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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Harmonic Oscillator Eigenket Notation

I'm reading the $3^{\mathrm{rd}}$ edition of Sakurai and Napolitano's Modern Quantum Mechanics, and I have a brief question about the notation used to describe the eigenstates of the harmonic ...
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Is amount of substance fundamentally a scalar quantity? (in the mathematical sense of scalar)

Reading the SI (and ISO) standard for units and quantities, I'm currently puzzled by something very subtle. If I can see and understand why we talk about scalars, vectors, and tensors in the context ...
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Distinguishing different senses of 'vector' [closed]

Two mutually orthogonal unit vectors acting at a point $p$, produce a resultant, whereas the two orthogonal unit basis vectors at the origin do not, why?
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Vectors and vector bases [closed]

If a unit $x$ velocity and a unit $y$ velocity are simultaneously applied to a point $p$ at the origin, the point $p$ will move at speed $ \sqrt{2} \enspace $ along the line $y=x$, to point $A = [0,1]...
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Why do systems of $n$ coupled oscillators have $n$ normal modes?

Consider a linear system of $n$ differential equations with constant coefficients corresponding to a physical scenario where I have $n$ coupled oscillators (like $n$ masses attached by springs in ...
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Connecting asymptotic scattering solutions to short-distance numerical potentials

Problem I'm dealing with a one dimensional quantum mechanical scattering problem in a finite region, say $x\in\left[0,L\right]$. At first, this problem is defined as a finite-difference problem, i.e., ...
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Eigenvalue of transfer matrix in Shankar (3.3.4), p.34

In the book of Quantum Field Theory and Condensed Matter written by Shankar, (3.3.4), p.34, there defined a transfer matrix $$T=\left( \begin{array}{cc} 1 & e^{-2K} \\ e^{-2K} & 1 \\ \end{...
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Is quantum cloning $|\psi\rangle|\psi_1\rangle|\psi_2\rangle|C\rangle\to e^{i\alpha}|\phi\rangle|\psi\rangle|\psi\rangle|C'\rangle$ prohibited?

I think the no-cloning theorem is too restrictive, as in, $$|\psi\rangle |\phi\rangle\to e^{i\alpha}|\psi\rangle|\psi\rangle \tag{1}$$ does not allow for any arbitrariness in the final state. Instead, ...
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Measuring quantum states without violating no-cloning

In Nielsen and Chuang exercise 2.64, the following problem is given: Suppose Bob is given a quantum state chosen from a set $\{ \lvert \psi_1 \rangle, \ldots , \lvert \psi_m \rangle \}$ of linearly ...
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Transpose of a bilinear in Einstein notation

In Einstein notation we can take generic 1-vectors $x, y$ and (1,1) tensor $M$. As we know $x_{\mu}$ represents $x^{T}$, i.e. row vector (a co-vector), while $x^{\mu}$ is a column vector. So we can ...
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Does $\exp(-i \theta \sigma_m \otimes \sigma_n)$ represent a rotation operator?

It is well known that $\exp(-i \sigma_k \theta)$ where $\sigma_k$ $(k=x,y,z)$ is a Pauli matrix, represents the rotation operator about $k$-th axis. What physical interpretation does $\exp(-i \theta \...
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Is the following map linear over the space of density matrices?

I have a map $\mathcal{N}$ from the space of two-qubit subnormalised density matrices $\mathcal{S}(\mathcal{H}_2 \otimes \mathcal{H}_2)$ to itself (positive operators with trace between 0 and 1). ...
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Why do most introductory texts on QM use the Schrödinger formulation rather than Heisenberg's matrix mechanics? [closed]

I know they're mathematically equivalent, and that makes intuitive sense, seeing as linear differential equations can in general be solved using matrices and other linear algebra approaches. In fact, ...
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Problem with proving the invariance of dot product of two four vectors

I am having a spot of trouble with index manipulation (its not that I am very unfamiliar with this, but I keep losing touch). This is from an electrodynamics course - we're just getting started with 4 ...
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Why this definition of the trace?

It is common in quantum mechanics textbooks (e.g. Ballentine page 15) to define the trace of an operator as the sum of its diagonal elements in an orthonormal basis in particular. Why is this ...
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Resolution of force vectors

So, I have had this query for like the longest time, ever since I first studied this topic. So, take a force vector $F$ that is making an angle of say, 30 degrees with the horizontal axis. Now, I want ...
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Technical question in how to correspond operators to dynamical variables [closed]

I have following question. Here is the book I use for QM. I understand so far until the step of 3.43. Can somebody tell me how he arrived at the step of 3.43? I tried to expand the eigenvector in 3.42 ...
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Do the states in a decomposition $\rho=\sum_i p_i |\phi_i\rangle\!\langle \phi_i|$ need to be orthonormal?

On Wikipedia it says: Let $\mathcal H_S$ be a finite-dimensional Hilbert space, and consider a generic (possibly mixed) quantum state $\rho$ defined on $\mathcal H_S$, and admitting a decomposition ...
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Condition of the product of 2 operators being an observable [duplicate]

I'm trying to understand a bit the conditions of operators commuting, or themselves being an observable. Here I have the operator $\hat{A}$ which has Eigenvalues $-1,+1$ and Eigenstates $|u_1\rangle, |...
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What is the Point Spread Function of a pinhole camera?

Is the geometric spread of a pinhole camera a uniform blur convolution? Pinhole cameras have blur due to a. wave effects (diffraction) and b. ray effects (geometric properties). This question ignores ...
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Existence of unitary operator that transforms phase gates

Let me first introduce the entire problem: Let $H$ be an Hermitian operator, $W$ be an Unitary operator and let $S$ be the standard phase gate: $\begin{pmatrix}1 & 0 \\ 0 & i\end{pmatrix}$. ...
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Unitary equivalence between Hermitian operators

Take two (non-zero) Hermitian operators $A$ and $B$. I want to proof that there exists no unitary operator $W$ such that: $$W^{\dagger}AW = A + B$$ For my research I proved this for some specific case ...
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Inverting the Propagator [closed]

I would like to know how I can invert the following expression: $$ S^{\mu\nu}=(\!\!\not{p} -m)\eta^{\mu\nu}+\gamma^{\mu}\!\!\!\not{p}\gamma^{\nu}+m\gamma^{\mu}\gamma^{\nu} \ , $$ to get $(S^{−1})^{\mu\...
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Simultaneous eigenkets of two Hermitian and anti-commutative linear operators [closed]

I'm working through the problems at the end of the first chapter of the third edition of Sakurai and Napolitano's Modern Quantum Mechanics, and I've hit a snag with problem 1.18: "Two Hermitian ...
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How to prove formula for contraction of a vector with a Multivector?

I am currently reading "Space-Time Algebra" by David Hestenes and the following proof is given for the formula of contraction of a vector $a$ and multivector $b_1 \wedge b_2 \wedge b_3 \...
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Understanding conclusion of Proof that a matrix that commutes with all gamma matrices is proportional to the identity matrix

I am trying to understand why a matrix $M$ that commutes with all gamma matrices $\gamma^\mu $ is proportional to the Identity matrix. I am following example 3.18 in Voja Radovanovic's book on solved ...
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Quantum state in continuous basis [duplicate]

If I have an arbitrary state $|\psi\rangle$ and want to represent it in a continuous basis, for example the position basis in $x$-direction, I will get $$|\psi\rangle = \int dx\, \langle x|\psi\rangle|...
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On the definition of positive linear superoperators on Hilbert spaces

Consider a linear map between linear maps of a Hilbert space, $\mathcal{E}: \mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathcal{H})$. The standard definition I have encountered is that $\mathcal{E}$ is ...
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Hawking & Ellis: typo on page 16?

On page 16 of The Large Scale Structure of Space-Time (1973) by Hawking and Ellis, they describe the basics of tangent spaces. This line appears near the top of the page: Thus the tangent vectors at $...
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Coherent state basis

I'm learning about coherent states in a more in depth lesson the the quantum harmonic oscillator. Coherent states are eigenstates of the lowering operator. In my head this is just saying: since any ...
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Quaternions as rotation generators

The following exercise appears in Geometric Algebra for Physicists by Chris Doran and Anthony Lasenby in section 1.8. The unit quaternions $i, j, k$ are generators of rotations about their ...
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The Heisenberg Picture [closed]

In the Heisenberg picture, observables, rather than states, undergo unitary evolution. Can an observable turn into an $un$observable as a result of unitary time evolution? That is, If $U$ is a unitary ...
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How do you decompose a general tensor into a sum of outer products?

Dirac's book "General Theory of Relativity" says on p. 2 that a general rank-2 tensor can be written as a sum of outer products: $$ T^{\mu\nu} = A^\mu B^\nu + A'^\mu B'^\nu + A''^\mu B''^\nu ...
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Matrix representation of Grassmann variables and Berezin Integrals

In this question, the problem of finding matrix representations for a set of Grassmann variables is discussed. How can this representation be used in Berezin integrals or Grassmann derivatives? Can ...
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Clarification on operators and completeness relation

Given the momentum operator $$\hat{p} = -i\hbar \partial_x$$ we can apply the completeness relation to get $$\hat{p} = (-i\hbar \partial_x)(\sum_{a'}\vert a' \rangle \langle a' \vert) = -i\hbar(\sum_{...
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Why are the coefficients of vectors (that are not eigenstates) the probability amplitudes?

In the adiabatic approximation, we assume that: are the solutions of the eigenvalue problem: We know however that the ψn(t) 's here may not be actual solutions to the time dependent Schrodinger ...
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Covariant vectors and change of basis matrix

I'm studying the basics of special relativity and I've been having some difficulties with the concept of contravariant and covariant vectors. Now, at least in the basic way in which we introduce them ...
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Hamiltonian as a matrix and its elements [closed]

Let us consider an electron in an infinitely deep one-dimensional potential well of thickness L with zero potential energy at the bottom of the well. The normalised eigenfunction solutions to this can ...
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Distributing operators inside of the bra and kets confusion

I'm reading Griffiths and he has this section where he states that $|\hat{Q}f\rangle$ is mathematical nonsense and that really we should write $\hat{Q}|f\rangle$, where the latter makes more sense to ...
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Quantum Particles on a circle and Circulant matrices [closed]

I have $n$ particles on a circle with the Hamiltonian \begin{equation} H = \sum_{n=1}^N \frac{p_n^2}{2m} + \frac{1}{2}m\omega^2 \sum_{n=1}^N (x_{n+1}-x_n)^2 \end{equation} I need to find the energy ...
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Binary number as quantum observable variable

I am currently reading this article about quantum computations on set of N two-state ions. There author associates ground state with number zero $|0>$ and excited state of ion with number $|1>$ ...
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Why is the trace of the outer product of two states equal to the inner product of the two states?

Why is it that, given two quantum states $|\psi_1\rangle$, $|\psi_2\rangle$, $$\mathrm{Tr}(|\psi_1\rangle\langle\psi_2|) = \langle\psi_2|\psi_1\rangle \quad $$ I went through the equation with the ...
1 vote
5 answers
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Mathematical explanation of bra-ket notation in quantum mechanics

$\newcommand{\hp}[1]{\hphantom{#1}}$ We have the entangled state of two pairs of qubits: $$ |\psi \rangle =\frac{1}{2}|0011\rangle-\frac{1}{2}|0110\rangle-\frac{1}{2}|1001\rangle+\frac{1}{2}|1100\...
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What do special unitary groups $SU(n)$ represent geometrically?

It's frequently said that special orthogonal group $SO(n)$ represent rotations in $n$-dimensional space. What do $SU(n)$ groups represent?
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Quantum mechanics: "Representation" vs. "basis"

I am confused about the difference between the terms "representation" and "basis" of a state or operator. For example, Let us have eigen-kets of Hamiltonian $H$ denoted by $|\phi_n\...
4 votes
1 answer
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Notion of Co- and Contravariance in Dirac-Notation

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$ A (...
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Use Index Notation properly when indices are already used in identifying which bases is the matrix metric calculated with respect to

I am wondering how to apply the usual linear algebra to the rather unfamiliar case of 'matrices' with indices in special relativity or even general relativity. In particular, consider$$f=\sqrt{-\det\...
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Question on notation of eigenvectors of position operator in quantum mechanics

From the first postulate of quantum mechanics we known that the vector $|\psi\rangle$ is the mathematical entity that says, intuitively, "in a time $t$, the (state of a) system is a vector". ...
5 votes
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How to find a separable decomposition of a separable density matrix?

If I have a bipartite system of two qubits $A$ and $B$, and the density matrix $\rho$ is separable, how do I decompose it into its separable parts? That is, give $\rho$, expand it as follows: $$\rho = ...
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Can an operator map a state into a number? [closed]

In second quantization, it is often claimed $$ a_{\lambda} | \Omega \rangle = 0, \cdots \tag{2.1} $$ where $a_{\lambda} $ is an annihilation operator, $| \Omega \rangle$ is the vacuum, $0$ is the ...

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