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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26 views

Working out the properties of the Special Unitary Lie algebra $su(n)$

Given the special unitary group, we can define $$ M = -iX = \begin{bmatrix} b_{11} & -ia_{12}+b_{12} & ... & -a_{1n} + b_{1n} \\ ia_{12}+b_{12} & b_{22} & ... & ... \\ ... &...
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3answers
82 views

Problem with matrices in Dirac notation

Let $|q\rangle$ be the eigenvectors of the position operator, let $|\psi\rangle$ be a state and let $\hat{p}$ be the momentum operator. In my book it's stated that i can interprete the quantity: $$\...
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0answers
78 views

What really are eigenstates? [closed]

As far I've understood/misunderstood, there exists a State Vector in an infinite Hilbert Space. When an operator acts on an eigenstate it yields an eigenvalue times the eigenstate. What is the ...
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1answer
36 views

Verifying the Gaussian Transformation of $exp\left\{\frac{1}{2}\sum_{i,j} S_i J_{ij} S_j\right\}$ from “Advanced Mean Field Methods”

The book Advanced Mean Field Methods mentions the following equation as a result of a "simple gaussian transformation". $$ exp\left\{\frac{1}{2}\cdot\textbf{s}^T \cdot \textbf{J} \cdot\textbf{s}\...
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0answers
21 views

Phase portrait with python using Jordan and eigenvalues

I want to make a phase portrait on python. I have 3 Ode's. I know the Jordan canonical form. And i know the eigenvalues. So i know when it is stable. Does anybody know how I can make a portrait using ...
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1answer
41 views

Measuring a state in a basis other than eigenbasis

Suppose I have a state expressed in its eigenbasis as follows. $\rho = \sum_i\lambda_i\vert i\rangle\langle i\vert$. It is now measured in some other basis $\{\vert x\rangle\}$ that is distinct from ...
1
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1answer
39 views

Self-adjoint operators on $\mathbb{C}^2$

I want to show that self-adjoint operators in a complex space of dimensions 2 does form a real vector space. It seems a very simple questions, but then I realize that I am not sure of how to represent ...
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0answers
26 views

How do we construct the matrix representation for three Grassmann numbers?

I want to know how we find or construct the matrix representations for Grassmann numbers. For example, we can see from https://en.wikipedia.org/wiki/Grassmann_number: Grassmann numbers can always be ...
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0answers
29 views

Approximate eigenvectors of complex value eigenvalue [migrated]

I was studying some basic theory to approximate differential equations near their fixed points and some question regarding eigenvectors and eigenvalues occurred to me, here it is: If you have $2\...
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1answer
52 views

Do finite matrices in quantum mechanics disobey the commutation relation?

For the canonical commutation relation $[\hat{x},\hat{p}]=i\hbar\hat{\mathbb{1}}$ to be true, the operators $\hat{x}$ and $\hat{p}$ are to be described by infinite dimensional matrices only, cf. e.g. ...
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1answer
64 views

Question about the definition for the scalar magnitude of a symmetric 2nd-rank tensor in a given direction

The scalar magnitude $S$ of a symmetric 2nd-rank tensor $S_{ij}$ in a given direction having direction cosines $l_i$ is given as: $$\tag{1} S=S_{ij} l_i l_j$$ This result is obtained by starting with ...
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1answer
36 views

Convex combination of product states cannot be purified to a product state

Suppose we have two distinct states $\rho,\sigma\in \mathcal{H}_A$. Define the following state $$\omega = \frac{1}{2}(\rho^{\otimes n} + \sigma^{\otimes n}) \in \mathcal{H}_A^n$$ Let $\mathcal{H}_R\...
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0answers
21 views

Weight function in inner product

Until know I thought that the definition of the inner product between two functions $f(\vec{r})$ and $g(\vec{r})$ with the same domain $D:[a,b]$ was: $$\int_a^b f\cdot \overline{g} \cdot d\vec{r}$$ ...
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0answers
28 views

A Relation between quantum systems

The following is from 'Discerning Fermions' by Muller & Saunders (2008: 532): Let $|\phi_{1}\rangle, |\phi_{2}\rangle, ..., |\phi_{d}\rangle$ $\in$ $\scr{H}$ be an eigenbasis of $\scr{H}$ ...
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0answers
61 views

Can the time-reversal operator of a two-level system be represented by a $2\times2$ matrix?

I am studying the time-reversal symmetry in the context of topological insulators. As usual, the minimal non-trivial model to be considered is a two-level system with Hilbert space $\newcommand{\ket}[...
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1answer
29 views

Numerical way of finding energy spectrum of $N$-body Schrodinger equation

For a single particle trapped in a potential, one can discretize the Time Independent Schrodinger Equation and hence find the eigenvalues of the corresponding Hamiltonian by diagonalising numerically. ...
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0answers
36 views

Finding the eigenvalues of a matrix with particular symmetry [migrated]

I have a matrix for which I want to get some analytical equations of the eigenvalues. The matrix is given as \begin{align} \mathbf A &= \begin{pmatrix} \epsilon_a & 0 & 0\\ 0 & \...
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2answers
52 views

Representation of operator as block matrix

I've two operators that commute: $A=\begin{pmatrix} 2 & 0 & i\\ 0 & 1 & 0\\ -i & 0 & 2 \end{pmatrix}$ and $B=\dfrac{1}{2}\begin{pmatrix} 3 & -i\sqrt{2} & i\\ i\sqrt{2} &...
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2answers
40 views

Product of non-commuting operators

I want to expand the product: $$\left(\hat{A}_{1}+\hat{A}_{2}\right)\left(\hat{B}_{1}+\hat{B}_{2}\right)$$ $\hat{A}_1$ and $\hat{B}_1$ are operators both working on the same particle, and do not ...
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0answers
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How can I express a two body HFB hamiltonian in a quasi particle HFB base?

The problem is the flowing: Let $H$ be the standard two body Hamiltonian: $$H=\sum_{ab}t_{ab}c_{a}^{+}c_{b}+\frac{1}{4}\sum_{ab}v_{abcd}c_{a}^{+}c_{b}^{+}c_{c}c_{d}$$ Were {$c_{a}^{+}c_{a}$} is the ...
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1answer
82 views

Landau mechanics - Normal modes of oscillation

In Landau's Mechanics book there's a section in which he explains small oscillations in systems with $s \geq 1$ degrees of freedom. He writes the kinetic and potential energies as $$ T = \sum_{i, k} \...
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1answer
71 views

Is the complete Hilbert space of a system, sum of the spans of position, momentum, energy, angular momentum spaces, etc?

What I understand is the following. 1) There is an abstract ket ( vector) which contains all the information about the system and it lives in an abstract vector space, the Hilbert Space which might ...
3
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0answers
44 views

Best way to find $SU(2)$ transformation representation in if I know the desired output? [closed]

I have two doublets in $SU(2)$: $\begin{pmatrix} 1 \\ 1\\ \end{pmatrix}$ and $\begin{pmatrix} 1 \\ i\\ \end{pmatrix}.$ What is the easiest way to figure out the vector $\vec{\theta} $ in the $SU(2)$ ...
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2answers
110 views

What does kets $|0\rangle$ or $|1\rangle$ exactly mean in linear algebra?

In a textbook of quantum mechanics, the author frequently makes use of notations like $|0\rangle$ and $|1\rangle$. What I am curious about is their dimensions. How can you have a matrix representation ...
2
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2answers
104 views

Meaning of $\langle q_n | E \rangle$ when $E$ is energy but $q_n$ is not

In this lecture of Prof. Binney (go to 15:40), he is explaining that if we have a system with a state of constant energy, then the expectation value of any observable of that system remains unchanged ...
2
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2answers
73 views

If $\langle\psi_{AB}\vert\rho_A\otimes\rho_B\vert\psi_{AB}\rangle = 0$, then $\langle\psi_{AB}\vert\rho_{AB}\vert\psi_{AB}\rangle = 0$

Let $\rho_{AB}$ be some bipartite quantum state. Let $\rho_{A}$ and $\rho_{B}$ be the marginal states. I am reading some notes where the following statement is made. The support of $\rho_{AB}$ is ...
6
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1answer
216 views

How to incorporate the uncertainty of the model coefficients in the prediction interval of a multiple linear regression

I'm dealing with the modeling of small experimental physics data sets (specifically the stickiness of glue-compounds). As most experimental work does not generate thousands of samples, but rather a ...
2
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1answer
44 views

Linear independence of a set of states, and the non-vanishing determinant of the matrix comprised of their inner products

So I have a very basic question in linear algebra, but I'll phrase it in terms of QM. Suppose we are given a set of $N$ states $\{ | \psi_i \rangle\}$. Construct the $N \times N$ matrix $$\mathcal{...
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0answers
47 views

Conservation of signs of eigenvalues of metric tensor

The metric of an arbitrary spacetime can be transformed into the Minkowski metric because $$g_{\mu \nu } = J_{\mu }^{\alpha } J_{\nu }^{\beta} \eta_{\alpha \beta }\tag{1}.$$ Gives ten equation for ...
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9answers
2k views

Are force vectors members of a vector space?

Vectors in vector spaces depend only on their size and direction. Force vectors, for example, depend also on their location. Opposite force at different locations, for example, do not annihilate each ...
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0answers
39 views

Differential of a metric determinant

In Landau Lifschitz (Volume II): We now derive an expression for the contracted Christoffel symbol $\Gamma_{ik}$ which will be important later on. To do this we calculate the differential $dg$ of the ...
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1answer
47 views

Two expressions of kinetic energy of rotation [closed]

The moment of inertia matrix for rigid body in general case is $I= \begin{bmatrix} I_{xx} & I_{xy} & I_{xz}\\ I_{xy} & I_{yy} & I_{yz}\\ I_{xz} & I_{yz} & I_{zz} \end{bmatrix} $...
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1answer
60 views

Operator acting on bras

I need some help. Suppose, $\hat{\textbf{A}}$ and $\hat{\textbf{B}}$ are operators and $|\psi\rangle$ is any state, so that $$ \hat{\textbf{A}}|\psi\rangle=a|\psi\rangle. $$ And I wonder if this ...
3
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1answer
109 views

How can I prove this relation? [closed]

What I want to prove : $$R_{ijrs}=\partial_r\Gamma_{ijs}-\partial_s\Gamma_{ijr}-\Gamma_{rj}^k\Gamma_{iks}+\Gamma_{sj}^k\Gamma_{ikr}$$ According to Riemann-Christoffel tensor, the covariant composant ...
7
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7answers
2k views

Why can vector components not be resolved by Laws of Vector Addition? [closed]

A vector at any angle can be thought of as resultant of two vector components (namely sin and cos). But a vector can also be thought of resultant or sum of two vectors following Triangle Law of ...
1
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2answers
65 views

How & why does the law of vector addition work? [closed]

Our teacher explained vector addition to us. He explained to us the triangle law of vector Addition. I have two questions: He said the vector $\vec{R}$ is the resultant vector, which means that ...
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1answer
54 views

Division of two operators (or polonomials of operators) in quantum mechanics

Consider a function of an operator $\hat{A}$, which is like follows $$ f\left(\hat{A}\right) = \frac{a + i b\hat{A}-c\hat{A}^2}{3-\hat{A}} $$ where $a$, $b$ and $c$ are complex numbers. My question ...
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1answer
56 views

Guessing eigenvalue solution

I am reading The Theory of Magnetism I, by Mattis. In Chapter 2, he proposes the following eigenproblem: $$ \left ( \begin{matrix} V & U \\ U^\dagger& V \end{matrix} \right ) \left ( \begin{...
3
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0answers
39 views

Inverting the matrix $\eta_{uv}(p^2-i\epsilon)-(1-\frac{1}{x})p_u p_v$

In Weinberg Quantum Theory of Fields Volume 2, p.23, it says that $\eta_{uv}(p^2-i\epsilon)-(1-\frac{1}{x})p_u p_v$ where $0 <x \leq 1$ has the inverse matrix $[\eta_{uv}+(x-1)\frac{p_u p_v}{p^2}]/(...
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2answers
90 views

Dirac expression derivation

In Quantum Mechanics, 2nd Edition by Davies & Betts on page 78 it states that there is a symmetry implied by the following Hermitian operator equation: $${\displaystyle \int \phi^{*}(A \psi)d \,\...
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1answer
53 views

Eigenvectors in anisotropic media

I have several questions: 1) First, In the susceptibility tensor, when it's diagonalized, i don't understand the physical significance when the off diagonal terms are zero. $$P_x=\epsilon_0\chi_{11}...
1
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2answers
80 views

Inner product: operation between vectors from the same vector space or between vectors from a vector space and its dual space (Ex: bras and kets)?

I am taking my first steps in learning quantum mechanics and am learning about Dirac's bra-ket notation. I am trying to understand what the inner product is. My understanding so far: the inner ...
2
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1answer
36 views

What transformation gives a Weyl-like representation by flipping $\gamma^0$ and $\gamma^5$?

The usual Weyl representation of the Dirac matrices is defined like this: $$\tag{1}\gamma_W^a = T_W \, \gamma^a \, T_W^{-1},$$ where \begin{align}\tag{2} T_W &= \frac{1}{\sqrt{2}} (1 + \gamma^5 \, ...
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1answer
61 views

General Gaussian integral in Peskin and Schroeder [closed]

I'm working on the Eq. (9.24) in Peskin & Schroeder. I tried to derive it but I have difficulties. I canʻt follow this step: $$ \left(\prod_{k} \int d \xi_{k}\right) \exp \left[-\xi_{i} B_{i j} \...
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vote
2answers
216 views

Cauchy-Schwarz inequality in Shankhar's Quantum Mechanics

I'm trying to understand proof of this inequality. But I have some problems. So, Shankar starts a proof with definition a new vector $|z \rangle$: $$ |z \rangle = |v\rangle - \frac{\langle w|v \...
2
votes
1answer
86 views

Linear algebra with Dirac notation

I have some questions about linear algebra. Let's say $\{|v_1 \rangle, |v_2 \rangle, |v_3 \rangle \}$ are orthonormal basis of the $\mathcal{V}(\mathbb{C})$. Then, let's define two vectors $$ |a \...
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2answers
47 views

Understanding the inverse in the definition $(\tilde{\Pi}_gf)(v)\equiv f(\Pi^{-1}_gv)$

I'm trying to understand the representation $\tilde{\Pi}$ induced from the fundamental representation $\Pi$, defined as $(\tilde{\Pi}_gf)(v)\equiv f(\Pi^{-1}_gv)$ for $g\in G,\hspace{1mm}f\in\mathcal{...
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0answers
28 views

Looking for a freeware software/app that can solve eigenvalue problems symbolically

I'm taking a quantum mechanics course and my homework involves extremely tedious algebra to solve symbolic eigenvalue problems. I'm looking for a software that I can give matrices with symbolic ...
0
votes
0answers
37 views

How to prove that all eigenstate $|0\rangle,|1 \rangle,…| n\rangle $ are non-degenerate? [duplicate]

How to prove that all eigenstate $|0\rangle,|1 \rangle,...| n\rangle $ are non-degenerate where $a^{\dagger}a|n \rangle =n|n \rangle $
1
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1answer
84 views

Commutator of $B$, $C$ vanishes if $A$, $B$, $C$, $AB$, $AC$ are Hermitian

Suppose 3 operators $A$, $B$, $C$ are Hermitian operators. Assume $A$ has a non-degenerate spectrum, and $AB$ and $AC$ are also Hermitian. Show that $$[B,C] = 0$$ From the conditions $A$, $B$, $C$, $...

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