# 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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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...