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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.

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Permanent operation's result

N-body fermionic systems are constructed by Slater determinant, and it is equal to Vandermonde polynomial. Are there any special polynomial for the permanent which is used to construct N-body ...
Abdülcanbaz's user avatar
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Decomposition of $(x \pm i y) \, Y_{l m}$ and $z \, Y_{l m}$ on spherical harmonics

Using the various algebraic properties of the associated Legendre polynomials $P_l^m(u)$ and of the spherical harmonics $Y_{l m}(\theta, \varphi)$, I was able to decompose the following expressions, ...
Cham's user avatar
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Algebra in (yet another) Lorentz transformation derivation

I would appreciate help with some of the physics/algebra steps in this derivation as presented in this MIT OCW lecture https://ocw.mit.edu/courses/8-20-introduction-to-special-relativity-january-iap-...
No infinity's user avatar
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Calculating Eigenkets of Perturbed Matrix for Second-Order Correction

Q: Find the eigenvalues of the 3x3 symmetric matrix $H$ using perturbation theory where all of the elements on the diagonal of $H$ are an order greater than the elements not on the diagonal. We can ...
PineappleThursday's user avatar
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Problem with logarithm of tensor product of matrices

In the book of From Classical to Quantum Shannon Theory, in exercise 11.8.1, there is a property of logarithm of a tensor products of two matrices, defined as follows: $$\log ( A \otimes B) = \log(A) \...
JasonWS 's user avatar
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Rotation of Pauli Vectors with $SU(2)$ reproduces the $SO(3)$ matrix. but do all $SU(2)$ matrices reproduces $SO(3)$?

So we can write the $SU(2)$ matrices multiplication as this. $$\begin{bmatrix}\alpha&\beta\\-\beta^*&\alpha^*\end{bmatrix}\begin{bmatrix}z&x-iy\\x+iy&-z\end{bmatrix}\begin{bmatrix}\...
abx_pradB's user avatar
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Why does applying Ladder operators change the eigenfunction?

When applying a ladder operator to a spherical harmonic function, it spits out the function with a lower or higher magnetic quantum number. My question is how does this abide by the classical ...
ajox3412's user avatar
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Regarding Energy Eigenstate and Position Eigenstate

I am solving problem 14.4. (a) of Schwartz's Quantum Field Theory and the Standard Model. It is related to the simple harmonic oscillator in quantum mechanics. It asks the eigenstate of the position ...
Jaeok Yi's user avatar
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Operators algebra for quantum mechanics [closed]

I am taking my first quantum mechanics course and I am a bit lost in operators algebra. These are the main questions I have: Why can we write this kind of equations? $$ Ô \psi = o\psi $$ What I mean ...
Tymothée Waldner's user avatar
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Rotating a system

brekely physics book chapter 2 page 30 , a question about rotating a system by $ \frac{\pi}{2} $ around the z axis clockwise direction and writing vectors according to the new axis after rotation ...
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Intuition behind dampening as a rotation in the complex plane

I'm currently building my intuition behind the meaning of the variables within the general second-order damped harmonic oscillator frequently taught in engineering: $$ \ddot{x} + 2\zeta\omega_n\dot{x} ...
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Question about the identity operator and the bosonic ladder operators

Consider a self-adjoint operator $B$, such that for each mode $a_1,...,a_n$ [of a quantum bosonic system with Hilbert space $\cal H$ given by the corresponding Fock space] we have $B a_i B^\dagger = ...
Noobgrammer's user avatar
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Understanding equation for eigenvalues of a Hamiltonian

I'm reading the paper Hamiltonian Truncation Study of Supersymmetric Quantum Mechanics. I'm not understanding a claim they make about the eigenvalues of a certain Hamiltonian. In particular, how eqn 3....
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Scattering matrix vs. unitary transformations

In quantum optics, the input/output bosonic modes at a beam splitter transform according to the scattering matrix $$ \begin{pmatrix} a_1 \\ a_2 \\ \end{pmatrix} = \dfrac{1}{\sqrt{...
m137's user avatar
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What is the difference between a Lindbladian and a Liouvillian in open quantum systems?

As far as I know, when we try to write the Lindbladian equation in a generalized nice operator basis we get Liouvillian. Is this correct? What are the differences between them?
Rishwi Thimmaraju's user avatar
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Pauli matrix exponentials [closed]

Just a short query to confirm my understanding. Given the Pauli-X operator $\hat{X}$ and it's eigenstates $|+\rangle:=\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and $|-\rangle:=\frac{1}{\sqrt{2}}(|0\...
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Exponential of the metric tensor

Exponential of an arbitrary matrix can be written as $$e^A = \displaystyle\sum_{n=0}^\infty \dfrac{A^n}{n!}$$ In Einstein notation, how this expression will look like? In Einstein notation, what ...
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Determinant of Rank-2 Tensor using Levi-Civita notation

In my Professor's notes on Special Relativity, the determinant of a rank-two tensor $[T]$ (a $4\times 4$ matrix, basically) is given using the Levi Civita Symbol as: $$T=-\epsilon_{\mu\nu\rho\lambda}T^...
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Landau levels in symmetric gauge, what is the constraint on the quantum numbers?

After solving the Schrödinger equation for the charged particle in a constant and homogeneous magnetic field, using the symmetric gauge $\vec{A} = \frac{B}{2} (-y, x, 0)$, we could find the Landau ...
Cham's user avatar
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Why do QM books point out that $S^2$ commutes with $S_x$, $S_y$, and $S_z$?

The spin angular momentum magnitude squared operator: $$S^2=S_x^2+S_y^2+S_z^2=\frac{3\hbar^2}{4} \begin{pmatrix}1&0\\0&1\end{pmatrix}$$ Obviously $S^2$ commutes with everything, so why do QM ...
hbar's user avatar
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What can we say about the eigendecomposition of quantum channels?

It is known that quantum channels, being CPTP maps, map density operators to density operators. And thus, they can be seen as superoperators. Similar to operators, where eigenstates and eigenvalues ...
ironmanaudi's user avatar
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Einstein Summation Convention Confusion

My textbook: The second bit confuses me. I asked a question on this site yesterday (Moment of Inertia tensor confusion) which involved the moment of inertia tensor and the term $$r_{i}r_{j}$$ The ...
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What does the identity operator mean in Quantum Mechanics? [duplicate]

I'm new to quantum mechanics, and I am beginning to study Dirac notation, but I do not understand the significance or meaning of the following equation: $$\sum_n\left|e_n\right\rangle\left\langle e_n\...
cookiecainsy's user avatar
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Finding the Cartesian position of the lunar ascending node

What I'm trying to derive The Cartesian position of the lunar ascending node relative to the true equator and equinox of date reference frame. My issue is I'm getting a bit tripped up with reference ...
Hunter's user avatar
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Can we project into Pauli size sectors efficiently?

Suppose that we have a system of $L$ qubits and some Hermitian operator $\mathcal{O}$ acting on the system. We can expand $\mathcal{O}$ in Pauli strings $P_{n}$: $$\mathcal{O} = \sum_{n=0}^{4^{L}}c_{n}...
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$SU(3)$ adjoint representation and irreducibility

Consider the Gell-Mann matrices, with $$ \lambda_3 = \operatorname{diag}(1, -1,0), \quad \lambda_8 = \frac{1}{\sqrt{3}}\operatorname{diag}(1, 1, -2), \quad, ... \ , $$ they span the Lie algebra $\...
user31415926's user avatar
3 votes
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Numerical procedure for finding steady eigenstate for a non-Hermitian effective model

I am working with an effective model describing an exciton-polariton system. I need to solve the Schrödinger equation (eigenvalue problem). To describe the presence of losses the diagonal terms are ...
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How can a vector field in $E^3$ be represented by a linear combination of only 2 basis vectors?

In Chapter I.7 of "Einstein Gravity in a Nutshell", Zee introduces the concept of covariant derivatives. I am confused by the first line in this section (see below) as it appears that we can ...
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Identifying avoided crossings

Consider the following spectrum This spectrum represents the evolution of the energy levels of a certain molecule in its ro-vibrational ground state as a function of the magnetic field. Such graphs ...
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Is this matrix equal to the identity?

In the context of this larger problem, I have a question about the meaning of the matrix $W$ here. The time reversal operator $\Theta$ can be written $$\Theta = UK$$ and also as $$\Theta = U'K'$$ ...
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A confusion about the inner product keeps disturbing me [duplicate]

I get very confused about the concept of inner product. When an inner product is defined in a vector space, $\mathbb{V}$, don't we define it as an operation between two vectors from $\mathbb{V}$ ...
Solidification's user avatar
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Levi-Civita symbol and Einstein notation algebra

I am reading "An Introduction to the Theory of Piezoelectricity" 2nd edition by Jiashi Yang I am trying to understand a derivation for an equation that uses multiple Levi-Civita symbols. The ...
MikeM's user avatar
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2 answers
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Conflicting definitions of vector conjugate in QM

Let $e$ be a finitely matrix representable operator. In physics, specially in quantum mechanics (QM), it is customary to define the conjugate operator $e^{\dagger}$, as the adjoint or the Hermitian ...
physicsrev's user avatar
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What are examples of qubit channel with non-singular Choi state?

Are there any obvious (ideally physically reasonable) examples of qubit channels that give rise to non-singular Choi states? I've been exploring the Choi state of a variety of qubit channels but find ...
Theoreticalhelp's user avatar
2 votes
2 answers
544 views

Change of basis in bra-ket notation [duplicate]

In the post Change of Basis in quantum mechanics using Bra-Ket notation , the accepted answer explores the relationship between an arbitrary operator $\hat{x}$ and another named $\hat{u}$, such that $\...
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Integral expansion of a ket [closed]

I'm taking my first quantum physics course and I've been seeing a lot of stuff like $|\psi\rangle = \int d\phi |\phi\rangle \langle \phi | \psi \rangle$ and $|\psi\rangle = \int d^3 r |r\rangle \...
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When an eigenvalue is degenerate, are there always other operators which distinguishes the degenerate states? [duplicate]

Familiarity with QM tells us that when an eigenvalue of an operator $\hat{A}$ is degenerate i.e. more than one eigenfunction of $\hat{A}$ has the same eigenvalue, there is usually another operator (or ...
Solidification's user avatar
2 votes
2 answers
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What is the equivalent of Slater determinants for superconductors

Suggestions to better formulate my question are also highly appreciated. To better explain my question let me start with a very familiar case: Suppose we have a two-band insulator described by $$H = \...
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3 answers
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Is Jones calculus a "calculus" in the proper mathematical sense? [closed]

I've come to understand "calculus" as the mathematical study of continuous changes in a mathematical function or physical system. Differential and integral calculus are broad examples of ...
BenjaminDSmith's user avatar
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1 answer
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Is the Hamiltonian for the transverse field Ising model Hermitian?

I'm watching these lectures in Condensed Matter Physics. At Lec. 13, the lecturer introduces the transverse field Ising model with the Hamiltonian $$H = - J \sum_i \sigma_i^x \sigma_{i+1}^x - h \sum_i ...
Níckolas Alves's user avatar
1 vote
1 answer
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Adjoint and index notation in Weyl field context

In the answer to a question I previously asked, the following manipulation was done but I don’t understand it$.$ $$ (U_{jm}\psi_m)^\dagger=\psi_m^\dagger U_{mj}^\dagger $$ aside from the context from ...
JohnA.'s user avatar
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5 votes
4 answers
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Can you convert a state vector from a certain basis to any other basis?

I am in the beginning of my first Quantum Mechanics class, and I just learned about state vectors. From my understanding, the representation in a basis is related to the probability distribution of ...
toomanyfeet's user avatar
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1 answer
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Is there a difference between the spectral decomposition and the orthonormal decomposition of a matrix?

I was studying quantum information from Nielsen and Chuang's book and I got a little bit confused because sometimes they use the terms "spectral decomposition" and "orthonormal ...
Anis Younes's user avatar
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Implementation of Hamiltonian coupling to a bath

I want to study a system coupled to a bath, however I do not fully understand how to implement/think of the Hamiltonian. For simplicity say the bath is given by a spin chain (PBC), e.g. Ising-like $$...
qising's user avatar
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1 answer
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Non-Abelian anomaly: why does non-Hermitian operator have complete basis of eigenvectors?

In section 13.3 of his book [1], Nakahara computes the non-Abelian anomaly for a chiral Weyl fermion coupled to a gauge field by making use of an operator $$ \mathrm{i}\hat{D} = \mathrm{i}\gamma^\mu (\...
xzd209's user avatar
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6 votes
1 answer
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Inverse of anti-symmetric rank 4 tensors?

I am trying to find an inverse of a tensor of the form $$M_{\mu\nu\rho\sigma}$$ such that $M$ is anti-symmetric in the $(\mu, \nu)$ exchange and $(\rho, \sigma)$ exchange. The inverse should be such ...
Dr. user44690's user avatar
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The similarity transformation of Pauli matrices

I am looking for a transformation $S$ of the Pauli matrices ($X,Y,Z$) such that \begin{align*} S^{-1}XS=Y, S^{-1}YS=Z, S^{-1}ZS=X. \end{align*} Simply put, my question is a cyclic transformation of ...
Kitchen's user avatar
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2 answers
156 views

Relation between eigenvalue equation for an operator and for its square

Consider a time indipendent Schrodinger problem: $$\hat{H}\psi_E(p) = E \psi_E(p)$$ with suitable boundary conditions. We know that $\psi_E$ are the eigenfunctions of $\hat{H}$. If we now consider the ...
LolloBoldo's user avatar
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2 votes
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Unable to re-derive Inverse Transfer Matrix

I don't get the same result as the book when computing the inverse of this matrix. For context, I'm rederiving some equations from a book on acoustic waves in periodic structures, specifically a ...
korokame's user avatar
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Rigorous proof of Fluid Flux with Linear Algebra

In fluid mechancics the flux is the volume that passes through a surface in a unit time. To do the easiest case to then build up to the integral, I want to look at a square surface in a time-invariant,...
16π Cent's user avatar

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