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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114 views

Anti-commutator version of Zassenhaus formula

The Zassenhaus formula goes like $$ e^{t(X+Y)}= e^{tX}~ e^{tY} ~e^{-\frac{t^2}{2} [X,Y]} ~ e^{\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )} ~ e^{\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X],Y] + 3[[[X,Y],Y],...
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824 views

Schmidt decomposition of entangled state [closed]

I have a problem with some homework our teacher assigned. I have to find the Schmidt decomposition of the entangled state $$\lvert\psi\rangle_{A,B}=\frac{1}{2}\lvert0\rangle_{A}\lvert0\rangle_{B}-\...
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137 views

Feynman diagrams for eigenvalue perturbation theory

I posted this question in MathOverflow but was not lucky with the answers, so wil try here. Suppose I have a matrix given by a sum $$A=D+\epsilon B$$ where $D$ is diagonal and $\epsilon$ is small, ...
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3answers
292 views

What is the role of determinant and trace of matrices in physics? [closed]

There is vast area of physics where we have to use matrices.It is not only to do the mathematical problems in physics but also to produce a physical realization of an operation. I think matrices carry ...
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315 views

Charge conjugation in chiral representation

I'm reading Maggiore's book and I got to the part of charge conjugation symmetry for Dirac spinor. I get that the definition of charge conjugation is representation-dependent, however I couldn't find ...
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1answer
223 views

Property of the adjoint operator in the array element

In Quantum Mechanics how can I prove this property? $$<\psi|A^{\dagger} |\phi>=<\phi|A|\psi>^{*}$$
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103 views

Demonstration of a property of the adjoint operator in Quantum Mechanics

I would like to demonstrate the following property: $$(|\psi\rangle\langle\phi|)^{\dagger}=|\phi\rangle\langle\psi|$$ The other properties that concern the adjoint operator I have already been ...
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1answer
94 views

Are there multiple equivalent ways of writing an eigenstate?

So my problem here is that I'm confused about how to solve for the eigenstates corresponding to certain eigenvalues. For my problem I have the Hamiltonian $$ H=E_0 \begin{pmatrix} 3 & 5i \\ -5i ...
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1answer
63 views

Constructing an arbitrary 2-Qbit state

I am reading a book on quantum computing. The author is constructing an arbitrary 2-Qbit state from unitary transformations. I need help understanding on step in his logic. He starts by noting that ...
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793 views

The formal way to simultaneously diagonalise two Hermitian operators that Commute?

Given two Hermitian operators $A$ and $B$, such that $[A,B] = 0$, if one [or both] of these operators are degenerate, how does one define a formal way of going about simultaneously diagonalising both ...
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116 views

Is a basis vector always unit-length in a wave function?

I'm currently studying wave functions and I came across an assertion, that $$\psi(x) = \left<x \middle| \psi \right>$$ is a projection of $\psi$ onto $x$. The vector projection being defined ...
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4answers
284 views

Any 'Metric' That Takes a Ket to a Bra?

Due to glaring similarities between 'the vectors and one-forms of Riemannian geometry' and 'the kets and bras of linear algebra', I am curious as to whether there exists an object analogous to the ...
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1answer
107 views

Are there linear operators and vector spaces in classical physics?

Linear operators and vector spaces form the backbone of the operator formulation of quantum mechanics. I want to ask are there operators in classical physics too? Are these operators defined on some ...
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762 views

What is exactly eigenfunction in quantum mechanics? [closed]

What is exactly eigenfunction in quantum mechanics i understand eigenvalue and eigenvectors but does it mean boundary condition in quantum mechanics
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141 views

Doubt about a discussion of Tensors and coordinate independence

Consider the following observation: "The importance of tensors is due to the fact that they offer the opportunity of the coordinate-independent description of geometrical and physical laws. Any ...
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1answer
157 views

How to write a research paper properly [closed]

I'm a student, 15yrs, in 10th standard. I live in India. I want to become a pure mathematician or a Theoretical Physicist. I want to know what is the correct protocol to write a research papers in ...
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1answer
318 views

Are eigenvalues in quantum mechanics related to eigenfunctions (in the PDE sense) or to linear algebra and eigenvectors?

I'm in 10th grade and a beginner in the amazing world of quantum physics, I want to become a mathematician but I like quantum mechanics as well. The eigenvalues in Schrödinger wave equation used to ...
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224 views

Determining a real qubit Choi matrix given two states

I've been curious recently about considering quantum channels whose Choi matrices are strictly real in the computational basis. Given the Choi matrix of a quantum channel $$C=\sum_{i,j=1}^d \mathcal E(...
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2answers
124 views

State of a linear vector space

I was going through a lecture on linear vector space. It embodied all the examples of a linear vector space like it could be function spaces or vectors or real entities. But what exactly is meant by ...
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2answers
292 views

How does a linear operator act on vectors that are not eigenvectors of the operator?

I feel like I have the wrong end of the stick in thinking about quantum state vectors, and so wish to try and clear this up. If I prepare a single spin in the up state (along the z-dimension), and ...
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542 views

Commutativity vs Compatibility

As far as I know, two compatible observables have a complete set of common eigenvectors, and using this fact, one can prove that their corresponding operators are commutative. Well now is the converse ...
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1answer
126 views

How do tensors point in multiple directions?

I have been learning about tensors recently but I am still confused about one thing. I understand that tensors are just objects whose elements transform in a fixed way so that the object is coordinate ...
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223 views

Why is it that a coupled mass-spring system will always produce a diagonalizable matrix?

If you take a system like the one in the image, and you do the $y=x'$ trick to turn it into a first-order system of equations ($x_{1}$ or $x_{2}$ being the displacement of the mass $m_{1}$ or $m_{2}$ ...
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2answers
429 views

Which phase shift gate form is correct?

I am trying to figure out the matrix for a gate I'm going to be implementing - a mirror, i.e., a $180^{\circ}$ phase shift. Quantiki gives $$R(\theta)=\begin{bmatrix}1&0\\0&e^{2\pi i\theta}\...
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Has the asymptotic theory of eigenvalues of infinite matrixes already been applied to vibrations analysis?

My question is reffering to the masses/springs model of a material, like the one presented in this article http://www.laserpablo.com/baseball/Kagan/UnderstandingCOR-v2.pdf. If one treates a long ...
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1answer
155 views

How can eigenvectors of a Hermitian matrix be entangled?

You have a tensor product space $H_1 \otimes H_2$. Any vector $w$ in this space has a Schmidt decomposition: $$ \mathbf{w} = \sum_{i} \alpha_i \mathbf{u_i}\otimes\mathbf{v_i} $$ Vector $w$ is not ...
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68 views

Bounds on the effect of strong coupling

I am interested in bounding the effects of system-environment interaction. Suppose I have an initial state $\rho \in \mathcal{H}_S \otimes \mathcal{H}_E$ where the system and environment might be ...
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1answer
116 views

A silly doubt on notation of Hartle's GR book

Quoting from the text [...] Evaluating (20.46) in such coordinates is just like evaluating the derivative of a function (20.1): $$\left( \mathbf{\nabla_t v}\right)^\alpha = t^\beta \frac{\...
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1answer
178 views

Some beginner doubt about tensors

I have a question about tensors. Okay, you could say that this question would fits more properly on Math.StackExchange but, here it's more cozy. Anyway: A vector, as we all learned in first year, ...
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1answer
333 views

Proving that unitary transformations preserve orthonormality in bracket notation

I'm trying to prove that unitary transformations preserve orthonormality for quantum chemistry course. There is a proof for this in many mathematical texbooks, but non of them is dealing with matrix ...
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2answers
434 views

Zee's use of Kronecker Product in “QFT in a Nutshell” to represent Dirac matrices

In his book Quantum Theory in a Nutshell (2nd edition, p. 94), Zee describes the Dirac gamma matrices and lists a representation using Pauli matrices and the identity matrix. For example he writes $$ ...
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1answer
496 views

Fermionic statictics in $SU(2)$ slave-boson representation

One of the $SU(2)$ slave-boson decompositions has been introduced by X.-G. Wen and P. A. Lee in PRL, 76, 503 (1996). (A generic recipe for constructing the SU(2) slave-particle framework has been ...
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108 views

How to prove that $\hat{A}\phi = \phi^2$ is a linear operator?

In my textbook it is written that an operator $\hat{A}$ is linear if it satisfies the condition: $$ \hat{A}(c\phi)= c\hat{A}\phi $$ Then they ask us to prove whether $\hat{A}\psi=\phi^2$ is a linear ...
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1answer
55 views

Finding matrix representation of an operator given an eigenproblem [closed]

I have to find the matriz representation of $L_z$ to finish a problem and I was given these two informations: $$L_z|+>=\hbar|+> \ L_z|->=-\hbar|-> $$ and $$|+> = \frac{1}{\sqrt{2}} \...
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535 views

Inverse of a Hermitian Operator

The momentum operator $p = i\frac{\mathrm d}{\mathrm dx}$ (with $\hbar = 1$) is hermitian. Hence its imaginary exponential i.e. $U = e^{ip}$ must be unitary. $U$ being a unitary operator, must have a ...
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143 views

Matrix of any special unitary transformation in two dimensions

I want to show that every special unitary transformation in two dimensions can be written as the matrix $$ U = \left(\begin{array}{cc} e^{i(\delta + \varphi)}\cos\theta & i~e^{i(\delta - \varphi)}...
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1answer
420 views

How to prove this inequality for the Hamiltonian operator?

I am trying to prove the following: $$\langle\psi|\hat{H}|\phi\rangle\langle\phi|\hat{H}|\psi\rangle-\langle\psi|\hat{H}|\psi\rangle\langle\phi|\hat{H}|\phi\rangle\leqslant0.$$ I tried some ideas ...
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367 views

Diagonalizing a fermionic Hamiltonian

My question somewhat builds off of this answer. For a fermionic Hamiltonian and diagonilzaition of the form $$H = \sum_{i,j} G_{ij}a_i^\dagger a_j = A^\dagger G A = A^\dagger U^\dagger D U A = F^\...
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1answer
110 views

What eigenvector-like tools are there for analyzing tensors of rank three and higher?

If I have a rank-two tensor that I want to analyze ─ say, an electric quadrupole moment, or a moment of inertia ─ it can often be very easy to analyze by moving to its principal-axes frame: one ...
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1answer
497 views

Finding correct zeroth order eigenstates in degenerate perturbation theory when degeneracy is not lifted by first order correction

I have recently learned that if the degeneracy is not lifted in the first order in degenerate perturbation theory, one has to diagonalize a different matrix that plays a similar role to the first ...
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1answer
511 views

How does the addition of two wavefunctions develop in time?

Two time dependent wavefunctions: $\Psi _1(t)= \psi_1*exp(\frac{-i * E_1}{\hbar}*t)$ $\Psi _2(t)= \psi_2*exp(\frac{-i * E_2}{\hbar}*t)$ Both a solution to the timeindependent (note "in") ...
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2answers
112 views

Minimum uncertainty Gaussians as basis for Hilbert space?

In a class on QM the lecturer (near min. 47) briefly says that gaussians of minimum uncertainty can form a basis for Hilbert space, meaning that any element of the space can be expressed as a linear ...
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260 views

How can I compute the eigenstates of a tight-binding Hamiltonian describing a system?

I have the following set-up in a 3-site tight-binding system: \begin{align} i\hbar\frac{dc_1}{dt}&=-Ac_2,\\ i\hbar\frac{dc_2}{dt}&=-Ac_1-Ac_3,\\ i\hbar\frac{dc_3}{dt}&=-Ac_2, \end{align} ...
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1answer
192 views

Eigenvalues of the Klein-Gordon operator

If I've understood what I've read correctly, the eigenvalues of the Klein-Gordon (KG) operator $\Box+m^{2}$ are $-p^{2}+m^{2}$, but how does one show this? Naively I assumed that the eigenfunctions ...
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2answers
358 views

Significance of using Eigenvalues / Vectors in QM?

A fundamental idea in Quantum Mechanics is that observable quantities are represented by linear, hermitian operators. Why is it that we represent distinguishable states as the eigenvectors of ...
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1answer
576 views

Product of operators, eigenvalues

If I've two hermitian operators let's say A and B,then their eigenfunctions(vectors) form a basis... If I now take a product of both of them and create a new operator AB (composition of both), that ...
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37 views

How is the Gastner-Newman equation implemented to create value-by-area cartograms? [closed]

There is a paper called "Density-equalizing map projections: Diffusion-based algorithm and applications" by Michael T. Gastner and M. E. J. Newman, which explains their algorithm (which is based in ...
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144 views

Transformation of a covariant tensor and general interpretation

If we consider a coordinate transformation defined by the rule $$x^{\mu}\rightarrow x'^{\mu}=x'^{\mu}(x)$$ If we consider the old coordinates as functions of the new ones, then the coordinate ...
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1answer
287 views

Eigenstates and Standard Deviation of an Operator

I have been asked to prove that, for an Hermitian operator A, its standard deviation $\Delta_{\varphi}$ in a certain state $|\varphi\rangle$ is equal to $0$ if, and only if, said state is an ...
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33 views

How do I compute the 4-component efficiency vector for an Nx4 Mueller Matrix where N > 4?

I am trying to use this paper by Manuel Collados to better understand polarization analysis to determine the efficiency of generalized Mueller Matrices. Equation #8 (on page 5 of the paper) works only ...