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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Stinespring dilation using ancilla in mixed state?

The Stinespring dilation theorem states that any CPTP map $\Lambda$ on a system with Hilbert space $\mathcal{H}$ can be represented as $$\Lambda[\rho]=tr_\mathcal{A}(U^\dagger (\rho\otimes |\phi\...
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Integral of eigenkets (QM)

I'm reading Dirac's book about QM. I reached the point (in my edition on page 37) where he tells it is possible, given the eigenkets of an observable, express any other ket in function of them (as he ...
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Matrix elements of operators in position representation

In a lecture note, it is written $$ T_{ij} = \langle \phi_i| \hat{T} | \phi_j \rangle = \int d^3 \vec{r} \phi_i^*(\vec{r}) T(\vec{r}) \phi_j(\vec{r}) $$ How to obtain the second integral form from ...
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Troubles in Dirac's “Principles of quantum mechanics”

I'm reading the Dirac's book about QM and I am finding troubles understanding a proof of a theorem (in my edition at page 32), which says that "there are so many eigenkets of $\xi$ that any ket ...
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Theorem in Dirac's “Principles of quantum mechanics”

I'm pretty new to quantum mechanics and after reading the Susskind's book I dived into the Dirac's one. I've managed to understand until this theorem has been enunciated (in my edition at page 32): &...
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Cross product and determinants [migrated]

The determinant of a $3 \times 3$ matrix gives the volume of the parallepiped formed by 3 vectors. With the cross product, say to find the torque, we find the $3 \times 3$ determinant value of ...
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Relation between the boundedness and discreteness of conjugate operators

I have two very general questions about operators in quantum mechanics. Suppose $A$ and $B$ are self-adjoint operators associated to conjugate physical quantities (e.g. position/momentum), meaning ...
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1answer
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What will happen to Hamiltonian matrix, eigenvalues and eigenvector on a non-eigenbasis?

In a simple example, Most of the Hamiltonians are talked about on its eigenbasis or a basis that can be transformed from the eigenbasis. With this, the eigenvalues do not change even if on a different ...
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Diagonalization using LAPACK [migrated]

Say, we have a Hamiltonian which for simplicity does not mix particle hole sectors. It is just a simple Hamiltonian in real space as shown, $H=\sum_{ij,\sigma} A(i,j)(c_{i\sigma}^{\dagger}c_{j\sigma} +...
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Why is the multiplication of the metric and an inverse metric the Kronecker delta?

I am having a hard time understanding \begin{align*} \delta_{\beta}^{\alpha}=g^{\alpha\nu}g_{\beta\nu}\\ \end{align*} equality. I understand the situation where the indices are the same and the ...
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Why is Minkowski metric diagonal?

Why is the Minkowski metric a diagonal in a 4x4 matrix? What does the diagonal do?
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Can QM be used to model 2 state systems with more than 4 linearly independent observables?

Suppose I have a system (e.g., a particle) and I have different physical measurement apparatus which can act on it. Each of the measurement apparatus (observables) has 2 distinct labeled outcomes, ...
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If two operators say $D$ and $B$ commutes then why a non-degenerate eigenfunction of operator $D$ is also an eigenfunction of operator $B$?

I have following derivation which is not understandable for me and I am unable to understand it.Consider a eigenfunction-value equation $$D{\Psi}=d{\Psi} $$ Now $B$ operates on above equation and ...
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Why $\Theta^2 = 1$ suggests the $\Theta$-invariant space only consists of 1-dimension subspaces, for an antilinear operator $\Theta$?

Let $S$ be a subspace of the Hilbert space, and let $S$ be invariant under an $\textit{antilinear}$ operator $\Theta$, i.e. for every $\vert\psi\rangle$, $\Theta\vert\psi\rangle \in S$. I learned that ...
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Solving the matrix Schrödinger equation

One can solve the Schrödinger equation by diagonalizing the Hamiltonian $H$. Due to limited memory, we truncate $H$ up to $N$. Now I red here on slide 8, that increasing $N$ cant lead to higher ...
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What is the procedure of “flattening of Hamiltonian”?

Can someone explain what exactly is spectral flattening of a Hamiltonian? I hear about it in the course, but it was never explained and every article on the Internet also speaks about it as if it was ...
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Total derivative of the inner product of a state with itself?

So, in computing the derivative $\frac{d}{dt} \langle \psi| \psi \rangle = 0$ one can (to my knowledge) go about it by computing the integral in the position basis, $$\langle \psi | \psi \rangle = \...
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Dynamics of a System via Orthogonal Diagonalization

About a week ago, I was working with the time evolution of a density operator for a given initial state $\rho_0$ given a specific Hamiltonian $\mathbf{H}$. My fist approach was to find an operator $\...
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Confused over a line in L&L Quantum Mechanics (non-relativistic) involving linear algebra

So I was reading Quantum Mechanics: Non-relativistic Theory by Landau and Lifshitz when I came across this line (Edition 2, page 10): $$\hat{f} \Psi = \sum_n{a_n f_n \Psi_n},$$ where $\Psi = \sum_n ...
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Prove $\exp (-i \phi(\hat{n} \cdot \vec{\sigma}))=\cos \phi-i(\hat{n} \cdot \vec{\sigma}) \sin \phi$ using certain property

I'd like to prove $$e^{-i \phi(\hat{n} \cdot \vec{\sigma})}=\cos \phi-i(\hat{n} \cdot \vec{\sigma}) \sin \phi$$ using $$\sigma_{i} \sigma_{j}=\delta_{i j} I+i \varepsilon_{i j k} \sigma_{k},$$ where $\...
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With reference to orthonormal bases, can someone explain this please? [closed]

$e_i$ and $e_j$ are orthonormal bases. $a$ is a vector; $$a = \sum_{i=1}^N a_i e_i. $$ Question is, how the operation below equals $a_j$ \begin{align}\langle e_j|a\rangle&= \sum_{i=1}^N\langle e_j|...
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Defining the inverse of a tensor via the adjugate tensor

My professor definied the adjugate of a tensor $\mathbf{t}\in T^{1}_{1}(E)$ (E is just a vector space of dimension n) by defining its components as $adj(\mathbf{t})^{a}_{b}=\frac{1}{(n-1)!}\...
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Diagonalizing eigensystem to find normal modes of coupled oscillator [closed]

I've two equations of motion that arose in a coupled oscillators in a magnetic field $\rightarrow$ continuum problem in classical mechanics: \begin{eqnarray*} -\omega^2 X &=& - \omega_0^2 X (2 ...
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Definitions of Determinants and Permanents in QFT

I have been recently reading a QFT book called: "QFT For the Gifted Amateur". It states in footnote 4 on p. 40 that the determinants and permanents of matrices can be defined as follows: $$ \...
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Does the matrix representation of a logic gate change according to the basis states?

To expand upon my question, allow me to introduce a problem that I'm attempting: Suppose $\lvert+\rangle = \frac{1}{\sqrt{2}}(\lvert0\rangle + \lvert1\rangle)$ and $\lvert-\rangle = \frac{1}{\sqrt{2}}...
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Conjugate complex of linear operators in quantum mechanics

I'm pretty new to quantum mechanics (I would like to understand it broadly as an hobbyist). I'm trying to reading Principles of Quantum Mechanics by Dirac. I've found difficult to understand a ...
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What is the meaning, in tensor notation, of the dot-within-a-circle $\odot$? Usually shown as an exponent?

This $\odot$ notation is e.g. used in this Phys.SE posts: General relativity: Why don't these two differentials commute? Why isn't there a second baryon octet?
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Conceptual question on eigenvectors in quantum mechanics

Page number 1 on Quantum Many-Particle Systems by John W. Negele and Henri Orland says the following about quantum mechanical position eigenvector $|r\rangle$ & momentum eigenvector $|p\rangle$ in ...
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Can mixed order tensor operations be calculated using matrix algebra?

Is there any reference on whether expressions of mixed order tensors, for instance: $$a = v_iT^{i}_{.j}u^j$$ which evaluate to the same scalar (irrespective of the order of these terms above) ...
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In what situation would we combine tensors from different points in a tensor field?

In my lecture notes it emphasises that a tensor $C^{abc}_{def} = A^{abc}_{def} + B^{abc}_{def}$ is only also a tensor itself if $A^{abc}_{def}$ and $B^{abc}_{def}$ are evaluated at the same point in a ...
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Quantum Mechanics without Complex Numbers [duplicate]

I have been studying some Lie theory recently and I came across the idea of representing complex numbers using matrices, e.g. $$1= \begin{pmatrix} 1 & 0\\ 0 & 1\\ \end{pmatrix} , i= \begin{...
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Linear algebra as a gauge theory

Is linear algebra a gauge theory? Is the gauge transformation a change of basis? This was the explanation that I received: "Take the principal bundle to be the frame bundle $LM$ of your space $M$...
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What are the Lorentz Transformations between polar coordinates? Or can Lorentz Transformations be Non-Linear?

This question rises from the comments on @G Smith's answer's to this question https://physics.stackexchange.com/a/603032/113699 Precisely I was trying to understand the Lorentz Transformations between ...
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Examples of the physical significance and importance of matrix diagonalization and eigenvalues for first year undergraduates? [closed]

To a student of physics, who is only exposed to the techniques of mathematical physics and read classical mechanics at the undergraduate level, but not quantum mechanics yet, how can we explain the ...
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Normalizing eigenvectors [closed]

Over the course of this quantum class I'm taking I've run into issues with properly normalizing my eigenvectors. Here is my TA's explanation of this particular example is done. I am lost as to where ...
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Meaning of The Eigenvalues and Eigenvectors of a Quantum Operator

This is more a check to ensure I know the physical meaning of eigenvectors and eigenvalues in quantum mechanics, and to ask the general community if this is wrong: On some observable, represented by ...
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Constant curvature for maximally symmetric spaces

I'm working through Walds GR Textbook and while reading chapter 5 I stumbled upon the question Proving constant curvature. However, my question is how do we prove that $L$ is symmetric? It is ...
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Derivation of Cayley-Klein parameters in Goldstein

In his derivation for Cayley-Klein parameters, Goldstein introduces the matrix $$\mathbf{Q}=\begin{pmatrix}\alpha & \beta \\ \gamma & \delta \end{pmatrix}$$ and says that the following unitary ...
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Transforming the metric between coordinate systems

The following is taken from Ta Pei Cheng's Relativity Gravitation and Cosmology book, pg. 283. For a coordinate transformation of $x^a \rightarrow x'^{a}$, the metric tensor transforms as $$g_{ab}'=g_{...
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Could I represent a two dimensional reciprocal lattice by just one integer index?

By considering the following reciprocal lattice vector $$G_{n,m}=n\vec{b}_{1}+m\vec{b}_{2}$$ where if we consider that this is from a triangular lattice, the cartesian form of this vector will be $$G_{...
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What's the difference between a linear operator and a tensor?

In Quantum Mechanics we mostly use linear operators which often represent a physical observable, In relativity we mostly work with tensors to describe how things change. But both objects are (multi)-...
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Does the first-order energy correction in the degenerate case equals to the eigenvalues of the perturbation matrix?

According to Griffiths, the degenerate perturbation theory says that the first-order corrections to the energies are the eigenvalues of the perturbation matrix. Griffiths solves for the eigenvalues in ...
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Angular change with camera reference point/vector change

Note: All measurements are taken/derived from 2D coordinates based on the video camera. Arm is a shorthand for angle between the hip to the armpit to the elbow. For all intents and purposes, arm can ...
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Is the Pusey-Barrett-Rudolph (PBR) theorem only applicable in two dimensions?

In the Pusey-Barrett-Rudolph (PBR) paper, https://arxiv.org/abs/1111.3328 it is shown that two non-colinear pure states $|\psi_0\rangle$ and $|\psi_1\rangle$ cannot have overlapping ontic supports (...
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Dirac Notation and Coordinate transformation of a function

Lets say we have a 1 dimensional system with coordinate $x$ and the associated operator $\hat x$ with eigenstates $|x\rangle$. A function of $x$ is defined as $$ f(x) = \langle x |f\rangle \tag{1} $$ ...
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The Physical Meaning of Projectors in Quantum Mechanics

Let $O$ be a single-particle observable for a system, and $|L\rangle$ and $|R\rangle$ two orthonormal eigenstates of $O$. You may imagine that the system consists in two photons, and $|L\rangle$ and $|...
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Symmetric upper index tensor representation of $SU(2)$

Zee writes in Appendix B of his QFT book, "For $SU(2)$, because the antisymmetric symbols $\varepsilon_{\mu\nu}$ and $\varepsilon^{\mu\nu}$ carry two indices, it suffices to consider only ...
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Showing $SU(N)$ matrices commute with conjugate transpose

$SU(N)$ is the group of all $N\times N$ matrices that satisfy $$ \mathbb{U}^\dagger\mathbb{U}=1~~,\quad\text{and}\qquad \det \mathbb{U}=1~~. $$ Denoting the $\mu$-row and $\nu$-column entry in $\...
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How to prove that $\sum_{n=1}^{\infty} | \phi_n \rangle \langle \phi_n | = \hat {I}$?

Imagine discrete orthonormal basis made of infinite set of kets $|\phi_1\rangle , ..., |\phi_n\rangle,...$ Completeness or closure of the basis is given by: $\sum_{n=1}^{\infty} | \phi_n \rangle \...
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Linearity of Adjoint operator

I was looking through my quantum mechanics textbook and found the following property of adjoint operators: $$(\hat A+\hat B)^\dagger = \hat A^\dagger +\hat B^\dagger,$$ where $\hat{A}$ and $\hat{B}$ ...

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