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Questions tagged [linear-algebra]

To be used for linear algebra, and closely related disciplines such as tensor algebras and maybe clifford algebras.

96 questions with no upvoted or accepted answers
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21
votes
0answers
586 views

Linear response theory for Gross Pitaevskii equation

I am trying to linearize the following GP eq: \begin{equation} i\partial_{t}\psi(r,t)=\left[-\frac{\nabla^{2}}{2m}+g\left|\psi(r,t)\right|^{2}+V_{d}(r)\right]\psi(r,t) \end{equation} The ansatz for ...
9
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0answers
103 views

Does the trace distance specify a unique state

In quantum information, we frequently use the trace distance (see definition) to look at how similar two states are. If I had a known complete set of states $\{\rho_i\}$ and some unknown state $\...
5
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0answers
202 views

Is there a proof that the number of eigenstates is countable for a bound system?

When you solve Schrödinger equation for a free particle with no boundary conditions your eigen states are indexed by quantum number $k \in \mathbb R $ and $\mathbb R$ isn't countable but if you add a ...
4
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0answers
68 views

Bounds on the effect of strong coupling

I am interested in bounding the effects of system-environment interaction. Suppose I have an initial state $\rho \in \mathcal{H}_S \otimes \mathcal{H}_E$ where the system and environment might be ...
4
votes
1answer
495 views

Fermionic statictics in $SU(2)$ slave-boson representation

One of the $SU(2)$ slave-boson decompositions has been introduced by X.-G. Wen and P. A. Lee in PRL, 76, 503 (1996). (A generic recipe for constructing the SU(2) slave-particle framework has been ...
3
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2answers
58 views

Differentiation of the determinant $g$

Let $g$ be the determinant of the metric tensor. I want to derive the following equation $g_{,\nu}=gg^{\lambda \mu}g_{\lambda \mu,\nu}$. It is said that $gg^{\lambda \mu}$ is a cofactor, but I can't ...
3
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2answers
103 views

Recovering symmetry in coupled oscillators

Consider a pair of LC oscillators, one with capacitance $C_1$ and inductance $L_1$ and the other with capacitance $C_2$ and inductance $L_2$. Suppose they're connected through a capacitor $C_g$. We ...
3
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0answers
72 views

Why are neutrino flavour eigenstates expressed in terms of the elements of the complex conjugate of the PMNS matrix?

If we have $$ \begin{pmatrix} \nu_e\\ \nu_{\mu} \\ \nu_{\tau} \end{pmatrix} =\begin{pmatrix} U_{e1} & U_{e2} &...
3
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0answers
222 views

Why is it that a coupled mass-spring system will always produce a diagonalizable matrix?

If you take a system like the one in the image, and you do the $y=x'$ trick to turn it into a first-order system of equations ($x_{1}$ or $x_{2}$ being the displacement of the mass $m_{1}$ or $m_{2}$ ...
3
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0answers
176 views

Two-Band k.p Model is not Hermitian for imaginary wavevectors

In E. O. Kane's original work on Zener Tunneling, he uses a two-band $k\cdot p$ model for the semiconductor bandstructure: $$H=\begin{pmatrix}E_g+\frac{\hbar^2k^2}{2m_0}&(\hbar/m)kp\\(\hbar/m)kp&...
3
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0answers
179 views

Fock Subspaces and Weight Vectors

This is my first time taking a physics course (I'm a mathematics major), so I'm encountering a lot of new things, which I'm kind of expected to know. In particular, how to work with Bosons. I've got ...
2
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0answers
41 views

Are the diagonal entries of the S-matrix in 1D equal?

Looking at the S-matrix of some potential barrier in one dimension one can show that is has the form $$ S = \begin{bmatrix} u & v \\ v & w\end{bmatrix} $$ with $u, v, w \in \mathbb{C}$ ...
2
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1answer
83 views

Cross product of vectors

I am unable to comprehend the following lines given in page 657 of Shankar's Principles of Quantum mechanics: One tricky point: The cross product is defined to be orthogonal to the vectors in the ...
2
votes
0answers
315 views

Charge conjugation in chiral representation

I'm reading Maggiore's book and I got to the part of charge conjugation symmetry for Dirac spinor. I get that the definition of charge conjugation is representation-dependent, however I couldn't find ...
2
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0answers
144 views

Transformation of a covariant tensor and general interpretation

If we consider a coordinate transformation defined by the rule $$x^{\mu}\rightarrow x'^{\mu}=x'^{\mu}(x)$$ If we consider the old coordinates as functions of the new ones, then the coordinate ...
2
votes
2answers
467 views

Determine the point at which moment vector is zero on a 3D body

I have information about total force and moment on a body for three points, whose coordinates I know. From this information I would like to determine the point at which moment would be zero. ...
2
votes
1answer
45 views

Proton spin independent fine structure “Hamiltonian” $W_f$

To find the perturbation correction (fine structure) in the case of a degenerate energy $E_n^0$, we can diagonalize the operator $W_f^n$, the restriction of $W_f$ to the eigen-space associated to $E_n^...
2
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0answers
213 views

commutation relations for operators in projected subspaces

I am looking for a consistent re-definition of commutators for certain operators when I work in a projected subspace. Basically, I have a spin defined in terms of 4 Majorana operators $b_{x}$, $b_{y}$,...
2
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0answers
67 views

Dropping vertices of an overdetermined statics system graph

In statics, the problem of determining the tensions of K cables that connect a structure made of N points and keep all points in static equilibrium implies $ND$ systems of equations, where $D$ is the ...
2
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0answers
137 views

How coordinate system shifting is related to similarity transformations?

I know that coordinate system shifting can be represented using matrices. But how exactly are similarity transformations related to coordinate shifts ?
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0answers
22 views

Finding an equivalent shape for a given mass and 3 mass moments of inertia

So I apologize if this is just impossible, but I was wondering if there was a way to find say, the dimensions of a box of a given density that would have the same mass and moments of inertia of ...
1
vote
1answer
30 views

Proving that motion of an $n$ dimensional oscillator can be written as a linear combination of “sine waves”

Here is a related question which might provide some context: LINK. Let's consider an oscillator with equation of motion in $n$ dimensions: $$ \frac{d^2}{dt^2} \vec{x} = K \vec{x}. $$ Given that $\...
1
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0answers
60 views

Is the basis of the occupation numbers of bosonic system orthonormal?

I am interested in the calculation of a correlation function in the Fock space of a system of $N$ bosons. As a trace it would be convenient for me to sum over the elements of the occupation numbers ...
1
vote
1answer
242 views

Finding basis of Schmidt decomposition

How do you find the basis for the Schmidt decomposition when given a state of multiple qubits? For example, if you have the systems $A,B$ and $C$, how do they correspond to eigenvectors of a density ...
1
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0answers
42 views

Definition of Alternating $(k,0)$- and $(k,l)$-tensors

I know that one can define the alternating subspace of $(0,l)$-tensors in a straightforward way. These are the renowned $l$-forms. However, I have been searching in the literature for a definition of ...
1
vote
1answer
49 views

Multilateration of Sound in 3D Space

TL:DR - How can you find the 3D coordinates of a emitter than transmits an impulse signal? STORY: I'm working on something to improve my bird-watching. I've got a camera that can take pictures of ...
1
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0answers
70 views

Why is hermitivity not preserved when Fourier transforming a matrix?

Consider a somewhat big Hamiltonian matrix $H$. I wanted to get its momentum representation, so I took the discrete Fourier transform or more specifically applied a FFT algorithm on it using Python. ...
1
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0answers
272 views

Time Reversal Operator in Sakurai's Book

In Sakurai's Modern Quantum Mechanics, Eq(4.16) says that $$\begin{align} |\alpha\rangle &= \sum_{a^\prime}|a^{\prime}\rangle\langle a^{\prime}|\alpha\rangle \overset{K}{\rightarrow} |\tilde{\...
1
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0answers
130 views

What is the trace of the (spatial) projection tensor?

In the discussion on gravitational waves in Moore's A General Relativity Workbook, Box 33.4 on page 391 introduces the projection operator $$P^j_m\equiv\delta^j_m - n^jn_m$$ where $\vec n$ is a unit ...
1
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0answers
84 views

Doubt in the transformation in R Shankar's Quantum Mechanics

On page number $19$ of the book Principles of Quantum Mechanics by R. Sankar, The author describes a transformation, essentially a rotation of the axes by an angle of $\frac{\pi}{2}$ counterclockwise, ...
1
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0answers
224 views

Determining a real qubit Choi matrix given two states

I've been curious recently about considering quantum channels whose Choi matrices are strictly real in the computational basis. Given the Choi matrix of a quantum channel $$C=\sum_{i,j=1}^d \mathcal E(...
1
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0answers
76 views

Has the asymptotic theory of eigenvalues of infinite matrixes already been applied to vibrations analysis?

My question is reffering to the masses/springs model of a material, like the one presented in this article http://www.laserpablo.com/baseball/Kagan/UnderstandingCOR-v2.pdf. If one treates a long ...
1
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0answers
535 views

Inverse of a Hermitian Operator

The momentum operator $p = i\frac{\mathrm d}{\mathrm dx}$ (with $\hbar = 1$) is hermitian. Hence its imaginary exponential i.e. $U = e^{ip}$ must be unitary. $U$ being a unitary operator, must have a ...
1
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0answers
367 views

Diagonalizing a fermionic Hamiltonian

My question somewhat builds off of this answer. For a fermionic Hamiltonian and diagonilzaition of the form $$H = \sum_{i,j} G_{ij}a_i^\dagger a_j = A^\dagger G A = A^\dagger U^\dagger D U A = F^\...
1
vote
1answer
62 views

Apparent analogies between statements from linear algebra and covariant tensor calculus

When using covariant tensors in relativity or particle physics, there are some statements that seem like analogues of statements known from linear algebra. For example, if we have a symmetric real-...
1
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0answers
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}\...
1
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0answers
64 views

Physical significance of $s$-sparse and efficiently row computable Hermitian matrix

In section II of Quantum algorithm for solving linear systems of equations by Harrow et al, the authors defined the matrix $A$ as $s$-sparse and efficiently row computable. $s$-sparse means having at ...
1
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0answers
152 views

Proof that a Hermitian Matrix is not defective?

I am taking an introductory course into Quantum Mechanics. To me to seems pretty simple to prove most properties of Hermitian operators. However, I am stuck at an edge case, proving that if an ...
1
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0answers
54 views

When trying to see what symmetries an operator generates, how do you “decide” what coordinate to apply it to?

Suppose I have $\hat{O}_{1}=-i\hbar\partial_{x}$ then \begin{eqnarray} e^{-i\gamma\hat{O}_{1}/\hbar}x\,e^{i\gamma\hat{O}_{1}/\hbar}=x+\gamma \end{eqnarray} and \begin{eqnarray} e^{-i\gamma\hat{O}_{...
1
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0answers
149 views

What is the relationship between the formal definition of a tensor and the frequently discussed notion of a “higher order matrix”?

I've been doing some self study on the principles of tensors & manifolds in preparation for a first course in general relativity. I tend to learn better when presented with the full mathematical ...
1
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0answers
538 views

Learning how to use Levi-Civita symbol

I've recently started my second course in Quantum Theory and am now often required to prove more complex commutation relations. I'm aware that the Levi-Civita symbol often makes this sort of thing a ...
1
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0answers
2k views

Change of Basis For Pauli Matrix From Z Diagonal to X Diagonal Basis

I want to find a matrix such that it takes a spin z ket in the z basis, $$ \lvert S_z + \rangle_z $$ and operates on it, giving me a spin z ket in the x basis, $$ U \lvert S_z + \rangle_z = \...
1
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0answers
59 views

When is the product of a hermitian unitary and another unitary hermitian?

I have a Hermitian unitary $\hat{H}$ and I want to know, if $\hat{U}$ is some other unitary, when is $\hat{H}\hat{U}$ a Hermitian unitary? Specifically, what are the conditions on $\hat{U}$? I know ...
1
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0answers
63 views

Random quantum systems with asymmetric Lifshitz tails?

For a quantum mechanical system with a periodic Hamiltonian (Schrödinger operator) $H$, let $N(E)$ be its integrated density of states, i.e. the fraction of eigenvalues in the spectrum $\sigma(H)$ ...
0
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0answers
33 views

Doubt about the “kernel of Einstein's equations”

From a coordinate-free point of view, we can rewrite the Einstein Field Equations $R^{\mu}\hspace{0.5mm}_{\nu} - \frac{1}{2}R \delta^{\mu}\hspace{0.5mm}_{\nu}=:G^{\mu}\hspace{0.5mm}_{\nu} = 8\pi T^{\...
0
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0answers
24 views

Hermitian phase operator and quantum harmonic oscillator

I need to apply a hermitian phase operator $\dfrac{1}{\sqrt{1+\hat{a}^\dagger\hat{a}}}\hat{a}$ to the nth harmonic oscillator state, but I have no idea how to interpret the square root of a ...
0
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0answers
33 views

How to find the hermitian adjoint and inverse of an operator?

Suppose I have a translation operator defined as: $$ \hat{T_a}\Psi(x)=\Psi(x+a) \, . $$ Now, how do I find the hermitian adjoint operator as well as its inverse?
0
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0answers
10 views

Taking out % contribution of a zero order state from an eigenvector - dipole calculation

I am doing an analysis of a theoretical spectroscopy calculation. I take an eigenvector (nx1) and dot it with many zero-order dipole vectors (nx3) to get the dipole contribution to my new eigenstate. ...
0
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0answers
18 views

Infinite Coupled Masses, symmetry, and the simultaneous diagonal theorem for infinite dimensional vector spaces

In The Physics of Waves by Georgi, in Chapter 4, we show that, in a coupled system of masses connected by springs, a transformation that preserves some symmetry $S$ commutes with $K^{-1}M$. From my ...
0
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0answers
32 views

Proof of skewsymmetry of electromagntic function in Minkowski spacetime

I have been studying special relativity from the Gregory Naber's book: "The geometry of Minkowski spacetime" and I found a very strange proof. In Section 2.1, just before of equation 2.1.2. the book ...