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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Why consider only direction cosines?
Why are these called direction angles?
Why do we consider only direction cosines and not direction sines or tans.
What is its actual significance?
And How to use them?
Why are they called ...
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Diagonalization Of $(\sigma_x+\sigma_y)$
Can this matrix $(\sigma_x\pm\sigma_y)$ be diagonalised? Clearly, if $\sigma_x$ is diagonalized by a similarity transformation $S_1\sigma_x{S_1}^{-1}$, then $\sigma_y$ can't be diagonalized by $S_1$, ...
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Use of 'complete' as in 'complete set of states' or 'complete basis'
Question. In the context of QM, I hear the phrases 'complete set of states' and 'complete basis' (among other similar expressions) thrown around rather a lot. What exactly is meant by 'complete'?
...
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Conceptual difficulty in understanding Continuous Vector Space
I have an extremely ridiculous doubt that has been bothering me, since I started learning quantum mechanics.
If we consider the finite dimensional vector space for the spin$\frac{1}{2}$
particles, ...
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Non-Euclidean spaces in Quantum Mechanics
In quantum mechanics, I have been going through basics of the subject. In general the space of quantum states is Hilbert space (which is Euclidean - I presume). Being just curious, are there any ...
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Representation of operators in quantum mechanics
For which systems we represent the Hamiltonian as a differential operator and for which system we represent it by a matrix? Can the momentum be represented by a matrix operator?
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Linear Operators and their representations
I am currently learning Quantum mechanics on a slightly advanced level. I am curious in knowing if there are Linear Operators (Linear Maps) in the Hilbert Space (finite dimensional ones) that don't ...
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Why does the pure shear term / strain deviator tensor have non-zero entries on the main diagonal?
In a textbook of mine an operation is performed, of which I think the goal is to get zeros on the main diagonal of a matrix (the matrix represents strain). But im not sure that is the goal and Im also ...
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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 ...
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Countable Matrix Representation
In my quantum mechanics class, my professor explained that the Hamiltonian along with position and momentum operators can be represented by matrices of countable dimension. This is especially usefull ...
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Lorentz boost matrix in terms of four-velocity
As I understand it, the value of a 4-vector $x$ in another reference frame ($x'$) with the same orientation can be derived using the Lorentz boost matrix $\bf{\lambda}$ by $x'=\lambda x$. More ...
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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 &...
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Is the spin-singlet state also a Resonating-Valence-Bond(RVB) state?
The spin-singlet state of a lattice spin-1/2 system is defined as $S_x\Psi=S_y\Psi=S_z\Psi=0$, where $S_\alpha=\sum S_i^\alpha(\alpha=x,y,z)$ are the total spin operators, in other words, a spin-...
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Dimension of the space of solutions in an electric circuit
Consider an electric circuit with dc sources ( voltage and current) and resistors. Write down the equations. In the most general case, the solution of the system is not unique. The set of solutions ...
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The eigenspinors for the spin operator in the $x$-direction?
$$S_x= \frac{\hbar}{2}\quad\begin{pmatrix}0&1\\1&0\end{pmatrix}\quad$$
$$S_x{X_+}^{(x)}=\frac{\hbar}{2}{X_+}^{(x)}$$
How can I find the eigenvalue of $S_x$?
My book says
$$
\left| \begin{...
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1D Ising Model (NN and NNN interactions) with 2 transfer matrices
I've tried an alternative solution for finding the partition function of this model.
So is what I've done correct? If it isn't then please prove and explain why not. (I'm pretty sure I made a mistake ...
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A naive question about the Second Quantization?
Let's consider a single-particle(boson or fermion) with $n$ states $\phi_1,\cdots,\phi_n$(normalized orthogonal basis of the single-particle Hilbert space), and let $h$ be the single-particle ...
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Can we show that time is orthogonal to space?
It's easy to show that the time we measure is "in a different direction" from the space directions we measure. However, it's not immediately obvious to me that these directions are orthogonal.
How do ...
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Boson calculus and Maximum Weight State
I'm just going over a few past exams for tomorrow, and I've come across a question that I'm having quite a bit of difficulty with.
Let $\left|0\right\rangle$ denote the Fock vacuum state so that $...
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Determinant expansion
I've seen in a few textbooks now a useful looking expansion procedure for determinants, but I don't understand the details of it. Here is precisely the example I'm thinking of (Ex. 7.6). I don't ...
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Properties of Lorentz transformation generator?
In chapter 2 of Srednicki, the author defines:
$$
U(1+\delta \omega) = I +\frac{i}{2h}\delta \omega_{\mu \nu} M^{\mu \nu}
$$
where the $M^{\mu\nu}$s are hermitian operators and are the generators of ...
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Fock Subspaces and Weight Vectors
This is my first time taking a physics course (I'm a mathematics major), so I'm encountering a lot of new things, which I'm kind of expected to know. In particular, how to work with Bosons.
I've got ...
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When is an operator subspace the span of Kraus operators?
Let $A$ and $B$ be finite dimensional Hilbert spaces, and let $\mathcal{L}(A \to B)$ be the space of linear operators from $A$ to $B$. Say that a subspace $K \subseteq \mathcal{L}(A \to B)$ is a span ...
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Are the symmetry operators well defined in the context of Projective Symmetry Group(PSG)?
Consider the Schwinger-fermion approach $\mathbf{S}_i=\frac{1}{2}f_i^\dagger\mathbf{\sigma}f_i$ to spin-$\frac{1}{2}$ system on 2D lattices. Just as Prof.Wen said in his seminal paper on PSG, the ...
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Positive cone of operators: if two selfadjoints $a$, $b$ obey $a^2 + b^2 =1$, must they commute?
The question relates to the structure of the positive cone of operators, in C*-algebra.
If $a$ and $b$ are selfadjoints such that $a^2 + b^2 = 1$ can one prove $a$ and $b$ commute? What one derives ...
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Couple Masses - Change in Basis
I'm having trouble with the linear algebra used to solved a coupled mass problem.
$\ddot{x}_1 = -(2k/m)x_1 + (k/m)x_2$ and $\ddot{x}_2 = (k/m)x_1 - (2k/m)x_2$
Shankar then sets the equation up in ...
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A simple question on the projected wave function?
For example, consider a spin-1/2 AFM Heisenberg Hamiltonian $H=\sum_{<ij>}\mathbf{S}_i\cdot\mathbf{S}_j$, and we perform a Schwinger-fermion($\mathbf{S}_i=\frac{1}{2}f^\dagger_i\mathbf{\sigma}...
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Saturation of the Cauchy-Schwarz Inequality
Going to as little details as possible, here is a statement from Wald's text on QFT in curved spacetimes(I am not quoting the book)
He considers two vector spaces ${\cal S}$ and ${\cal H}$.
Note - ...
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Vectors with more than 3 components
I have some confusion over Vectors, Its components and dimensions. Does the number of vector components mean that a vector is in that many dimensions? For e.g. $A$ vector with 4 components has 4 ...
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What do up-left orthogonality has in common with up-down and what is their relationship?
I am familiar with the true (or general) notion of orthogonality, given in the Linear Algebra and Pythagoras theorem derived from the $\vec x \cdot \vec y = 0$. I have also recently got to know that ...
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Is length/distance a vector?
I have heard that area is a vector quantity in 3 dimensions, e.g. this Phys.SE post, what about the length/distance? Since area is the product of two lengths, does this mean that length is also a ...
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Geometrical Representation Grover algorithm
I am studying the Grover algorithm and in my and others lectures, I've come across this picture.
If the dimension of the computational basis is greater than 2, why does the evolution algorithm have ...
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Eigenvector Grover Operator
I have a question about the eigenvectors for the evolution operator of Grover's algorithm. Let $U=R_DR_f$, where
$$\begin{align*}
R_D &= 2|D\rangle\langle D| -I_N , \\ R_f &= I-2|x_0\rangle\...
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Eigenvalue $a_n$
Q1:
In Zetilli's book page 166 (ch. "Postulates of QM", eq. 3.1) i encountered an expression $\hat{A}|\psi\rangle = a_n|\psi_n\rangle$. I know this is an eigenvalue equation, but i have seen another ...
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Why is this not a realisable operation on a quantum system?
Let $\rho = \begin{bmatrix}\ 1&0 \\ 0&0 \end{bmatrix}$, $\rho' = \begin{bmatrix}\ 0&0 \\ 0&1 \end{bmatrix}$, $\rho'' = \dfrac{1}{2}\begin{bmatrix}\ 1&1 \\ 1&1 \end{bmatrix}$ (...
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Hamiltonian in 2-dimensions? [closed]
I am trying to construct a Hamiltonian for a system in 2 dimensions using Matlab.
I am not sure how this Hamiltonian will look like in matrix form.
If somebody can help me visualize this matrix that ...
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What're the relations and differences between slave-fermion and slave-boson formalism?
As we know, in condensed matter theory, especially in dealing with strongly correlated systems, physicists have constructed various "peculiar" slave-fermion and slave-boson theories. For example,
For ...
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Confused on Calculating Resistance Distance Matrix
I am trying to create a computer program to compute the equivalent resistance over any points on any rectangular set of resistors (all with a resistance of 1 ohm). It seems that the resistance ...
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The issue on existence of inverse operations of $a$ and $a^{\dagger}$
I have asked a question at math.stackexchange that have a physical meaning.
My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators and $[a,a^\dagger]=1$. I want to prove that ...
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Math of eigenvalue problem in quantum mechanics
I learned the eigenvalue problem in linear algebra before and I just find that the quantum mechanics happen to associate the Schrodinger equation with the eigenvalue problem. In linear algebra, we ...
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Algorithm for identifying planes in a Bravais Lattice
I have a lattice with Lattice Vectors $(\vec{t}_1,\vec{t}_2,\vec{t}_3)$ which are NOT orthogonal in general.
How can I identify the atoms/unit cells that belong to a plane - that is normal to a given ...
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Schriffer Wolff Transformation - for first order change in eigenvalues
Step 1
Let me formulate the problem to convey my notation.
I have a matrix $A$ which is hermitian - and is diagonalisable by a transformation
$$ U_A A\,\,U_A^{-1} = A_{diag}$$
Now the matrix is ...
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Eigenvalues of a mean correlation matrix (integral over correlation matrices with arbitrary density)
Consider a stationary dynamic system with state $s(t)$ and correlation structure described by $C_{ij}(\tau)=\mathbb{E}[(s_i(t+\tau)-\bar{s_i})(s_j(t)-\bar{s_j})]$. Given an arbitrary density function $...
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Jacobi's matric formulation for tensors
Hi I am trying to derive the E field equation and am stuck using the Jacobi formula, is this correct: $$\delta \det g_{\mu \nu} = Tr(ADJ(g_{\mu\nu}\delta g_{\mu\nu})=\det(g_{\mu\nu})Tr(g^{\mu\nu}\...
user21119
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Once I have the eigenvalues and the eigenvectors, how do I find the eigenfunctions?
I am using Mathematica to construct a matrix for the Hamiltonian of some system. I have built this matrix already, and I have found the eigenvalues and the eigenvectors, I am uncertain if what I did ...
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"An operator is hermitian". Implications?
Alastair Rae states that there are 4 postulates of Quantum Mechanics in his text on the subject matter. The first part of his second postulate can be stated as:
Every dynamical variable may be ...
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Validity of Bogoliubov transformation
In condensed matter physics, one often encounter a Hamiltonian of the form
$$\mathcal{H}=\sum_{\bf{k}}
\begin{pmatrix}a_{\bf{k}}^\dagger & a_{-\bf{k}}\end{pmatrix}
\begin{pmatrix}A_{\bf{k}} &...
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Cyclic rotation of a Matrix
What is meant by a cyclic rotation of a matrix, specifically in proving that j*k-k*j=i*l
where j,k,l are cyclic rotations of the Pauli spin matrices sigma x, sigma y, and sigma z
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Hamiltonian in position basis
Let $ H = \frac{-h^2}{2m}\frac{\partial^2 }{\partial x^2}$. I want to find the matrix elements of $H$ in position basis. It is written like this:
$\langle x \mid H \mid x' \rangle = \frac{-h^2}{2m}\...
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Applying angular velocity to a rotation matrix
I have a very simple question. In our project we store an object's orientation as a 3x3 matrix which holds the orthonormal base of that object's local space. For instance if the object is aligned with ...
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