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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2 answers
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Dirac bra-ket operator notation [closed]

a simple question about Dirac notations: Let $|a\rangle = \begin{pmatrix} a_{1} \\ \vdots \\ a_{n} \end{pmatrix}$ and $|b\rangle = \begin{pmatrix} b_{1} \\ \vdots \\ b_{n} \end{pmatrix}$. Then what is ...
3 votes
2 answers
80 views

Can a Matrix Have EigenBras? [closed]

I only need a yes or no, but I cannot find anything online. I know a matrix $A$ can have eigenkets found by using. $$A\psi=\lambda\psi.$$ However, I was wondering if my matrix $A$ was Hermitian, could ...
0 votes
1 answer
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Geometric interpretation of tensor product

I am not sure if this question is better suited for math SE, but given that I am asking for a geometrical (ideally physical) interpretation, I figured it would be best asked here. I am looking for a ...
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2 answers
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Does each vector in $su(3)$ represent a different kind/type of gluon (infinite kinds/types of gluons); or, are they all considered the same kind/type?

According to Does gluons have names?, it seems that there is no way to give names to gluons because they are not "always the same". So, it seems to imply that the same gluon can change its ...
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2 answers
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How to assign a value to an observable when the statevector is not an eigenvector of the operator?

We get the value of an observable $A$ for a given state $|\lambda\rangle$ of a system from the eigenequation $\hat{A} |\lambda\rangle = \lambda |\lambda\rangle$ where $\hat{A}$ is the operator ...
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2 answers
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Why can't we take the larger angle in Cross Product?

Suppose, $\vec{A}$ and $\vec{B}$ are inclined with each other at an angle of $\phi$. Then, their cross product would be defined as $$\vec{A}\times\vec{B} = |A||B|\sin(\phi)\hat{n}$$ My Physics ...
0 votes
1 answer
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Determinant and inverse of metric tensor in Eddington-Finkelstein coordinates

I need to make sure I'm not going crazy here. We define the metric of Eddington-Finkelstein (EF) coordinates $(v,r, \theta,\phi)$ as $$g=-\left(1-\frac{2m}{r}\right)dv^2+2dvdr+r^2d\Omega^2$$ where $d\...
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4 votes
1 answer
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How can the position representation make sense with compatibility of addition? (Dirac Notation)

According to the definition of complex inner product is that: $$⟨\psi|\phi_{1} + \phi_{2}⟩ = \left<\psi|\phi_{1}\right> + \left< \psi| \phi_{2} \right>, \forall \psi, \phi.$$ This implies ...
1 vote
1 answer
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For $n\in\mathbb{N}$ is the $n$th power of a hermitian operator always hermitian?

It would seem to me that $\hat{Q}^n$ being a hermitian operator when $\hat{Q}$ is one, with $n$ being a positive integer, is not true in general. After all: if $n$ is uneven then it is impossible to ...
1 vote
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(Wald's GR book) Isotropy implying constant curvature

Context In Wald's GR book, p. 94 of chapter 5, he gives an argument for why isotropy and homogeneity implies constant curvature. In summary, it goes like this: We can take the induced metric, $h_{ab}$,...
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2 answers
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How do I diagonalize this Hamiltonian? [closed]

I have a Hamiltonian $$H = \frac{p^2}{2}+\frac{\omega^2 q^2}{2}+\frac{\gamma}{2}(qp + pq)$$ which I have to diagonalize, i.e., find $a$ and $a^\dagger$ as linear combinations of cannonical $p$ and $q$ ...
1 vote
0 answers
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Help with conmutator identity with angular momentum and vector [closed]

I need to prove this identity: $ [ \textbf{J}^2, \textbf{J}\times \textbf{V}] = 2i\hbar( \textbf{J}^2 \textbf{V} - ( \textbf{J} \cdot \textbf{V}) \textbf{J}) $ Where $ \textbf{J}$ is an angular ...
3 votes
1 answer
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Similarity transformations in QFT

I am trying to understand the gaps in my knowledge that prevents me from completely understanding quantum field theory. Sometimes I ask pretty basic questions, but please excuse me if I make a blunder....
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1 answer
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Density Matrices in Quantum Mechanics

I have a question about the physical meanings of various matrices expressed in Dirac bra-kets. I take it that $\frac{1}{2}|A\rangle\langle A| + \frac{1}{2}|B\rangle\langle B|$ can be interpreted as a ...
1 vote
0 answers
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Choi-Jamialkoski Theorem in Phase damping channel [closed]

I am trying to replicate the solution I have to this problem provided by the instructor in the class where I am trying to use the Choi-Jamialkoski theorem to prove that Phase damping channel is ...
0 votes
2 answers
68 views

$F$ transforms like a tensor $\Rightarrow B$ transforms like a pseudo vector

Notation: In the following $*$ is the hodge operator from $\Lambda^1(\mathbb R^{1\times 3})\cong \mathbb R^{1\times 3}$ to $\Lambda^2(\mathbb R^{1\times 3})\cong A\subset\mathbb R^{3\times 3}$ (or its ...
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4 votes
3 answers
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Diagonalising the Hamilton operator, why does this magic work?

Let the Hamilton operator $H= \omega_1 a_1^\dagger a_1 + \omega_2 a_2^\dagger a_2 + \frac{J}{2} (a_1^\dagger a_2 + a_1 a_2^\dagger)$ be given, of course $a_j$ and $a_j^\dagger$ are the creation and ...
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Simultaneous Measurement of Anti-commutative Operators

In quantum mechanics, two variables $A$ $B$ can be observed simultaneously if they commute with each other, i.e. $[A, B]=0$. From what I learned from courses, this is established by two facts: 1: ...
2 votes
0 answers
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What math do I need for physics? [closed]

I'm in 9th grade. I've gone through linear algebra, multivariable calculus, differential equations, and statistics. I'm attempting to get better at physics and maybe try out for the International ...
1 vote
1 answer
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Outer Product Other form [closed]

The outer product of a ket $|\psi\rangle$ with a bra $\langle\phi|$ according to the textbook Quantum Computing Explained by D. McMahon, behaves likes an operator. He illustrates this by applying an ...
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2 votes
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Parametrization of Unitary matrices

Is there any book to follow along with "The Unitary and Rotation Groups" by F.D. Murnaghan" for the first two chapters concerning the parametrization of general $n \times n$ unitary ...
-1 votes
1 answer
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Can someone redirect me to relevant mathematics? [closed]

These are two paragraphs from Chapter 3 of Principle of Quantum Mechanics by P.A.M. Dirac. I need to know what the relevant mathematics its referring specifically, I have some idea not proper enough ...
1 vote
1 answer
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Finding common eigenvectors for two commuting hermitian matrices [closed]

Let $A = \begin{bmatrix} 1 &0 &0 \\ 0& 0& 0\\ 0&0 &1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 &0 &1 \\ 0& 1& 0\\ 1&0 &0 \end{bmatrix}$ ...
0 votes
2 answers
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Change of basis, matrix and operators

If $U$ is an unitary operator written as the bra ket of two complete basis vectors i.e $U=\sum_{k}\left|b^{(k)}\right\rangle\left\langle a^{(k)}\right|$ Then $U^\dagger=\sum_{k}\left|a^{(k)}\right\...
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1 answer
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Kitaev Chain - Obtaining a real-orthogonal matrix that block-diagonlises the Kitaev Chain

I encounter a subtle problem regarding the Kitaev Chain. In Kitaev framework, he tried to express the Hamiltonian into real-orthogonal basis. Suppose the Majorana system is described by $$ H = \frac{i}...
0 votes
1 answer
35 views

Relation between diagonal and off-diagonal entries of Hermitian Operator

I am started doing a project in Quantum Chemistry and stumbled upon a problem which I can not seem to find the answer to. As the title suggests, I am looking for a relation between the diagonal and ...
1 vote
2 answers
65 views

Understanding dot product in quantum mechanics [closed]

Let's say we have a two-state-system with state $\vert 1\rangle$ and state $\vert 2\rangle$. From my understanding one can assume the base vectors of this system to be $\vert1\rangle \mapsto (1,0)^\...
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1 answer
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How to solve for the scattering solution of following Schrodinger equation?

Suppose you have non-relativistic fermions scattering off a delta function potential. It is an easy job to solve $H=-\partial_x^2+\epsilon \delta(x)$ by starting with an eigenfunction of the form $\...
1 vote
1 answer
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What's the contraction for non-adjacent fields?

In section 8.2 of Coleman's QFT lectures, he introduces the definition of contraction of two fields, where $T$ denotes time ordering and the colons normal ordering. Then he proceeds to contraction in ...
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4 votes
2 answers
365 views

Are the linear Lie groups matrices, tensors, or both?

In some ways, this is a question about notation. In my experience, I have only seen the classical Lie groups — such as $\operatorname{GL}(n,\mathbb{R})$, $\operatorname{SL}(n,\mathbb{R})$, $\...
7 votes
3 answers
1k views

How to avoid paradoxes about time-ordering operation?

(Original title: is time-odering operator a linear operator?) I'm confused with two formulas, one of which is $$ \mathcal{T} \exp \left [-\frac{\mathrm{i}}{\hbar} \int_{t_0}^t \mathrm{d} t' \hat{H}_I(...
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9 votes
1 answer
251 views

Are the instantaneous eigenstates of a time-dependent hamiltonian continuous?

I am trying to understand the adiabatic theorem. I can follow the proofs that are given in Wikipedia (https://en.wikipedia.org/wiki/Adiabatic_theorem) but there seems to be a hidden assumption. For a ...
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1 vote
1 answer
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Find projection operators degenerate energy eigensubspaces [closed]

A given system has Hamiltonian $H=\sum_{i=0}^{n}\sigma^{(i)}_{z}$, where $\sigma^{(i)}_{z}$ are the usual Pauli matrices. Now I want to find the corresponding $n+1$ projection operators corresponding ...
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1 vote
1 answer
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Checking if a state is an eigenstate of $L^2$ and $L_z$ without performing calculations

If I have a given state $\psi$ which is a linear combination of spherical harmonics, and I am asked if its an eigenstate of $L^2$ and $L_z$, is there a way to do it without using the eigenvalue ...
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Issue when trying to represent an operator as a matrix

We say that a ket $|V\rangle$ can be expresed in an orthonormal basis $(|e_1\rangle,|e_2\rangle,...|e_n\rangle)$ as : $$|V\rangle = \sum_i^n v_i |e_i\rangle$$ where $v_i = \langle i|V\rangle $ for a ...
1 vote
3 answers
267 views

Confusion about how adjoint matrices operate on state vectors

My understanding is that for an inner product in state-space, since we want the value to always be a real number we say that $$\langle\psi|\phi\rangle= {\langle\phi|\psi\rangle}^* $$ where * denotes ...
1 vote
1 answer
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The abstract state of a particle

I recently started learning about quantum physics. In the book, Quantum physics by H.C. Verma, the author explains that there are many ways to represent the state of a particle. The wave function $\...
2 votes
1 answer
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How should I interpret eigenvectors in second quantization?

a) I would like to ask, if knowledge about eigenvectors in second quantization is important and what do they mean? Let's just say, I create Fock space [(NumberOfSites)x(Permutations) matrix], then I ...
-2 votes
1 answer
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A linear algebra exercise from Griffiths "Introduction to quantum mechanics" [closed]

(Edited so that it obeys the rules of homework questions) I am stuck on this linear algebra problem from Griffiths's "Introduction to quantum mechanics". Can somebody give me some guidance? (...
2 votes
1 answer
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A relation for adjoint representation of $U(N)$ acting on product of matrices and $SU(2)$ generators

Is the following relation true, and if so, what is the property that makes it so? \begin{align} \sum_{i=1}^3\mathrm{tr}\left([U^{-1}L_iU]\phi[U^{-1}L_iU]\phi\right) \stackrel{!}{=} \sum_{i=1}^3\...
0 votes
1 answer
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A problem with Griffiths "Introduction to Quantum Mechanics" linear algebra

I have a problem at understanding the way linear transformations are used in Griffiths Introduction to Quantum Mechanics. My knowledge about linear algebra is basic. He is making a reference to the ...
0 votes
0 answers
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Spherical harmonics How do they span eigenspaces?

When trying to find common eigenstates of $L^2$ and $L_z$, we find the eigenstate $Y_m^l (\theta, \phi)$ My question is, if $m_1$ and $\lambda_1 = l_1(l_1+1)$ both have multiplicity $3$, then there is ...
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1 vote
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Question about the operators in quantum mechanics [duplicate]

I am confused about the operators in quantum mechanics and even the way they are used, their symbols, etc. Is there any book or anything that I can study so that I can fully understand them before ...
0 votes
1 answer
30 views

Diagonalizing Operators Simultaneously [duplicate]

Suppose we have a Hamiltonian operator $\hat{H}$ and another operator $\hat{A}$ such that $[\hat{H},\hat{A}]=0$. Then, if the spectrum of $\hat{H}$ is non-degenerate, from my understanding the ...
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0 votes
1 answer
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What specifically about the torque vector is perpendicular? Is the torque vector like this only so that it works smoothly with linear algebra?

What specifically about the torque vector is perpendicular? Is the torque vector like this only so that it works smoothly with linear algebra? The only explanation I get usually is "because it's ...
1 vote
1 answer
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How can I rewrite the coupling between a system and a bath in terms of Hermitian operators (Deriving Lindblad eqn)?

I'm trying to derive a Lindblad equation for a system where there is tunneling between a bath and a reservoir. This means that my interaction Hamiltonian is $H_I =\Sigma_s A^†B_s + B_s^†A$, where $A^†...
0 votes
2 answers
56 views

Is there a way to carry out the eigenstates and eigenvalues of annihilation operator using only its matrix form?

Knowing the matrix elements of annihilation operator, can I solve the eigenvalue problem without using operator method? I got stuck when I try to compute its eigenvalue, because the eigenvalues of a ...
1 vote
2 answers
83 views

Why is the Schrödinger Equation valid for the component functions (wave function) of state vectors?

I'm new to quantum mechanics and confused about the way the Schrödinger equation is used (more general eigenvalue equations of observables). Let's take the time-independent Schrödinger equation (...
0 votes
0 answers
55 views

Eigenbasis of Hamiltonian and momentum operators

I was taught that, if two Hermitian operators commute, they share the same eigenbasis. Since the Hamiltonian and momentum operators commute, am I right in concluding that they share the same basis of ...
0 votes
1 answer
52 views

Matrix formulation of the momentum operator

For a quantum state $\Psi=c_{1}\psi_{1}+c_{2}\psi_{2}$ with momentum eigenstates $\psi_{1}$ and $\psi_{2}$, the action of the momentum operator $\hat{p}$ is given by $$\hat{p}\Psi=p_{1}c_{1}\psi_{1}+...

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