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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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9
votes
0answers
103 views

Does the trace distance specify a unique state

In quantum information, we frequently use the trace distance (see definition) to look at how similar two states are. If I had a known complete set of states $\{\rho_i\}$ and some unknown state $\...
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1answer
104 views

How is every finite linear combination of eigenstates of H a bound state?

This is all i could find in my lecture notes about this. Its not very useful tbh. How can i show this? Thanks
3
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1answer
330 views

Second order perturbation of a degenerate system with no first order correction

Consider the following Hamiltonian, in arbitrary units: $$ H = \begin{bmatrix} 0 & 0 & g\\ 0 & 0 & g\\ g & g & 1 \end{bmatrix}$$ where $g<<1$. It is relatively ...
2
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2answers
84 views

It is correct to say that a tensor is simply a multidimensional array of related quantities? But what about a tensor as a transformation?

It is correct to say that a tensor is simply a multidimensional array of related quantities? More specifically a tensor is a collection or tuples of vectors where every vector in the tuple represent ...
3
votes
2answers
300 views

A useful identity for Gell-Mann $su(3)$ matrices?

We have the following beautiful result for Pauli $su(2)$ matrices $$(\vec{\sigma}\cdot\vec{a})(\vec{\sigma}\cdot\vec{b}) = \mathbb{I} ~\vec{a}\cdot\vec{b} + i (\vec{a} \times \vec{b}) \cdot \vec{\...
1
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2answers
153 views

Why do the density operators span the whole operator space $\mathcal{B}(H)$?

The convex set of density operators on a finite-dimensional Hilbert space $H$ defined by $$\mathcal{D}(H):=\{\rho\in\mathcal{B}(H)\,|\,\rho\geq 0,\, \operatorname{tr}\rho =1\},$$ This set is said to ...
2
votes
2answers
214 views

Matrix multiplication and tensorial summation convention

I'm reading this introduction to tensors: https://arxiv.org/abs/math/0403252, specifically rules concerning summation convention (ref. page 13): Rule 1. In correctly written tensorial formulas free ...
0
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0answers
55 views

Eigenvalues of Maxwell or Lorentz Matrices

It is actually a mathematical question. I encounter some problems on the properties of eigenvalues of Maxwell matrices $F^{\mu}_{\ \ \ \ \nu}$. If we are working in Euclidean space, the answer is easy,...
0
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1answer
178 views

Hadamard transformation acting on Y axis of the Bloch Sphere

By definition, Hadamard transformation (acting on a qubit) maps the unit vector in the $Y$ axis direction of the Bloch Sphere ($S^2$) to its negative, equivalent to a rotation of $\pi$ rad around $X+Z$...
1
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1answer
49 views

Multilateration of Sound in 3D Space

TL:DR - How can you find the 3D coordinates of a emitter than transmits an impulse signal? STORY: I'm working on something to improve my bird-watching. I've got a camera that can take pictures of ...
0
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1answer
101 views

Simple, $3 \times 3$ Hamiltonian with negative eigenvalues and $\langle H \rangle=0$

I have the following exercise: Consider a three-dimensional system whose Hamiltonian is described by the following matrix: $$\begin{bmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 &...
1
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1answer
55 views

Basis-free, non-power series definition of the exponential of linear operator?

Given an arbitrary linear operator $A$ (be it real, complex or whatever), how can the exponential of it ($e^A$) be defined naturally, without stuff like power series? The exponential for regular ...
0
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2answers
120 views

The Eigenvalue problem

In page 32 of R. Shankar's Principles of Quantum Mechanics is given the eigenvalue problem: We begin by rewriting Eq. (1.8.2)as $$(\Omega - \omega I)|V\rangle =|0\rangle \tag{1.8.3}$$ Opening ...
5
votes
2answers
240 views

Difference between physicist's vector and mathematician's vector

Mathematically a vector is defined as an element of vector space which obeys certain properties. While reading about the special theory of relativity, I came to know about another definition of ...
2
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1answer
83 views

Cross product of vectors

I am unable to comprehend the following lines given in page 657 of Shankar's Principles of Quantum mechanics: One tricky point: The cross product is defined to be orthogonal to the vectors in the ...
0
votes
1answer
71 views

Result of bra-kets with multiple spins

I'm working on an exercise where I'm calculating the transition probability of a system consisting of two spin-1/2 particles. This system has a Hamiltonian $$ \hat{H} = H_z (\hat{S}_{1x}+\hat{S}_{2x}) ...
0
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2answers
291 views

How to prove this identity for the complex conjugate of linear operator?

I want to prove the following identity: $$\langle v|\Omega^{\dagger}|u\rangle = \langle u|\Omega | v \rangle^*$$ How should I go about this? I believe I can prove it when $\Omega$ is hermitian, but ...
1
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0answers
70 views

Why is hermitivity not preserved when Fourier transforming a matrix?

Consider a somewhat big Hamiltonian matrix $H$. I wanted to get its momentum representation, so I took the discrete Fourier transform or more specifically applied a FFT algorithm on it using Python. ...
0
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0answers
37 views

Eigenstates of large system

How does one calculate the eigenvalues of a unitary 3 by 3 block circulant matrix where $$U = -\frac{i}{3} \begin{pmatrix} \Lambda_{1} & \Lambda_{2} & \Lambda_{3} \\ \Lambda_{3} & ...
4
votes
2answers
150 views

Construct an SO(3) rotation inside the two SU(2) fundamental rotations

We know that two SU(2) fundamentals have multiplication decompositions, such that $$ 2 \otimes 2= 1 \oplus 3.$$ In particular, the 3 has a vector representation of SO(3). While the 1 is the trivial ...
1
vote
1answer
94 views

Matrix representation of spin-1/2 system

Problem 1.19 from Sakurai's Modern Quantum Mechanics asks you to find $\left< (\Delta S_x)^2 \right> = \left< s_x^2 \right> - (\left< s_x \right>)^2 $ in the $S_z +$ state. However, ...
1
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3answers
478 views

Why vector decomposition works with forces?

We have this : I know. We decompose the red force and we deduce that the block will slide down on the inclide plane. But why a purely abstract concept as the one of vector works well in physics, and ...
-2
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1answer
76 views

Proving that $⟨\phi_{n} \mid a + b \mid\phi_{n} ⟩ = ⟨ \phi_{n} \mid a \mid\phi_{n} ⟩ + ⟨ \phi_{n} \mid b \mid\phi_{n} ⟩ $ [closed]

I am trying to prove to myself that this relation is true: $$⟨\phi_{n} \mid a + b \mid\phi_{n} ⟩ = ⟨ \phi_{n} \mid a \mid\phi_{n} ⟩ + ⟨ \phi_{n} \mid b \mid\phi_{n} ⟩ .$$ Where $a$ and $b$ are 2 ...
0
votes
1answer
110 views

How to determine basis transformation for state vectors given a Hamiltonian written in two bases?

A same system can be written in two basis: $$\psi^\dagger H_1\psi=\begin{bmatrix}c_k^\dagger&c_{k+\pi}^\dagger \end{bmatrix} \begin{bmatrix}A&iB\\-iB&-A \end{bmatrix} \begin{bmatrix}c_k\\...
0
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1answer
58 views

Another question about derivaton of Lorentz Tranformations

This question follows the same ideia of another question of mine: Doubts on derivation of Lorentz Transformations So, the problem here is the following phrase: The remain transformation equations ...
1
vote
0answers
272 views

Time Reversal Operator in Sakurai's Book

In Sakurai's Modern Quantum Mechanics, Eq(4.16) says that $$\begin{align} |\alpha\rangle &= \sum_{a^\prime}|a^{\prime}\rangle\langle a^{\prime}|\alpha\rangle \overset{K}{\rightarrow} |\tilde{\...
3
votes
2answers
2k views

What is the standard definition of “projector”, “projection” and “projection operator”?

What is the precise meaning of "projector", "projection" and "projection operator"? I always thougth those two terms are synonyms, but I have seen both used in a quantum optics paper where the former ...
3
votes
0answers
72 views

Why are neutrino flavour eigenstates expressed in terms of the elements of the complex conjugate of the PMNS matrix?

If we have $$ \begin{pmatrix} \nu_e\\ \nu_{\mu} \\ \nu_{\tau} \end{pmatrix} =\begin{pmatrix} U_{e1} & U_{e2} &...
0
votes
1answer
104 views

Unitary Matrix for Block Diagonalization

What is the method used to determine the unitary matrix that block diagonalizes the symmetry operations of a group? I'm sure there's multiple ways to go about it, but finding a source on how one block ...
0
votes
1answer
71 views

Question about transformation matrix for spin 1 particle

For spin one system the bases transformation of $y$ axis: $|1>_y=\frac{1}{2}|1>+i\frac{1}{\sqrt{2}}|0>-\frac{1}{2}|-1>$ $|0>_y=\frac{1}{\sqrt{2}}|1>+0|0>+\frac{1}{\sqrt{2}}|-1&...
4
votes
2answers
139 views

What is the difference between $T^\mu{}_\nu$ and $T_\nu{}^\mu$?

I do understand why the horizontal order matters for indices on the same vertical position, e.g.: $$T\left(V_{(1)},V_{(2)}\right) = T_\color{red}{\mu\nu}V^\mu_{(1)}V^\nu_{(2)} \neq T_\color{red}{\nu\...
0
votes
3answers
201 views

Equality of operators [closed]

$\def\bra#1{\langle#1|}$ $\def\ket#1{|#1\rangle}$ I'm trying to prove this little result. I'm sure its proof is very simple, but I'm stuck. Could you please give me a hint? Thanks. Let $\bra{\psi}A\...
1
vote
0answers
130 views

What is the trace of the (spatial) projection tensor?

In the discussion on gravitational waves in Moore's A General Relativity Workbook, Box 33.4 on page 391 introduces the projection operator $$P^j_m\equiv\delta^j_m - n^jn_m$$ where $\vec n$ is a unit ...
2
votes
1answer
136 views

Spin 1/2 system in the algebraic approach to quantum mechanics

I'm trying to understand the $\ast$-algebra approach to QM and QFT, and so I have decided to first try to understand how this works in one of the simplest systems: a particle with spin 1/2. This is ...
2
votes
2answers
128 views

Why does angular momentum point up for a counterclockwise rotation? Why not down?

I am a high school student, and lately, I am founding things a bit perplexing on some topics concerning cross product in physics. In angular momentum, we learned that the direction of an object ...
5
votes
3answers
283 views

Ladder operator identity for $\langle n | (a+a^\dagger)^k | m \rangle$

I would like to know if there is a convenient identity (and what it is) for $$\langle n | (a+a^\dagger)^k | m \rangle$$ where $| n \rangle, \, | m \rangle$ are energy eigenstates of a simple ...
0
votes
3answers
140 views

Postulates of inner product

In quantum mechanics, two fundamental properties of inner products (J.J Sakurai) Chapter 1.2, are: $\langle \alpha|\beta\rangle = \langle \beta|\alpha\rangle^*$ $\langle \alpha|\alpha\rangle \ge 0$ ...
-1
votes
1answer
109 views

What is norm of matrix element in Fermi Golden Rule

Fermi Golden Rule says: $\Gamma \propto |M_{ij}|^2$ I know how to get $M_{ij}$, but how do I proceed? How do I take a norm of Hermitian matrix? There is no clear (to me) definition in the internet ...
0
votes
2answers
329 views

Eigenvectors and eigendecomposition of Pauli matrices, why isn't there many?

Say we are finding eigenvectors of $\sigma _z$, the eigenvalues are $1,-1$ so filling into the eigenvalue equation $\sigma _z (a,b)=(a,-b)=1(a,b)$ and we find that $b=0$. I am confused about why we ...
0
votes
1answer
215 views

Constructing a CPTP-map on one density matrix using another

My question is: If one is given two density matrices $A$ and $B$, is there a way to use the first to construct a CPTP-map (quantum channel) acting on the on the other? I thought that Stinespring ...
0
votes
1answer
313 views

Why is the projection operator corresponding to $\tilde M$ given by $P_m\otimes I_B$?

Nielsen and Chuang, Chapter 2 (Box 2.6): Suppose $M$ is any observable on a system $A$, and we have some measuring device which is capable of realizing measurements of $M$. Let $\tilde M$ ...
0
votes
1answer
69 views

Which set of basis states can a quantum system of qubits actually collapse to?

I was watching a video on "How Does a Quantum Computer Work?". I'm confused about what they mean by: "Although the qubits can exist in any combination of states, when they are measured they must ...
0
votes
1answer
138 views

Uniqueness of simultaneous eigenstates of two linear operators

I was solving a homework problem where the question gives the representation of two operators in matrix form, in some arbitrary set of basis vectors. It then asks to find the simultaneous eigenstates ...
1
vote
0answers
84 views

Doubt in the transformation in R Shankar's Quantum Mechanics

On page number $19$ of the book Principles of Quantum Mechanics by R. Sankar, The author describes a transformation, essentially a rotation of the axes by an angle of $\frac{\pi}{2}$ counterclockwise, ...
1
vote
1answer
153 views

Requirements for the inner product in Hilbert space

I was reading the following text about the mathematical foundations of quantum mechanics when I stumbled upon the following conditions that the inner product must satisfy: I woud like to understand ...
-1
votes
1answer
149 views

Velocity vector $= 2î +3j +4k$. How is this vector 1D despite all the three axes being involved?

velocity vector = 2î + 3j + 3k How can this vector be 1D despite there being all the three axes involved
1
vote
1answer
59 views

Solve eigenvalue problem with known constraint on one of the Eigenvalues

I have the following problem and would appreciate any help. I have a real, symmetric matrix M given by $$M=\begin{pmatrix} m_{11} & m_{12} & m_{13} & m_{14} \\ m_{12} & m_{...
0
votes
1answer
107 views

Does the dimensions of the wavefunction vector in Hilbert space depend on the number of eigenfunctions it is a superposition of?

I've seen people say that wavefunctions represented as vectors in a Hilbert space can (but don't have to) have infinite dimensions. So if a state vector requires X number of basis eigenfunctions ...
7
votes
1answer
526 views

What's the physical meaning of the kernel of density matrix?

The kernel of this linear map is the set of solutions to the equation A x = 0, where 0 is understood as the zero vector. But what's the physical meaning of the kernel of density matrix?
0
votes
1answer
73 views

Assumption of the form of kinetic energy

In my lecturer's notes on Lagrangian mechanics in the chapter on normal modes they state the following: Instead of rigid constraints, let us now consider a situation where the constraints are ...