To be used for linear algebra, and closely related disciplines such as tensor algebras and maybe clifford algebras.

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3
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2answers
61 views

What is the difference between active and passive transformations in Quantum Mechanics?

I am trying to understand what each transformation means and what their differences are but many books that don't state which transformation they are referring to make it a bit confusing to understand ...
3
votes
1answer
53 views

Basis/Projection Notation Question Quantum Mechanics

Lets say you have a inner product between two state vectors with an operator in between A|X|B. I can write this as a summation over I and j as A|i i|X|j j|B (sorry for notation). But I don't ...
0
votes
1answer
47 views

Block Diagonal Matrix Shankar Quantum Page 45

On page 45 of Shankar's intro to qm (you can find a pdf of it online if you want) he says that a specific operator has a block diagonal form because when it operates on some element of an eigenspace ...
2
votes
2answers
352 views

Commuting operators and Direct product spaces

Under what conditions is the common eigenspace of two commuting hermitian operators isomorphic to the direct product of their individual eigenspaces? When can an eigenket $|\lambda$1$\lambda$2$\rangle$...
-3
votes
1answer
69 views

Conservation of energy in the form of y=kx +b?

I am doing an experiment on method of mixtures and specific heat capacities. During my experiment I have to use the graph as a means of finding the SHC of the subject. So I did the experiment with ...
0
votes
1answer
50 views

Can a quantum mechanical system have more than one wave-function?

I was told that a quantum mechanical system is completely determined by its wave function. But superposition principle says that given two wave functions of some system, a linear combination of them ...
-2
votes
0answers
24 views

Body Angular Velocity

I'm working with rotation matrices and body angular velocities and got a bit confused. I'm doing a yaw, pitch, roll rotation like here $$ \begin{bmatrix} cy & -sy & 0 \\ ...
5
votes
2answers
185 views

Why do we use orthogonal axes?

I have been asked several times that “why do we use orthogonal axes in coordinate systems?” and I was always replying that “because of simplicity”. But, today morning, someone asked me that question ...
0
votes
1answer
55 views

Resistance Distance in large electrical networks

I am not tremendously familiar with electrical circuits (I have some memories, but too long ago) and now I have come accross a problem where I need to compute the resistance distance in a graph. So, ...
3
votes
1answer
131 views

Confusion About Operators

Hello I am currently studying introductory QM and am confused about bases and operators. If I have an operator $\hat{Q}$, does this represent a change of basis matrix? In other words, does $\hat{Q} | \...
0
votes
0answers
40 views

Vector addition over orthogonal group [migrated]

I am working on a fun problem. The problem is to solve the following equation: $O_1x_1+O_2x_2=x_1+x_2$, where $x_1, x_2 \in \mathbb{R}^2$ are known and $O_1,O_2 \in \mathbb{O}(2)$ are unknowns. [$\...
0
votes
0answers
23 views

Are there explicit formulas for the eigenvalues and eigenvectors of a generic 4x4 density matrix? [migrated]

I have a 4x4 density matrix all of whose elements are nonzero. Its form is $$\begin{pmatrix} a & b & c & d \\ b^* & e & f & g \\ c^* & f^* & h & j \\ d^*...
2
votes
1answer
36 views

Find force centroid from forces and moments [closed]

A force $\vec F = (F_x,F_y,F_z)$ is applied to a point that its position vector is $\vec r = (x,y,z)$ generating a moment $\vec M = (M_x,M_y,M_z)$. Find position vector of the point ($\vec r$) if $\...
1
vote
1answer
58 views

What is the difference between a tensor, vector, and a matrix? [duplicate]

I'm currently going through notes on a physics course and I'm having trouble understanding the difference between a tensor, a vector, and a matrix. I know that a vector is a kind of tensor and that a ...
5
votes
1answer
70 views

Signature of $f: \Lambda^2(\mathbb{R}^4) \times \Lambda^2(\mathbb{R}^4) \to \mathbb{R}$, $f(\omega, \omega') = \omega \wedge \omega'$ [closed]

Define$$f: \Lambda^2(\mathbb{R}^4) \times \Lambda^2(\mathbb{R}^4) \to \Lambda^4(\mathbb{R}^4) \cong \mathbb{R}, \quad f(\omega, \omega') = \omega \wedge \omega'.$$ What is the signature of $f$? ...
5
votes
2answers
307 views

Determinant and adjunct of $k-\omega^2m$ in terms of natural frequencies

Given is a mechanical multiple degree of freedom system described by the following matrices and equation: mass matrix ${\bf{m}} = \left[\begin{matrix} m & 0 & 0 \\ 0 & m & 0 \\ 0 &...
10
votes
6answers
5k views

Linear Algebra for Quantum Physics

A week ago I asked people on this site what mathematical background was needed for understanding Quantum Physics, and most of you mentioned Linear Algebra, so I decided to conduct a self-study of ...
3
votes
2answers
75 views

Understanding basics of tensors

I am trying to understand tensors to learn General relativity. In the book that I am reading they claim that if the basis of a vector space undergoes a linear transformation $T$ then the components of ...
3
votes
2answers
90 views

Lagrange Multipliers and Virtual Work: Are Joos & Freeman wrong?

I have come to suspect that the treatment of virtual work in configuration space using Lagrange multipliers given here "Theoretical Physics, by Georg Joos & Ira M. Freeman, pg 114" is not correct. ...
1
vote
2answers
101 views

Pauli Matrices & 2D Rotation Operators?

I was doing a strange calculation with my teacher the other day: find the eigenvalues and eigenvectors of the 2D rotation operator. Intuitively, there should be no solution to this problem in $\mathbb{...
2
votes
1answer
73 views

Diagonalisation: Schmidt vs eigenvalue - when to use which?

In physics we encounter diagonalisation of matrices or operators in a variety of areas. But there are different kinds, the main two being Schmidt decomposition and eigenvalue diagonalisation. The two ...
0
votes
0answers
44 views

Extending projection operator to infinite-dimensional case

Hi I have a basic question regarding bra-ket notation. Given that $\{|e_n \rangle \}$ is a discrete orthonormal basis, $$\langle e_m | e_n \rangle = \delta_{mn}$$ then $$\sum_{n}|e_n \rangle \langle ...
-1
votes
1answer
67 views

Dot product in index notation [closed]

This is a question about a small exercise I am trying to do in order to check if I am correct. Such type of quantities can appear in propagators in QFT. Since I am not an index expert I need some ...
1
vote
1answer
46 views

Solving systems of equations using Levi-Civita and index notation?

I'm doing some self-studying out of Hughston and Tod's Introduction to General Relativity and I stumbled upon a few problems asking me to solve systems of equations using Levi-Civita and index ...
0
votes
1answer
71 views

Technique to Solve 2-Dimensional Linear Momentum Problem

Problem: After a completely inelastic collision, two objects of the same mass and initial speed stick together and move away at half their initial speed. Find the angle between their initial ...
1
vote
2answers
51 views

Determine the point at which moment vector is zero on a 3D body

I have information about total force and moment on a body for three points, whose coordinates I know. From this information I would like to determine the point at which moment would be zero. ...
7
votes
1answer
421 views

Kraus operator rank

All quantum operations $\mathcal{E}$ on a system of Hilbert space dimension $\mathcal{d}$ can be generated by an operator-sum representation containing at most $\mathcal{d^2}$ elements. Extending ...
3
votes
2answers
118 views

Clarification from Griffiths Introduction to quantum mechanics

A question from Appendix Linear algebra A.3 Matrices on page 441 "If you know what a prticular linear transformation does to a set of basis vectors, you can easily figure out what it does on any ...
0
votes
1answer
51 views

How is the range of $x\cdot s$ in Simon's algorithm restricted to $\{0,1\}$?

The final critical step of Simon's algorithm (before setting up a simpler system of equations to solve) involves taking advantage of the fact that $$x\cdot s \in\{0,1 \}$$ but for $x,s \in \{0,1\}^...
-1
votes
1answer
147 views

Meaning of Eigenvalues/Eigenvectors of a linear system of equations

I have a 41x41 system of linear equations (inhomogen) which I derived with Eureqa by describing the timecourse of fMRI haemodynamic data from a brain area as a function of the timecourses of 40 other ...
1
vote
1answer
62 views

Why superposition is useful just for linear functions?

I saw a problem which said that we have a bar between two walls and we increase the temperature. and as you know walls push a force to the bar so the length of it does not change. in the solution I ...
0
votes
1answer
104 views

Relationship between Quantum superposition and Uncertainty principle

I'm an amateur in quantum mechanics. I am confused after reading the following in the wikipedia article about quantum superposition: If the operators corresponding to two observables do not ...
1
vote
2answers
69 views

Exercise 18b in Schutz's First course in GR

The question is as follows: Show that a timelike vector and a non-zero null vector cannot be orthogonal. So we have a timelike vector $\vec{A}$, s.t $\vec{A}^2<0$; and a non-zero null vector, ...
4
votes
4answers
747 views

How do you find a particular representation for Grassmann numbers?

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
3
votes
2answers
86 views

Interpretation of Eigenvalues and Eigenvectors of an hyperbolic conservation law $\partial_t W + A \partial_x W = 0$

I read in a article dealing with a hyperbolic partial differential equations this statement : For any system of hyperbolic partial differential equations (pde), expressed as (1) $\...
1
vote
2answers
209 views

Eigenvalue physical meaning [closed]

What is the physical significance of eigenvalues or eigenvectors?? Please try to explain in very simple language simple harmonic oscillator , potential well could you support your answer by ...
1
vote
0answers
50 views

Proof that a Hermitian Matrix is not defective?

I am taking an introductory course into Quantum Mechanics. To me to seems pretty simple to prove most properties of Hermitian operators. However, I am stuck at an edge case, proving that if an ...
0
votes
0answers
60 views

What is the meaning of “closure is lost” for a set of kets (or any members of a vector space)?

This is the closure relation in Quantum Mechanics: $$\sum_i |i\rangle \langle i| = 1 $$ which I understand as "the sum of the projections onto the basis vectors leaves the projected vector unchanged"...
1
vote
1answer
133 views

Kronecker sum or direct sum?

When we write $$H=\sum_k H_k$$ in condensed matter physics, are we using Kronecker sum or direct sum? I think this is direct sum. However, Wikipedia says it is Kronecker sum. Can anyone give some ...
3
votes
1answer
65 views

Tensor indices and row and column labels of corresponding representation matrices

When reading undergraduate GR literature, I often see that the authors represent tensors ${\eta^\alpha}_{\beta}$, ${\eta^\beta}_{\alpha}$, $\eta_{\alpha \beta}$, $\eta^{\alpha \beta}$ as matrices. ...
0
votes
1answer
91 views

Clarification about two forms of the wave function

The wave function in the position representation is $\langle\ x\rvert\psi\rangle$ = $ \psi (x) $ , where $ \psi (x) $ are the continuous coefficients that multiply the orthonormal basis vectors, i.e, $...
0
votes
0answers
67 views

Electromagnetic scattering $T$-matrix in MATLAB

My problem with the inverse of the matrix T (in the photo) , the matrix consists of Bessel and Henkel functions in high order (the orders from 1 to 21), then the elements of this matrix arrived to ...
0
votes
0answers
20 views

Eigenvalues for correlation matrix which have the form of an harmonic function

I am trying to understand the written in the picture below. I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\...
1
vote
2answers
90 views

Negative powers of operators

This may sound like a strange question, but just to be sure: Suppose I have a general Hermitian operator in Hilbert space whose action on an eigenvector is given by $R|r\rangle = r|r\rangle$. Then, I ...
7
votes
3answers
494 views

How to find the logarithm of Pauli Matrix? [closed]

When I solve some physics problem, it helps a lot if I can find the logarithm of Pauli matrix. e.g. $\sigma_{x}=\left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right)$, find the matrix $...
2
votes
0answers
42 views

Two-Band k.p Model is not Hermitian for imaginary wavevectors

In E. O. Kane's original work on Zener Tunneling, he uses a two-band $k\cdot p$ model for the semiconductor bandstructure: $$H=\begin{pmatrix}E_g+\frac{\hbar^2k^2}{2m_0}&(\hbar/m)kp\\(\hbar/m)kp&...
0
votes
0answers
32 views

How many degrees of freedom to diagonalize the metric?

In A. Zee's Einstein Gravity in a Nutshell, he starts with the following expansion of the metric at some point $P$ of a Riemannian manifold, with coordinates $x^\mu$ that have the origin at $P$: $$ ...
0
votes
0answers
61 views

What is The most detailed Mathematical Methods for physics book? [duplicate]

I am taking a course in Mathematical Physics Junior Level (Undergraduate). We are working from Arfken's "Mathematical Methods for physicists", but i am finding trouble with it specially in the ...
0
votes
1answer
91 views

Is there an online course in Mathematical Methods for Physics, Covering Matrices and Vector Analysis? [duplicate]

I am taking a course in Mathematical Methods for physicsPhysics (Junior Level). We are working from > Mathematical Methods for physicists, George B.Arfken . I just need online resources ...
2
votes
2answers
118 views

Prove that a translation operator times a reflection operator is unitary and Hermitian [closed]

I am trying to prove some properties of the product of the (unitary) translation operator $\hat{T}(a)\psi(x) = \psi(x-a)$ and the (Hermitian) reflection operator $\hat{R} \psi(x) = \psi(-x)$. In ...