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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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?

$e_i$ 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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Simplifying Wave Algebra - Possible binomial theorem?

In my physics class today, we were looking at the expression $(1-(u/v))^{-1}$. In a single step, the professor showed that this expression equals $1+u/v$. How is that? Is it the binomial theorem? So ...
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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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147 views

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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Can one compute eigenvectors from submatrices obtained by removing successive rows and columns? [migrated]

In this paper, an 'eigenvalue-eigenvector relation' is (re-)discovered which expresses the eigenvectors of an $N$ by $N$ hermitian matrix $M$ in terms of its eigenvalues and its minors (minors are ...
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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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1answer
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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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On Unitary Equivalent Observables

In J.J. Sakurai's Modern Quantum Mechanics, he introduces the concept of 'Unitary Equivalent Observables'. If $|a^{'}\rangle$ and $|b^{'}\rangle$ are the orthonormal bases eigenkets of two non-...
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Physical meaning of eigenvalues in the heat equation problem

Let's consider the heat equation on a $\Omega \subset \mathbb{R}^2$ manifold with a boundary $\Gamma$, with initial and boundary conditions \begin{align} \dot{u}(\mathbf{r}, t) &= \Delta u(\mathbf{...
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Expectation of an Observable

If we have the complex scalar $\langle f|\Omega|g\rangle$ as in equation (2.73) of these course notes (where $\Omega$ is Hermitian), and want to evaluate it in the position basis, I would proceed as ...
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What does it mean for a function in one Hilbert space to be to be diagonal in a basis for a Hilbert space that is not a subspace?

I am following Richard Martin on interacting electrons. For independent electrons at zero temperature he finds that the time-ordered Green's function is given by $$ G(x_1,x_2;\omega) = \sum_{l} \frac{\...
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Combining Jacobians to get linear velocity

I have a robot composed of $n$ links. The robot is modular, so each link is itself a mini-robot. The problem is, I have a geometric (basic) Jacobian for each link $i$ which maps link $i$'s joint ...
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Confusion about the dimension of a Hilbert Space in Quantum Mechanics [duplicate]

In Quantum Mechanics, the quantum state of the physical system lives in an infinite-dimensional Hilbert space and can be written in terms of two different bases, the position basis (uncountably ...
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Raising and Lowering Operators of a Hamiltonian

Lets say that I have a Hermitian Hamiltonian $H$ with a non-Hermitian raising operator operator $A$ which satisfies \begin{equation} [H,A] = \Omega A, \quad \Omega \in \ \mathbb{R}_{>0}. \end{...
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Freedom in choosing elements/entries of an eigenvector

I want to understand why there is freedom in choosing entries of an eigenvectors on some instances. I will take up a particular Hamiltonian to explain this. $$H=H_0 \left[ {\begin{array}{ccc} 1 &...
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What does it mean to expand a function in its basis?

I was reviewing my quantum mechanics notes, and I was confused on what this expression meant: $$ |{\psi}\rangle = \sum_{i}|{\omega_i}\rangle\langle{\omega_i}|{\psi}\rangle $$ I understand that it's ...
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Why does Quantum Mechanics use Linear Algebra? [closed]

I am currently doing Linear Algebra in hopes of one day tackling QM, and I need some motivation now to continue in this pursuit. The University I attend set this as a pre-requisite for QM. Now I have ...
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Doubt in a solved example from Quantum Mechanics: Concepts and Applications by Nouredine Zettili [closed]

Question 3.7 b) from Quantum Mechanics: Concepts and Applications by Nouredine Zettili, on page no. 188 (solved examples) - I understand all the solutions mentioned therein but can't figure out why ...
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1answer
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Strange use of the mean value into the definition of operators

I am currently working on quantum mechanical wave packets and minimum uncertainty states, to be specific I am trying to prove that the minimum uncertainty state is represented by a gaussian. Anyway, I ...
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Unitary transformation of Dirac equation

Dirac equation is given by $$(i\gamma^\mu\partial_\mu-m)\psi=0.$$ The matrices $\gamma^\mu$ satisfy the relation $$\{\gamma^\mu,\gamma^\nu\}=\gamma^\mu\gamma^n+\gamma^\nu\gamma^\mu=2g^{\mu\nu},$$ ...
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5answers
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How to pick out eigenvectors after solving for eigenvalues?

I'm currently doing a bit of quantum mechanics, and I can't figure out how to pick out eigenvectors. Let me explain through an example. An operator $A= \begin{bmatrix} 1 &0 &0 \\ 0&0 &...
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1answer
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Why don't the specificities of quantum mechanics (like the necessity of complex number) appear in classical mechanics?

It's well known that classical mechanics is a crude approximation of reality, and that it can be derived from quantum mechanics. But if this is so, why is it not a linear theory, like quantum ...
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Simplifying a Bra-Ket Expression

Consider the following relations $$H_0|\psi_a\rangle = E_a|\psi_a\rangle$$ $$H_0|\psi_b\rangle = E_b|\psi_b\rangle$$ I am struggling then to understand why the following identity holds (its probably ...
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The Functional Determinants in Peskin and Schroeder (Eq.9.77)

I'm working on the Eq.9.77 in Peskin (page 304): To demonstrate this, we need only apply standard identities from linear algebra. First notice that, if a matrix $B$ has eigenvalues $b_i ,$ we can ...

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