The linear-algebra tag has no wiki summary.
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1answer
50 views
Hamiltonian in 2-dimensions?
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 ...
4
votes
1answer
79 views
Is orthogonality = mutual exclusiveness?
What does Euclidian orthogonality of 90° have in common with mutual exclusiveness?
It is paradoxical to me to hear that spin up state is orthogonal to spin down. If two elements are orthogonal, it ...
3
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2answers
146 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 ...
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0answers
44 views
What kind of object is a column vector of basis vectors? [closed]
Can you arrange orthonormal basis vectors $\hat{i}, \hat{j}, \hat{k}$ in a single 3D column vector? It's not clear to me what space this vector lives in. Is it a dyad in ...
3
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2answers
94 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 ...
3
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2answers
144 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
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2answers
309 views
2D - Kinematics - Linkage System using Vector Algebra
I have this question that I dont know how to solve correctly :
My question is, how do I find $V_B$ ? I will find the angular velocities myself, but I want to know the method to get $V_B$ ?
I know ...
1
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1answer
46 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
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1answer
30 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
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1answer
42 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 ...
8
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1answer
138 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 ...
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3answers
578 views
Should linear algebra and vector calculus from traditional courses be replaced with `geometric algebra`?
geometric algebra gives geometric meaning to linear algebra and much more.
it can provide a coordinate free geometric interpretation of spaces.
those who learn of ...
0
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1answer
28 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
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1answer
108 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 ...
2
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1answer
58 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}$$
Now the matrix is ...
3
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5answers
240 views
Math of eigenvalue problem in quantum mechanics
I learned the eigenvalue problem in linear algebra before and I just find that the quantum mechanics happen to associate the Schrodinger equation with the eigenvalue problem. In linear algebra, we ...
2
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1answer
73 views
Eigenvalues of a mean correlation matrix (integral over correlation matrices with arbitrary density)
Consider a stationary dynamic system with state $s(t)$ and correlation structure described by $C_{ij}(\tau)=\mathbb{E}[(s_i(t+\tau)-\bar{s_i})(s_j(t)-\bar{s_j})]$. Given an arbitrary density function ...
3
votes
1answer
82 views
Algorithm for identifying planes in a Bravais Lattice
I have a lattice with Lattice Vectors $(\vec{t}_1,\vec{t}_2,\vec{t}_3)$ which are NOT orthogonal in general.
How can I identify the atoms/unit cells that belong to a plane - that is normal to a given ...
0
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1answer
143 views
Jacobi's matric formulation for tensors
Hi I am trying to derive the E field equation and am stuck using the Jacobi formula, is this correct: $$\delta \det g_{\mu \nu} = Tr(ADJ(g_{\mu\nu}\delta ...
1
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1answer
142 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 ...
6
votes
1answer
190 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 ...
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4answers
611 views
Spring-mass system with two springs and three masses
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
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4answers
213 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 = ...
0
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1answer
74 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
2
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3answers
137 views
What is the general approach to calculating time of impact in 3D?
Given two objects a and b moving at fixed velocities, how would you determine (a) whether they will collide at all, and if so, (b) time of impact?
(Let us assume these are spherical bodies each with ...
1
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2answers
423 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 ...
6
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2answers
735 views
Difficulties with bra-ket notation
I have started to study quantum mechanics. I know linear algebra,functional analysis, calculus, and so on, but at this moment I have a problem in Dirac bra-ket formalism. Namely, I have problem with ...
1
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2answers
163 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 ...
0
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2answers
170 views
When and how do you represent a two body state as a tensor product?
I have read that in quantum mechanics, compound systems are constructed as tensor products.
But on page 177 of Griffith, for example, a two body wavefunction is introduced as
Psi ...
0
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1answer
57 views
Writing a tensor with respect to a particular basis
When defining tensors as multilinear maps, I am having trouble understanding why a tensor, let's say of type (2,1), can be written in the following way:
$$T = T^{\mu\nu}_{\rho} e_\mu \otimes e_\nu ...
-1
votes
2answers
135 views
Will a one year undergraduate course of Linear Algebra be enough for QM? [duplicate]
Possible Duplicate:
Linear Algebra for Quantum Physics
Can you get all/most of the knowledge you need of Linear Algebra for QM in a one year course? I know for certain my course also ...
3
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2answers
146 views
How to express continuous values as a matrix
Usually a quantity of a matrix is defined as the eigenvalues of the matrix. If so, how can anyone express continuous values, as in Schrodinger picture, into a matrix?
5
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5answers
477 views
Linear Algebra for Quantum Physics
A week ago I asked people on this forum what mathematical background was needed for understanding Quantum Physics, and most of you mentioned Linear Algebra, so I decided to conduct a self-study of ...
1
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2answers
361 views
Metric tensor and its inverse
Is it always allowed to represent the metric tensor $g_{\mu \nu}$ in General Relativity as a $4\times 4$ matrix?
If the last one is represented for example with a $4\times 4$ matrix ...
1
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1answer
203 views
Tensors: relations between physics and linear algebra
In continuum mechanics we use finite deformation tensors to exprime deformations in a point. The 9 components of the tensor (in reality 6 because of its symmetry) are defined as
$$
...
0
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0answers
10 views
Velocity Matrix Equation including throttle. [duplicate]
Possible Duplicate:
2D Car Physics including Throttle
For a simulation for testing on automatic cruise control, I came across with the equation.
To calculate the velocity for the next ...
4
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5answers
620 views
Show the Lorentz Transformation Matrices Have an Inverse
Assume the Lorentz transformations obey the relationship:
$g_{uv}\Lambda^u_{p}\Lambda^v_\sigma = g_{p\sigma}$, where $g_{uv}$ is the metric-tensor of special relativity.
Show then that a Lorentz ...
7
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3answers
617 views
What does the dual of a tensor mean (e.g. dual stress tensor in relativistic ED)?
I know what the dual of a vector means (as a map to its field), and I am also
aware of of the definition a dual of a tensor as,
$$F^{*ij} = \frac{1}{2} \epsilon^{ijkl} F_{kl}\tag{1}$$
I just don't ...
3
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1answer
86 views
Random Hankel matrix and eigenvalues distribution
I would like to know if there are any theoretical results on the distribution of the eigenvalues of Hankel matrices. I seek a result like the Marchenko–Pastur distribution for random matrices.
1
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0answers
29 views
Random quantum systems with asymmetric Lifshitz tails?
For a quantum mechanical system with a periodic Hamiltonian (Schrödinger operator) $H$, let $N(E)$ be its integrated density of states, i.e. the fraction of eigenvalues in the spectrum $\sigma(H)$ ...
1
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3answers
181 views
Anybody have example of two-qubit non-Pauli and non-Clifford quantum gate?
A lot of known quantum gates are in the Pauli group (I,X,Z,Y) or in the Clifford group (H,P,Cnot). I need examples of the quantum gates that aren't in this groups. Also, are there are matlab functions ...
1
vote
1answer
147 views
Proof of Pauli group preservation by Clifford group conjugation?
A well know result is that Clifford group preserve the Pauli group under conjugation or, in other words:
$C(P_{1} \otimes P_{2})C^{\dagger} = P_{3} \otimes P_{4}$, with $C \in$ Clifford group and ...
1
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0answers
61 views
Matrix separability preservation under conjugation?
Someone know any paper about matrix separability preservation under conjugation? A well know result is that Clifford group preserve the Pauli group under conjugation or, in other words:
$C(P_{1} ...
2
votes
1answer
400 views
Jarlskog Invariant and its mathematical origin
CP violation is present in the weak interactions if
There are no degeneracies in the up-quark/down-quark matrices
The Jarlskog invariant $J=Im(V_{us} V_{cb} V_{ub}^* V_{cs}^*)$ is nonvanishing
...
5
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2answers
200 views
Why are the solution coefficients for a harmonic oscillator proportional to minors of the determinant?
I'm studying the oscillations of systems with more than one degree of freedom from Landau & Lifshitz's Mechanics Third Edition (for those who have the book, my question corresponds roughly to ...
14
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4answers
2k views
Are matrices and second rank tensors the same thing?
Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified:
1-Are matrices and second rank tensors ...
0
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3answers
123 views
Dimension of vector resulting from tensorial product
I'm quoting what I found in a book about quantum computation:
Individual state spaces of $n$ particles combine quantum mechanically through the tensor product. If $X$ and $Y$ are vectors, then ...
2
votes
3answers
129 views
Vector space of $\mathbb{C}^4$ and its basis, the Pauli matrices
How do I write an arbitrary $2\times 2$ matrix as a linear combination of the three Pauli Matrices and the $2\times 2$ unit matrix?
Any example for the same might help ?
1
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1answer
457 views
Dimensional Analysis: Buckingham Pi Theorem Using a Matrix
I have the following problem - which is basically that $\omega = f(g,h,l)$ now the book claims that the two $\pi$ terms as follows $\pi_1 = \omega \sqrt{\frac{l}{g}}$ and $\pi_2 = \frac{h}{l}$. Now we ...
5
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3answers
501 views
What is a basis for the Hilbert space of a 1-D scattering state?
Suppose I have a massive particle in non-relativistic quantum mechanics. Its wavefunction can be written in the position basis as
$$\vert \Psi \rangle = \Psi_x(x,t)$$
or in the momentum basis as
...
