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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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 ...
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3
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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 ...
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1
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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 $\...
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1
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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 ...
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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? (...
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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\...
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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 ...
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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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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 ...
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1
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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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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 ...
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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^†...
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2
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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 ...
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2
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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 (...
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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 ...
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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
$$\hat{p}\Psi=p_{1}c_{1}\psi_{1}+...
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Weird question: elements of eigenvector as kinetic and potential energies
Assume we have $N$ particles each having some potential and kinetic energies. Denote the sum of kinetic energy as $\sum_i T_i = T$ and the sum of potential energy as $\sum_i V_i= V$. This is a closed ...
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Proving the uncertainty relation for the quantum covariance matrix
In quantum optics papers we often encounter this version of the Heisenberg uncertainly relation (for an $n$-mode quantum system):
$\sigma + \iota \Omega \ge 0$
Where $\sigma$ is the covariance matrix $...
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2
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Position of a point on a rigid body
I can not wrap my head around the following doubt:
How can we express (or prove the fact that) the position of a material point on a rigid body as the sum of a "traslational component" and ...
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Eigenkets of a two-state hamiltonian
I have a question related to this other question: Eigenenergies and eigenkets given the Hamiltonian. In it, OP is given the following hamiltonian:
$$
H=a(|1\rangle \langle1|-|2\rangle\langle2|+|1\...
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2
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Why are Eigenvectors of a 1D quantum ising hamiltonian real
I was modelling the 1D transverse quantum Ising model and made a Kronecker product loop to find the Hamiltonian of the system, for a given magnetic field configuration. Now, my question is that when I ...
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Measurement operator in a Bell experiment
I'm trying to figure out why a Bell experiment gives rise to the payoff (measurement) operator used in this paper on quantum game theory.
Two players are each in control of one half of an entangled ...
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Special relativity book which describes concepts using linear algebra notions
It seems so every idea of special relativity can be formulated quite nicely in Linear algebra notions such as the inner product matrix and change of basis matrices. However, I can't find a single book ...
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1
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Whats the meaning of the 1 Ket? [closed]
I am talking this one: $|1\rangle$. If I have 2 orthonormal states $|1\rangle$ and $|2\rangle$ in the 2D Hilbert space, does that imply the vector $\vec{\psi_n}=(1,2)$, if I would like to solve the ...
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2
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Equation for the simultaneous eigenfunction of three operators in spherical coordinates
If I'm considering the three operators $H,L^2,L_z$ with the condition $[H,L^2]=[H,L_z]=[L^2,L_z]=0$, I can find a complete set of simultaneous eigenfunctions. If I study this problem in spherical ...
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How the eigenvalue problem was solved?
In Gasiorowicz 3rd edition Chapter 3, I've tried to solve this problem
I checked the solution's manual,
When I tried to integrate it, the answer I got is
$$
\psi(x)=Ce^{x^2/2\lambda}
$$
Can you ...
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1
answer
54
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Factorization of the wavefunction in a central Hamiltonian problem
I am trying to understand the topic of the title. If I consider a central Hamiltonian, so an Hamiltonian of the form $H=\frac{p^2}{2m}+V(r)$ what are the logical steps that lead me to the known result?...
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Quantum mechanics, are simultaneous eigenstates to be intended always as a tensorial product of two eigenstates?
The question is the one of the title, let $\hat{O}_1$ and $\hat{O}_2$ two commuting operators: $[\hat{O}_1,\hat{O}_2]=0$, there is an orthonormal basis formed by their simultaneous eigenstates.
These ...
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What is an eigensystem? Could you provide a simple example? [closed]
Also, what is the difference between an eigensystem and the eigenspace?
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Force to Inflate a ball underwater [closed]
How much force is required to fully inflate (with air) a beach ball that is 6 feet in diameter at depths of 200 feet underwater?
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1
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Geometric multiplicity vs degree of degeneracy of energy levels
Let $A$ be a square matrix of order $n$. Given a fixed eigenvalue $\lambda$, we call geometric multiplicty the dimension of the associated eigenspace
$$g_{\lambda} = \dim(v\in\mathbb{K}^n : Av= \...
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Are there ways to find representations of matrices given an algebra?
Given an equation (or a set of equations) involving matrices, is there an algorithm to find possible representations of these matrices?
For example, we can consider a matrix $A$ such that
$A^2=\begin{...
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1
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Is the Lindblad equation invariant under the unitary transformation?
Let us say we have a Lindblad equation for the density matrix
$$
\dot{\rho}=-\frac{i}{\hbar}[H, \rho]+\sum_{i=1}^{N^{2}-1} \gamma_{i}\left(L_{i} \rho L_{i}^{\dagger}-\frac{1}{2}\left\{L_{i}^{\dagger} ...
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2
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How to write a matrix $\mathcal{M}$ such that $\mathcal{M} \boldsymbol{x}=\boldsymbol{\omega}\times\boldsymbol{x}$? [duplicate]
As is well known, it is possible to use the $\nabla$ operator as if it were a vector. Someone consider it an abuse of notation but surely something that works well and is very useful. Well, how is it ...
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1
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Energy (Hamiltonian) of Trial Wavefunction
Here I give a part of derivation of Hartree-Fock equations in case where basis functions (wavefunctions) are orthonormal and real: $$ \langle \psi_i | \psi_j \rangle = \langle \psi_j | \psi_i \rangle =...
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Square of Volume of Tetrahedron in Loop Quantum Gravity
In the subsequent statement about the geometry of Tetrahedron in Carlo Rovelli's book, there is a formula as follows:
$$V^2 = \frac {2}{9}(\vec L_1 \times \vec L_2)\cdot \vec L_3$$
Where $\vec L_a$ is ...
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Please explain statement in a book on Loop Quantum Gravity
In a book by Carlo Rovelli on Covariant Loop Quantum Gravity, I struggle to understand a statement on Tetrahedron as follows:
What is the dimension of the matrix?
How to derive the given matrix ...
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1
answer
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Pure and mixed density operators of a Schmidt decomposition
Suppose we have Hilbert space factorisable in to K subsystems
$$
\mathcal{H} = \mathcal{H}_1\otimes...\otimes\mathcal{H}_K
$$
in which we can express a pure state as the Schmidt decomposition
$$
|\...
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1
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Can the time-evolution operator be factorised if the Hamiltonian is a sum of two commuting operators?
Let the time-dependent Hamiltonian $H(t) = A(t) + B(t)$ for some quantum system be given as the sum of two time-dependent operators $A(t)$ and $B(t)$. Further, assume that $A(t)$ and $B(t)$ commute, ...
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$y$ Pauli Operators Eigenvectors - How are they orthogonal?
I am struggling to obtain that the eigenvectors of the Pauli $y$ operator are orthogonal, and would appreciate guidance on where I am going wrong. I have calculated the eigenvalues as: 1, -1
And ...
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1
answer
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What does eigenvalues of the Lorentz matrix represent physically speaking? [duplicate]
In special relativity, if we have a boozt in the x - direction, the relationship between the coordinates of the inertial frame of reference S, and the one of S' (moving with velocity v relative to S), ...
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Which of these is the logical way to establish tensors on a manifold?
You start by defining a vector space at each point of the manifold. The defining feature being the vector transformation law under change of co-ordinates.
Then you define dual vectors as linear ...
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Inner product evaluation in QM
On wikipedia on the page for inner product it states that for any two $x,y$ in a vector space $V$ the inner product $(\cdot , \cdot)$ satisfies $(ax, y) = a(x,y)$ where $a\in\mathbb{C}$.
The inner ...
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Srednicki eq. (1.27): $\left\{\alpha^{j}, \alpha^{k}\right\}_{a b}=2 \delta^{j k} \delta_{a b}$
Srednicki, QFT, p. 8 writes
$$\left\{\alpha^{j}, \alpha^{k}\right\}_{a b}=2 \delta^{j k} \delta_{a b}\tag{1.27}.$$
What does exactly $ab$ here denote?
Assume I have a matrix X [0 1]
[2 3] and does a ...
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2
answers
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Minimum number of non-coplanar forces required to keep an object in equilibrium
The minimum number of non-coplanar forces that can keep a particle in equilibrium is:
(a) 1
(b) 2
(c) 3
(d) 4
Answer given is option $(d)$ , i.e $4$. But can’t it be $(c)$ , i.e $3$ too?
Suppose I ...
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1
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Null vectors that aren't the zero vector in general relativity?
So I was trying to understand the null energy condition of $T_{μν}k^μk^ν≥0$ Where $k$ is an "arbitrary future-directed null vector" and couldn't really wrap my head around how the $k$ is ...
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Inner product question
Why is it valid to write that:
$$\langle\alpha|a^{\dagger}\alpha\rangle=\langle a\alpha|\alpha\rangle$$
where $\alpha$ is the lowering operator, $\alpha^{\dagger}$ is the raising operator, and $a\in \...
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1
answer
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Finding $\theta$ and $\phi$ when qubit state is $\frac{1}{\sqrt 2}[i ,1]^T$
Because we know the state of a qubit can be described as:
$$
|q\rangle=\cos{\frac{\theta}{2}}|0\rangle+e^{i\phi}\sin{\frac{\theta}{2}}|1\rangle\\\ \\
\theta, \phi \in \mathbb{R}
$$
How do I find the ...
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1
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Commutation relation confusion of ladder operators in Quantum Mechanics
Suppose that $X$ and $N$ are operators such that they follow the commutation relation $$[N,X]=cX$$ for some scalar c. In this Wikipedia article it is shown that if $|n \rangle$ is some eigenstate of ...
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