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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1answer
101 views

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 ...
3
votes
1answer
291 views

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?
1
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1answer
98 views

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 ...
0
votes
1answer
96 views

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 ...
2
votes
3answers
177 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 ...
4
votes
3answers
222 views

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 ...
1
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1answer
213 views

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 ...
1
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3answers
161 views

Normalization of basis vectors with a continuous index?

I have an infinite basis which associates with each point, $x$, on the x-axis, a basis vector $|x\rangle$ such that the matrix of $|x\rangle$ is full of zeroes and a one by the $x^{\mathrm{th}}$ ...
3
votes
1answer
118 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 ...
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2answers
215 views

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 ...
4
votes
2answers
127 views

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 ...
1
vote
2answers
184 views

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| ...
3
votes
1answer
432 views

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 ...
3
votes
1answer
153 views

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 ...
7
votes
2answers
266 views

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 ...
3
votes
1answer
60 views

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 ...
2
votes
1answer
91 views

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 ...
1
vote
1answer
168 views

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 ...
1
vote
1answer
135 views

How to understand the entanglement in a lattice fermion system?

Topological insulator is a fermion system with only short-ranged entanglement, what does the entanglement mean here? For example, the Hilbert space $V_s$ of a lattice $N$ spin-1/2 system is ...
3
votes
0answers
135 views

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 ...
5
votes
1answer
81 views

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 ...
3
votes
1answer
100 views

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 ...
2
votes
1answer
96 views

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 ...
1
vote
1answer
67 views

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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0answers
135 views

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 ...
4
votes
1answer
148 views

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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vote
4answers
436 views

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 ...
5
votes
1answer
365 views

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 ...
5
votes
2answers
309 views

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 ...
6
votes
3answers
2k views

What is the physical significance of the off-diagonal moment of inertia matrix elements?

The tensor of moment of inertia contains six off-diagonal matrix elements, which vanishes if we choose the principle axis of the rotating rigid body and the components of the angular momentum vector ...
1
vote
1answer
108 views

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 ...
0
votes
1answer
52 views

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 &= ...
0
votes
1answer
78 views

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 ...
3
votes
2answers
168 views

Why is this not a realisable operation on a quantum system?

Let $\rho = \begin{bmatrix}\ 1&0 \\ 0&0 \end{bmatrix}$, $\rho' = \begin{bmatrix}\ 0&0 \\ 0&1 \end{bmatrix}$, $\rho'' = \dfrac{1}{2}\begin{bmatrix}\ 1&1 \\ 1&1 \end{bmatrix}$ ...
-2
votes
1answer
122 views

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 ...
10
votes
1answer
584 views

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 ...
0
votes
1answer
104 views

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 ...
3
votes
1answer
200 views

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 ...
4
votes
5answers
381 views

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 ...
3
votes
1answer
189 views

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 ...
2
votes
1answer
75 views

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 ...
2
votes
1answer
122 views

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 ...
0
votes
1answer
206 views

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 ...
1
vote
1answer
195 views

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 ...
8
votes
1answer
715 views

“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 ...
0
votes
1answer
147 views

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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vote
4answers
555 views

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 = ...
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2answers
1k views

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 ...
2
votes
3answers
1k views

Spring-mass system with two springs and three masses [closed]

I'm trying to solve a system of springs and masses that is confusing me. First, the balls are all lined up linearly. Secondly, the ball in the middle has a smaller mass $m$ while the first and last ...
1
vote
2answers
253 views

Conservation of Linear Momentum with respect to a given direction

Is linear momentum conserved in any direction? More specifically, if you project all momentum vectors in a system onto another vector, will momentum be conserved? I know that momentum is conserved ...