# 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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### Why does the eigenvector I calculated for 𝑆𝑦 in the spin-1 case give an incorrect probability for measuring ℏ? [closed]

I've been working on a problem involving a spin-1 particle in state: $|\chi _{z}^{0}>=\begin{pmatrix} 0 \\ 1\\ 0 \end{pmatrix}$ Which corresponds to the $m_{z}=0$ state of the spin-1 particle. ...
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### How can I calculate derivative of eigenstates numerically?

I want to calculate $\langle n | \partial_{k_x} n \rangle$ where $| n \rangle \equiv | u_{n,\mathbf{k}} \rangle$ is the $n$-th Bloch eigenstate of a $6\times6$ Hamiltonian $H\equiv H(\mathbf{k})$. ...
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### Are projectors observable quantities in QM? [duplicate]

Given a certain quantum state $|\Psi\rangle$, then we can project any other quantum state $|\Phi\rangle$ on the first one by using the projector: $P_\Psi\equiv |\Psi\rangle\langle\Psi|$ in such a way ...
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### "There is a bra for every ket, but there is not a ket for every bra"

This statement is an excerpt from Quantum Mechanics (Cohen-Tannoudji), but I don't quite understand why it holds: given all our kets live in a certain Hilbert space $\mathcal H$, then all the bras ...
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### Parity eigenstates in massive $\mathcal{N}=1$ multiplet

In $\mathcal{N}=1$ SUSY we have two scalar states in the massive chiral multiplet. These are $\left|\Omega\right>$ and $\left|\Omega'\right> = \bar{Q}_1 \bar{Q}_2\left|\Omega\right>$, which ...
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### Goldstein Chapter 6 Question

I have a question about a potential error in the $3^{\mathrm{rd}}$ edition of Goldstein's Classical Mechanics. In their exposition in Chapter 6 of small oscillations, the authors obtain the usual ...
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### Rotation of spherical harmonics

I have a question about the rotation of spherical harmonics. In Wikipedia it is mentioned that if we make a rotation in 3D space: $R\vec{r}=\vec{r}'$,then the Spherical Harmonics can be written as a ...
101 views

### How to use the definition of a rank-$2$ tensor for this kind of examples?

Suppose that, a rank-$2$ tensor transforms as \begin{align} T'^{ij}=\frac{\partial x'^i}{\partial x^k}\frac{\partial x'^k}{\partial x^l}T^{kl}. \end{align} How to use this criterion to investigate if ...
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### Multiplication of $\mathrm{U}(3)$ matrices [closed]

On this page of this paper: I am unable to understand how they multiplied the $3\times 3$ $\mathrm{U}(3)$ matrix with $T_{3,2}$, which is a $2\times 2$ matrix, in Eqs. (26) to (28). Can anyone please ...
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### Is that right that about trace of the two density matrices multiply to each other? [closed]

We have: $$\mathrm{Tr}\,(\rho \rho^{\prime})=1$$ then, is it right to say $\rho=\rho^{\prime}$?
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### Is the tensor product injective on pure quantum state vectors?

I am reading lecture notes on quantum information/computing, and the tensor product of two pure qubit states $|b_1\rangle\otimes |b_2\rangle\in\mathbb{C}^{2\times2}$ was introduced as the "...
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### Permanent operation's result

N-body fermionic systems are constructed by Slater determinant, and it is equal to Vandermonde polynomial. Are there any special polynomial for the permanent which is used to construct N-body ...
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### Decomposition of $(x \pm i y) \, Y_{l m}$ and $z \, Y_{l m}$ on spherical harmonics

Using the various algebraic properties of the associated Legendre polynomials $P_l^m(u)$ and of the spherical harmonics $Y_{l m}(\theta, \varphi)$, I was able to decompose the following expressions, ...
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### Calculating Eigenkets of Perturbed Matrix for Second-Order Correction

Q: Find the eigenvalues of the 3x3 symmetric matrix $H$ using perturbation theory where all of the elements on the diagonal of $H$ are an order greater than the elements not on the diagonal. We can ...
1 vote
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### Why does applying Ladder operators change the eigenfunction?

When applying a ladder operator to a spherical harmonic function, it spits out the function with a lower or higher magnetic quantum number. My question is how does this abide by the classical ...
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### Regarding Energy Eigenstate and Position Eigenstate

I am solving problem 14.4. (a) of Schwartz's Quantum Field Theory and the Standard Model. It is related to the simple harmonic oscillator in quantum mechanics. It asks the eigenstate of the position ...
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### Operators algebra for quantum mechanics [closed]

I am taking my first quantum mechanics course and I am a bit lost in operators algebra. These are the main questions I have: Why can we write this kind of equations? $$Ô \psi = o\psi$$ What I mean ...
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### Rotating a system

brekely physics book chapter 2 page 30 , a question about rotating a system by $\frac{\pi}{2}$ around the z axis clockwise direction and writing vectors according to the new axis after rotation ...
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1 vote
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