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

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2
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0answers
45 views

“find eigenvalue for some fractional power of operator” [duplicate]

If $A$ is an operator and $Aφ=αφ$ ( $α$ is eigenvalue ) then $A^nφ=α^nφ$ and n is a positive integer. Question is that if $n=1/m$ ($m$ is a positive integer) then does it true that say ...
0
votes
0answers
25 views

Determinant of block matrix with off diagonals as vectors [migrated]

I've been given an $(n+1)\times(n+1)$ square matrix, which is written in the form of a block matrix with the following dimensions $ \begin{bmatrix} (1\times1) & (n\times1)\\ ...
1
vote
2answers
69 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 ...
3
votes
2answers
66 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
0answers
36 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
40 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 ...
1
vote
1answer
121 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
52 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
86 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
38 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
18 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}}\\ ...
1
vote
2answers
66 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
476 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 ...
0
votes
0answers
18 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. ...
2
votes
0answers
37 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: ...
1
vote
1answer
51 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 ...
0
votes
0answers
28 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
1answer
77 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
77 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 ...
0
votes
2answers
53 views

Griffiths use of a linear transformation on basis vectors. Need help with derivation

In Griffiths' intro to Quantum Mechanics 2nd edition, in the appendix A.3 which is a review on linear algebra and matrixes (on page 441), he states that a linear transformation on a set of basis ...
1
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0answers
24 views

LinAlg based physics textbooks [duplicate]

I'm in my second year studying physics, and ever since I took LinAlg, I've been noticing LinAlg-related concepts pop up all over the place, but it has never been presented directly as matrices, bases, ...
0
votes
0answers
60 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
2answers
202 views

General Theory of Small Oscillations and existence of solutions

For small oscillations, my textbook equation for amplitude says: $(V-\omega^2T) \cdot a=0$ where $a$ is a column vector in which each component $a_i$ is related to $q_i$ as $q_i=a_i\cos(\omega ...
1
vote
1answer
51 views

Unitary Transfomation from One Basis to Another [closed]

So we have two orthonormal linearly independent basis $\{ |\phi_1 \rangle, \dots, |\phi_n \rangle \}$ and $\{ |\psi_1 \rangle, \dots, |\psi_n \rangle \}$. We can express the basis vectors of the ...
0
votes
0answers
45 views

Why can differentiating a function carry it out from Hilbert space?

I was just doing a QM Griffiths Problem. I was able to get it correct, but I have a few questions. Let $f(x)=x^v$ be defined on $[0,1]$ where $v= 1/2$ then $df/dx = \frac{x^{-1}}2$ Then, we know ...
2
votes
1answer
107 views

Proving polarization identity for operators in complex vector spaces

In general for two operators to be equal, all their (matrix) elements must be equal $$A = B \rightarrow \langle \phi_1|A| \phi_2\rangle=\langle \phi_1|B| \phi_2\rangle$$ However, I am asked to show ...
1
vote
1answer
74 views

Definition of the “support” of the reduced density matrix

Some of the papers in condensed matter physics use the word "support" (space). For example, the following papers use the support especially for the reduced density matrix. ...
-1
votes
2answers
131 views

What's the proof that sum of all projection operators for orthonormal basis gives us identity operator? [closed]

\begin{align} \ I = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix} \end{align} \begin{align} \ {\sum_{i} |i{\rangle}}{{\langle}i|}=I \end{align} $I$ is identity operator, while the $|i{\rangle}$ ...
0
votes
1answer
43 views

Expressing operator in bra-ket notation, knowing just matrix elements

This is in context of unitary transformations between two othonormal bases $\{ | e_i \rangle\}$ and $\{ | \bar{e}_i \rangle\}$. I define $U$ by $$U| e_i \rangle = | \bar{e}_i \rangle $$ Now I can ...
3
votes
1answer
285 views

Many Body Physics: Hamiltonian block structure and Symmetries

Consider a many body problem of a small cluster, e.g. the 'Hubbard-Cluster' (albeit the question may be of relevance for other Hamiltonians as well): $$\mathcal{H}=\sum_{<ij>\sigma} t_{ij} ...
0
votes
0answers
29 views

Entanglement entropy for stabilizer states

A stabilizer state for a stabilizer set S for a system of qubits can be written as $\rho=\frac{1}{2^{n}}\sum_{g\epsilon S}g$ . If we take a bipartition A-B of our system and partial trace over A, ...
2
votes
2answers
121 views

Superposition in Quantum Mechanics

First of all, let $V$ be a vector space over the field $\mathbb{F}$. It is possible then to show, by Zorn's Lemma that there is a basis for $V$. The main point is that although basis are quite ...
4
votes
0answers
93 views

Is there a proof that the number of eigenstates is countable for a bound system?

When you solve Schrödinger equation for a free particle with no boundary conditions your eigen states are indexed by quantum number $k \in \mathbb R $ and $\mathbb R$ isn't countable but if you add a ...
1
vote
2answers
112 views

Transformation of four-velocity in special relativity

I am revising special relativity introducing more matrix form in the equation. Currently I am reading book in which transformation matrix is defined as $${\Lambda= \begin{bmatrix} \gamma & ...
2
votes
1answer
31 views

Proton spin independent fine structure “Hamiltonian” $W_f$

To find the perturbation correction (fine structure) in the case of a degenerate energy $E_n^0$, we can diagonalize the operator $W_f^n$, the restriction of $W_f$ to the eigen-space associated to ...
0
votes
2answers
96 views

$[A_1, H] =[A_2, H] = 0$ but $[A_1, A_2] \neq 0$?

I am having a difficult time understanding this problem. Suppose $[A_1, A_2] \ne 0,$ $[A_1, H] = 0,$ $[A_2, H] = 0.$ Show that the energy eigenstates of $H$ are in general ...
0
votes
0answers
41 views

Is there a physical interpretation of the alternating property?

A map from a vector-space to its base field is called "alternating" if each vector with repeated elements is mapped to zero. I've read that symplectic geometry is an important representation of ...
0
votes
1answer
42 views

Quantum beam-splitter matrix

I have seen the matrix for the action on a quantum beam splitter described in one of two ways: $$\begin{pmatrix} t_1 & r_2 \\ r_1 & t_2 \end{pmatrix}$$ (this appears in Quantum Optics by ...
1
vote
2answers
105 views

Find the covariant metric tensor from a given contravariant metric tensor

If given $$g^{\mu\nu}=\pmatrix{\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ ...
20
votes
5answers
4k views

What exactly is a dimension?

How do you exactly define what is and isn't a dimension? I heard somewhere that it is "anything you can move through" but if that is right, why wasn't time and space considered a dimension before ...
-2
votes
2answers
174 views

Linearity in Quantum Mechanics that make superposition possible

As a beginner in QM, all the video lectures that i have seen talk about superposing wave functions in order to get $\psi$. But from what i know from linear algebra, the system must be linear in order ...
1
vote
3answers
120 views

“Complete” confusion

The word "complete" seems to be used in several distinct ways. Perhaps my confusion is as much linguistic as mathematical? A basis, by definition, spans the space; some books call this "complete" -- ...
2
votes
1answer
110 views

Proofs on operator algebra [closed]

I'd like to ask the community to please verify the first two proofs below and help me get through the last one since I seem to be stuck. Thank you in advance. Proof 1: Given two noncommutting ...
1
vote
0answers
55 views

how to rotate scaled-vector (orientation) by scaled-vector (rotation)

Recently I seem to have gotten the physics-engine portion of my 3D simulation/game engine [apparently] working correctly. The most convenient way to store and compute position and orientation are in ...
1
vote
0answers
170 views

Solving a one-dimensional spring system using linear algebra [closed]

I'm trying to gain some intuition into how the following problem would be solved using linear algebra. The setup of the problem is a horizontal line of 4 springs and 3 masses: alternating spring, ...
1
vote
0answers
46 views

When trying to see what symmetries an operator generates, how do you “decide” what coordinate to apply it to?

Suppose I have $\hat{O}_{1}=-i\hbar\partial_{x}$ then \begin{eqnarray} e^{-i\gamma\hat{O}_{1}/\hbar}x\,e^{i\gamma\hat{O}_{1}/\hbar}=x+\gamma \end{eqnarray} and \begin{eqnarray} ...
2
votes
2answers
549 views

How to take partial trace?

$L$ is a linear operator acting on hilbert space $V$ of dimension $n$, $L: V \to V$. The trace of a linear operator is defined as sum of diagonal entries of any matrix representation in same input ...
0
votes
0answers
38 views

Physical significance of Cayley Transform

In the book on Quantum Mechanics by Capri (in Chapter 6), its said that an operator $A$ is self adjoint if the operator, $U$ given by $$ U = (A - i I)(A + i I)^{-1} = -(I+iA)(I-iA)^{-1} = -\text ...
6
votes
3answers
723 views

Does it mean anything if the commutator of an operator with the Hamiltonian is equal to the Hamiltonian?

Question says it all, really. I have $[\hat{H},\hat{O}]=-2i\hbar\hat{H}$. Does this mean that the operator $\hat{O}$ (an observable) is special in some way?
2
votes
2answers
196 views

When is matrix representation of a Hermitian operator invertible?

If I have a Hermitian operator $H:V \to V$ on a finite-dimensional vector space $V$, and I write down its matrix representation in some basis $B$ with matrix representation being $[H]_B$, then in what ...