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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Vector representation in dual space

I'm new to tensor analysis, and came across the topic of vectors and duals, and faced a massive confusion. Are vectors and duals different representations of the same object ? I had another doubt ...
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How to map the eigenvectors of a superoperator into the corresponding operators?

The set of linear operators acting on a $d$ dimensional Hilbert space, $H$ form a vector space, called operator space $\mathcal{L}(H)$. Elements of operator space are $d \times d$ matrices. Now the ...
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Dumb notation question about bra-ket

Is $\langle b | A^T | a \rangle = \langle a | A | b \rangle ^ *$ where $^*$denotes complex conjugation? Edit Maybe I should add some more info here and clear things up. First, $A$ is not necessarily ...
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59 views

Is there any importance to notations like $| \Omega V\rangle$, $|aV\rangle$?

I'm not really liking this notation. Before this notation, neither of bras and kets have any preference over the other. Either of $|V\rangle$ and $\langle V|$ can be understood as the adjoint of the ...
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1answer
51 views

Why not just complex conjugate bras and kets instead of Hermitian conjugate?

I read that one equation involving bras, kets and operators, implies another equation (its transpose conjugate), analogous to how one equation involving complex numbers implies its complex conjugate ...
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What is the easiest explanation of a tensor for a non-expert? (not a duplicate question, please read the body!) [closed]

I know this question has been asked before, but when I was looking into tensors for the first time I never found a good straightforward explanation of what a tensor is and why it is important. I tried ...
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20 views

Eigenvalues of system with 5 or more degrees of freedom?

When finding eigenvalues for a system consisting of a single particle, its position and velocity are used when making the system of equations. So that there is an equation like $\dot{x} =\begin{...
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3answers
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Book recommendation for Quantum Mechanics [duplicate]

I want to delve into Quantum Computation, and for that I need background in Quantum Mechanics and other relevant mathematical topics. Which Quantum Mechanics book/resource should I use for that ...
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1answer
55 views

SVD of $2\times 2$ matrix where entries have different units

I have the following matrix: $$ M = \begin{pmatrix} 1 + xy & y \\ x & 1 \end{pmatrix} $$ where $x$ has the unit $m^{-1}$ (per meter) and $y$ has the unit $m$ (meter). This matrix acts on real ...
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1answer
62 views

What is the point of decomposing the Kraus operators of a channel in terms of a fixed basis?

Say we have a linear map $\mathcal{E}$ describing the dynamics of a quantum system, $$\rho \rightarrow \mathcal{E}(\rho)$$ As expressed in the operator-sum representation, $$\mathcal{E}(\rho) = \sum_i ...
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1answer
29 views

Why is modulus of common ratio taken for convergence of geometric series? [closed]

S=a(1-r^n)/1-r, my question is why do we take |r|<1 for convergence geometric series, can r be negative and still converge or absolute value is needed to converge?
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1answer
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Understanding the definition Fock space

In my lecture we defined the Fock space as follows: Let $\mathfrak{h}_1,\mathfrak{h}_2,\dots $ be a sequence of separable Hilbert spaces. Let $\mathcal{H}_N=\mathfrak{h}_1\otimes \dots \otimes \...
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7answers
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Why do physicists use the Bra and Ket notation when mathematicians tend not to? [closed]

Please forgive me that this is a lay question. I realise it will not add any profound physics to stack exchange, and I ask only out of layman's curiosity. I saw a question on google from a physicist ...
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2answers
80 views

Why does entanglement negativity not satisfy the triangle inequality in the usual sense?

I am a bit puzzled, I’ve read in some places, like the original paper by Vidal, that $$ \mathcal{N}(\sum_n a_n \rho_n)\leq \sum_n a_n \mathcal{N}(\rho_n) $$ whenever $a_n \geq 0$ and $\sum_n a_n =1$. ...
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Proof of relation with coherent quantum states

According to a Wikipedia article (https://en.wikipedia.org/wiki/Coherent_state), The quantum state of the harmonic oscillator that minimizes the uncertainty relation with uncertainty equally ...
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0answers
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Dirac operator: Hermitian or antiHermitian parts

We know that $i \partial_t$ and $-i \partial_x$ are both Hermitian operators. If I understand correctly the Dirac operator $$ i \gamma^\mu \partial_\mu $$ contains the $\gamma^\mu$ such that in the ...
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1answer
70 views

Hilbert space algebra

I do not know the rules (mathematica rules) in order to perform the following calculation: Lets say we have a $2$ particle system. Each particle has its own eigenbasis: $|\phi_r\rangle$ is an ONS of ...
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1answer
27 views

A question on time taken in different frames of references

Let us say , a swimmer has to swim upstream and downstream a river , between two points which are at distance "L" from each other . The swimmer can swim in still-water at velocity "Vs&...
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1answer
135 views

Positive Matrices Representation

Is it true that any positive matrix $\hat{H}$ can be rewritten in terms of pauli operators ($\sigma_0, \sigma_1, \sigma_2, \sigma_3$) as: \begin{align} \hat{H} = \sum_{i,j}c_{ij} \sigma_i \otimes \...
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1answer
81 views

Analysis of a state vector $\,|\psi\rangle\,$ in the basis of eigenvectors of a $4\times 4$ Hamiltonian matrix

I have the following matrix \begin{equation} A= \begin{pmatrix} 0 &1  &0  &0  \\ 1 & 0 &0 &0\\ 0 &0&0&1\\ 0&0&1&0 \end{pmatrix} \end{equation} The ...
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1answer
39 views

Eigenvalue of a vector in a subspace [closed]

Consider, a quantum system has a hamiltonian with eigenstates $\{|\phi_1\rangle,|\phi_2\rangle,|\phi_3\rangle\}$ and associated eigenvalues $\{\lambda_a,\lambda_a,\lambda_b\}$. My notes state that any ...
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1answer
54 views

Eigenvectors for spin matrix along arbitrary direction

In Appendix A of this paper, the authors start from (equ. A1a and A1b) $$\sigma \cdot \hat{n} |\hat{n},+\rangle = |\hat{n},+\rangle $$ $$\sigma \cdot \hat{n} |\hat{n},-\rangle = -|\hat{n},-\rangle $$ ...
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How to invert this matrix?

Is there a smarter method for finding the determinant and inverse of the following matrix, without using the brute force procedure? (When I say brute force, it is to write the matrix with each term ...
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2answers
95 views

Eigenstates for $\vec{L}^2, L_z, L_x$ and $L_y$? [closed]

I am asked to find states $|j,m\rangle$ that are simultaneously eigenstates for $\vec{L}^2, L_z, L_x$ and $L_y$. I know that the $L_i$ operators do not commute and hence you cannot have a state $|\phi\...
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1answer
81 views

How can I invert this matrix? [closed]

How can I prove that the inverse of following matrix: $\Omega_{ij} = \omega \delta_{ij} +i \varepsilon_{ijk} v_k$ is, in fact $\Omega_{ij}^{-1} = \dfrac{\omega}{\omega^2-\vec{v}^2} \delta_{ij} -\dfrac{...
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2answers
354 views

Completeness in Quantum Mechanics

While studying Matrix representation I found a topic about completeness for a basis by the equation:$ ∑_{n}|ψ_n⟩⟨ψ_n|=1$ I don't understand the physical meaning behind it. Is it something to do with ...
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40 views

Finding eigenvectors for SHO lowering operator numerically in Python

I'm trying to reproduce coherent states for the quantum simple harmonic oscillator in Python. Griffiths says (2nd ed., Problem 3.35) these are eigenvectors of the lowering operator, while the raising ...
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1answer
43 views

Do linear theories have infinitely many solutions?

It is said that, in a linear theory, you can add any number of solutions to still find a solution. So, say I initially found three solutions to a linear theory, and I call them a1, a2, and a3. Now I ...
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6answers
263 views

Angular momentum commutation relations [duplicate]

The operator $L^2$ commutes with each of the operators $L_x$, $L_y$ and $L_z$, yet $L_x$, $L_y$ and $L_z$ do not commute with each other. From linear algebra, we know that if two hermitian operators ...
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How to compute the covariance error term in cosmology context?

Below the error on photometric galaxy clustering under the form of covariance : $$ \Delta C_{i j}^{A B}(\ell)=\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}}\left[C_{i j}^{A B}(\ell)+N_{i j}^{...
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2answers
65 views

The general wavefunction can be expanded in such eigenstates

Suppose we have solved for the energy eigenstates of some Hamiltonian operator $\hat{H}$. We call the energy eigenstates $\psi_n (x)$, where: $n=1$: $\psi_1 (x)$ is the ground states $n=2$: ...
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How to solve general wave equation and dispersion relation using Fourier series?

In this paper (open access), the authors used Fourier series with most general wave equation to find the dispersion relation. I am presenting some main equations as snippets to depict their solution. ...
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1answer
46 views

Quantum Harmonic Oscillator and Diagonalization

Suppose we want to find the eigenvalues and the eigenfunctions of the following 3D Hamiltonian: $$H=\frac{p_x^2+p_x^2+p_y^2}{2m}+\frac{1}{2}m \omega ^2(2x^2+2y^2+2xy+z^2)$$ Now: On my own, right now, ...
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Each wave function can be expanded in terms of an infinite number of linearly independent functions

Suppose we take a wave function for a infinite potential well. So our system can be represented by a wave function of sum of sine functions with n ranging from 1 to infinity. Is that the meaning of ...
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36 views

Stability analysis of a cubic characteristic polynomial

I have the following cubic characteristic polynomial describing some dynamical system: $$f(\lambda, b) = \lambda^{3} + 3\left(\frac{1}{2} + i \right)\lambda^{2} + \left( 3i-3b-\frac{1}{4} \right)\...
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1answer
65 views

Why do basis vectors transform covariantly?

I have some trouble understanding transformation rules of basis vectors. My question/goal is to obtain a mathematical derivation to see why basis vectors transform covariantly and other vectors (...
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21 views

What's the domain and range of time-reversal operator in QM?

I haven't found a rigorous definition of the time-reversal operator and here are my questions. What's the definition of a time-reversal operator? Is it an operator on $L^2(\mathbb{R})$? Are there any ...
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1answer
58 views

Condition for discrete spectrum and ladder operator

In my first course in quantum mechanics we have seen three operators with discrete spectrum: the Hamiltonian of an harmonic oscillator $\hat{H}$, the square of the angular momentum $\hat{L^{2}}$ and $\...
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2answers
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Representatives in QM

I'm reading Dirac's book about QM. I reached the chapter called "representations" where Dirac introduces how can bras, kets, and observables be decomposed using a base. I have found issues ...
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1answer
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How Does This Equation even work dimensional analysis: $λ=\frac{hc}{E}$

equation: ${λ=\frac{hc}{E}}$ units: λ = m E = ${J}$ h = ${\frac{J}{s}}$ c = ${\frac{m}{s}}$ trying to isolate to get ${λ = m}$: ${λ = \frac{hc}{E}}\Rightarrow{λ = \frac{Jm}{Js^2}}\Rightarrowλ =\frac{...
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1answer
104 views

Integral of eigenkets (QM) [duplicate]

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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1answer
56 views

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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563 views

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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53 views

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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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
57 views

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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1answer
63 views

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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3answers
1k views

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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1answer
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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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