hamiltonian I found where it is written as:
$$\frac{H}{\hbar}= \Omega_1|e\rangle \langle g_1| + \Omega_1^* |g_1\rangle \langle e| + \Omega_2|e\rangle \langle g_2|+ \Omega_2^* |...
0
votes
0answers
15 views
How to incorporate refractive index in transfer matrix method
I need to determine the TE and TM reflections at the interface between a uniaxial crystal and air, using a matrix-based method, such as the TMM.
I can do so using the Fresnel equation with ease, but ...
1
vote
1answer
86 views
Heisenberg Hamiltonian 2-Spin Terms in Matrix Representation
I am stuck on the interpretation/derivation of the 2-spin terms of the quantum Heisenberg model Hamiltonian.
In this model, our electrons, with spin up or down, are confined to sites on a lattice. ...
0
votes
1answer
28 views
Simplified computation of matrices for normal modes?
In normal modes, we often refer the total potential energy of the system to be:
$$V = q^T B q$$
where $V$ is the total potential energy, $q$ is the coordinates of the system and $B$ is just some ...
1
vote
0answers
31 views
For degenerate perturbation theory, how do we interpret the eigenvectors and eigenvalues of $\hat V$?
For the eigenvectors that are unmixed by the matrix $\hat V$, the eigenvalues are the energy corrections of this eigenbasis.
However, the eigenbasis tends to always be (as far as I'm aware) a linear ...
1
vote
1answer
108 views
Rotation matrix - levi-civita symbol
I'm trying to solve the following problem:
Given a rotation matrix $R_{ij}$, show that
$$n_k=\frac{-R_{ij}\epsilon_{ijk}}{\sqrt{(3-tr(R))(1+tr(R))}}$$
and that $$\sin(\phi)=-\frac{\epsilon_{ijk}...
1
vote
1answer
70 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
46 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
229 views
How to write an operator in matrix form?
Say I have the following operator:
$$\hat { L } =\hbar { \sum_{ \sigma ,l,p } { l } \int_{ 0 }^{ \infty }\!{ \mathrm{d}{ k }_{ 0 }\,\hat { { { a }}}_{ \sigma ,l,p }^{ \dagger } } } \left({ k }_{...
0
votes
1answer
31 views
How to construct a 2-partite matrix
Let's assume we have an internal hamiltonian $H_0 = \mid 1\rangle \langle 1\mid$.
Now let's assume we have two systems with identical Hamiltonians $H^1_0$,$H^2_0$ and I want to compute the joint ...
0
votes
1answer
80 views
Constructing a Hamiltonian for $N$-qubits
Let us assume we have a qubit with an internal Hamiltonian $H_0 = \sum_i \varepsilon_i |i\rangle\langle i|$.
Now let's assume we have 2 such qubits. How would their joint Hamiltonian look like?
I ...
0
votes
2answers
108 views
Rank of a density matix
I was just trying to understand the meaning of rank of a density matrix. I came across the following post, which says that the rank of density matrix is the number of non-zero eigenvalues. And for a ...
11
votes
2answers
312 views
Matrix elements of the free particle Hamiltonian
The Hamiltonian of a free particle is $\hat H = \frac{\hat p^2}{2m}$, in position representation
$$ \hat H = -\frac{\hbar^2}{2m} \Delta \;. $$
Now consider two wave functions $\psi_1(x)$ and $\psi_2(x)...
0
votes
1answer
208 views
Operators in Dirac notation and matrix representation (intuition)
So, I'm taking a QM 1 course, and we have reached a point where we used Dirac notation to solve two-level systems more efficiently, but our professor never really bothered to explain it further (he ...
0
votes
1answer
25 views
Relation of vector and scalar matrix elements for hadronic transitions
Consider a matrix element
$$
F_{\mu}(p_{h'}, p_{h}, \dots) = \langle h'(p_{h'})|\bar{q}_{i}\gamma_{\mu}q_{j}|p_{h}\rangle,
$$
describing transition of some initial hadron $h$ that contains a quark $...
2
votes
1answer
191 views
Explicit construction of $(\frac{1}{2}, \frac{1}{2})$ representation of Lorentz group
For the vector representation of the Lorentz group (actually the algebra), the $J^1$ generator is
$$J_1 = i \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 &0 \\ 0 & 0 & 0 ...
3
votes
1answer
212 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)
$$\quad V _ {...
1
vote
1answer
170 views
Tensor Product Explanation
I'm currently doing a research project involving 3 particle spins and have developed a simple function for the Hamiltonian:
I understand how to code my work but the physics behind it is unfamiliar ...
2
votes
2answers
323 views
How to find the coordinate representation of the kinetic operator?
From my professor's notes on statistical mechanics. $\left|\bf{k}\right\rangle$ is eigenstate of the hamiltonian of the free particle with periodic boundary conditions:
$$
\left\langle{\bf r}|{\bf k}\...
2
votes
2answers
228 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
133 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
648 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
159 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
33 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
81 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
341 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
302 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
416 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
286 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
209 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
80 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
802 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:
$$|\psi(t)\rangle=\left(\begin{matrix}
\psi_0(t)\\
\psi_1(t)\\
\psi_2(t)\\
\vdots
\end{matrix}\...
1
vote
0answers
29 views
Stiffness matrix issue
Good day All,
while trying to solve this exercice
I was puzzeld by the solution approach
indeed, they use the symmetry of the structure, they have made a cut on the hinge where the force F is applied ...
0
votes
2answers
502 views
Sign issue with the matrix of rotated elements (stiffness matrix)
Good day All
I have a doubt regarding the derivation of the following matrix
according to my basic understanding we want to go from the basis u1, v1, u2, v2, to the basis u'1, v'1,u'2 ,v'2, and for ...
-1
votes
1answer
108 views
What is norm of matrix element in Fermi Golden Rule
Fermi Golden Rule says:
$\Gamma \propto |M_{ij}|^2$
I know how to get $M_{ij}$, but how do I proceed? How do I take a norm of Hermitian matrix? There is no clear (to me) definition in the internet ...
0
votes
1answer
178 views
Hermitian Matrix
What is the geometrical significance of the Hermitian matrix? Actually what does the conjugate of the transpose represents. As a determinant if 3 by 3 expresses the volume.
1
vote
0answers
193 views
Finding a Lens System Matrix for a Planar Concave Thick Lens
How can I go about determining the lens system matrix for a planar concave thick lens? I analyzed that a light ray first enters the lens and refracts, then it goes through the lens, and then it ...
1
vote
1answer
98 views
Dimensions/Units and Matrix Inverses
I was going through a homework problem which is essentially a math review in a kinematics frame. This group of problems start as follows,
Given $a_x$, constant acceleration, and initial conditions, ...
2
votes
2answers
420 views
Which phase shift gate form is correct?
I am trying to figure out the matrix for a gate I'm going to be implementing - a mirror, i.e., a $180^{\circ}$ phase shift. Quantiki gives
$$R(\theta)=\begin{bmatrix}1&0\\0&e^{2\pi i\theta}\...
0
votes
1answer
1k 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
382 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 $$ L_\pm = L_x \...
0
votes
1answer
137 views
Representation of the Poincare algebra on the space of smooth functions
The following representation describes how a field $\varphi$ transforms under the Poincaré group $\mathcal{P}$.
$$\mathsf{S} : \left\lbrace \begin{aligned}
\mathcal{P} \times C^{\infty}(\mathcal{M})...
1
vote
2answers
932 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
197 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
331 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
301 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 ...
3
votes
1answer
118 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) | \...
