# Tagged Questions

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

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### The definition of transpose of Lorentz transformation (as a mixed tensor)

In the appendix of the textbook of Group Theory in Physics by Wu-Ki Tung, the transpose of a matrix is defined as the following, Eq.(I.3-1) $${{A^T}_i}^j~=~{A^j}_i.$$ This is extremely confusing for ...
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### Dropping vertices of an overdetermined statics system graph

In statics, the problem of determining the tensions of K cables that connect a structure made of N points and keep all points in static equilibrium implies $ND$ systems of equations, where $D$ is the ...
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### Learning how to use Levi-Civita symbol

I've recently started my second course in Quantum Theory and am now often required to prove more complex commutation relations. I'm aware that the Levi-Civita symbol often makes this sort of thing a ...
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### These two operators commute…but their eigenvectors aren't all the same. Why?

The Hamiltonian $$H = \left[ \begin{array}{cccc} a & 0 & 0 & -b \\ 0 & 0 & -b & 0\\ 0 & -b & 0 & 0\\ -b & 0 & 0 & -a \end{array} \right]$$ commutes ...
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### Forces exerted on legs of a table

This is actually for a computer science homework problem, but I haven't studied any physics in five years so I'm having a bit of trouble. I'm given a table of three legs with their coordinates and I ...
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### Examples of application of detour matrices in physics?

Are there any good examples of application of detour matrices in physics?
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### 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$...
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### What information does the trace of a matrix give?

I was recently thinking what information do we get from a matrix. So if we say the columns (or rows) of a matrix define the basis of a system, say vectors of 3 dimensional space. Then the determinant ...
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### Why eigenvector points to principal stress plane?

I can represent a tensor by a matrix. Suppose we are talking about a 2nd order tensor, and the matrix is therefore 3x3. If I find one eigenvector of that matrix; that vector represents normal vector ...
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### A question on partial trace and density matrix computation

Consider a Pure state of a two dimensional system $|\psi\rangle={1\over\sqrt{2}}(|e_1\rangle|e_1\rangle+|e_2\rangle|e_2\rangle)$ where $\{|e_i\rangle\}$ is an orthonormal basis. Could any one just ...
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### Dimension of separable state

Please can you help me to understand how the dimension of the set of separable states is $\dim \cal H_1 + \dim \cal H_2$? This is the relevant passage: So far, we have assumed implicitly that the ...
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### $\exp(i\alpha\hat {\bf n}\cdot{\bf \sigma} )=\cos\alpha I+i(\hat {\bf n}\cdot{\bf \sigma})\sin\alpha$

Could anyone tell me $\hat {\bf n}\cdot{\bf \sigma}$ is defined in such way? In the book they have not defined what is $n_z,n_x,n_y$. It is from Quantum Computing: From Linear Algebra to Physical ...
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### How to define tensor contraction without referring to summation?

The textbook defines a tensor to be an element in $(T^*)^kÃ—T^lâ†’R$. It then expresses tensors as arrays of components with respect to a certain basis, and defines tensor contraction using summation ...
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### Geometry topics in physics [closed]

I'd like to learn modern physics at an advanced level, but since I've no access to university, I'm self-teaching, and appeal to the Internet for information about what to study and how. Currently, I'...
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### Physical applications of matrices and determinants

Other than notation devices, I don't see any direct application of matrices/determinants in physics. For example, they are just a different way to write a partial derivative and determinants find if ...
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### Why are the energy eigenstates realized in atomic transitions?

I have a question like "Why is it often assumed that particles are found in energy eigenstates?", it is a little different, though. When one solves the hydrogen atom, one can use a polynomial Ansatz ...
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### Complex semi-definite programming

I'm doing some calculations and I want to simulate them in python or matlab (or whatever). However I use hermitian matrices and I don't really manage to find a library which enables me to calculate ...
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### Link between Quantum and Classical Mechanics [duplicate]

In classical mechanics we have momentum as generator of translation by following definition: $$f(x+\delta x)=f(x)+[f(x),p]\delta x+....$$ I was wondering whether using this relation and commutation ...
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### Differences between wave function and set of orthonormal wave functions?

I'm reading a QM book. It first says for wave function: "The state of a physical system (or particle) is completely specified by an entity associated with it called a wave function, Î¨ , that in ...
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### Straight line on 8-axis [closed]

How would a straight line appear in a 8-axis co-ordinate system. Will it continue to look like a straight line or will it appear like zig-zag line. We can see that straight line appears like a dot in ...
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### How are matrices used to represent quantities, and what is the meaning of a matrix?

So I'm reading this text on Quantum Mechanics, and it goes through a few chapters that I understand fairly well including probability. But then it says that all quantities, like position and energy ...
I am still getting used to the braket notation. Is this manipulation correct? $$\frac{-\hbar^2}{2m}\int_{-\infty}^{\infty}\frac{\partial^2\phi_n^*}{\partial z^2} \phi\,dz = \int_{-\infty}^{\infty}\... 0answers 115 views ### How to make a base tranformation for a linear operator in QM? [closed] I have 2 bases A and B with the following kets: Base A: |a_1\rangle and |a_2\rangle Base B: |b_1\rangle = \frac{1}{\sqrt2} \cdot(|a_1\rangle + i\cdot|a_2\rangle) |b_2\rangle = \frac{1}{\sqrt2}... 2answers 182 views ### What does \left|x,t\right> actually mean (Heisenberg picture)? I am pretty much confused with this notation I believe. The Heisenberg states are denoted by \left|x,t\right> and the Schrodinger states are given by \left|x(t)\right>. It seems like both of ... 0answers 81 views ### Trouble with proof about operators in QM I would like to complete the following exercise: Prove that if the operators P_i satisfy P_i^{\dagger} = P_i and P_i^2 = P_i, then P_iP_j=0 for all i\neq j. From P_i^2 = P_i I ... 1answer 515 views ### What do matrices in the Gaussian orthogonal ensemble look like? I've been reading a fair amount about quantum chaos, and random matrix theory comes up a lot. I get that they're looking at the distribution of eigenvalues from an ensemble of random matrices, but I ... 1answer 118 views ### Characteristic polynomial of a Matrix In fact, this problem is more likely to be a math problem. When I read a paper(http://arxiv.org/abs/0707.2875), the author includes the characteristic polynomial for a type of matrix A_k with ... 1answer 84 views ### Converting between (abstract) linear operators and their position representations Just as we have an abstract state vector |\psi\rangle and its position representation \psi(\vec{x}) = \langle \vec{x} | \psi \rangle, how do we transform between a linear operator, say H, that ... 1answer 128 views ### Some question about symplectic transformation I read Arnold's book Mathematical Methods of Classical Mechanics and come across with three problems in page 229. 1.Let \lambda and \bar{\lambda} be simple (multiplicity 1) eigenvalues of a ... 1answer 134 views ### Rotation operator for a point in a coordinate system linearly derived from Cartesian coordinates For some experimental and practical reason, I have created a new coordinate system in the form$$x^\prime_i=T_{ij}x_j where $T_{ij}$ isn't a square matrix. $x_i$ is standard Cartesian coordinates, ...
Physicists define the trace of an operator $\rho$ as the follows, $Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$ where B is some orthonormal basis, and this quantity is ...