The linear-algebra tag has no wiki summary.
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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 &= ...
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1answer
39 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 ...
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1answer
102 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}$ ...
8
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1answer
117 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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1answer
17 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
105 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 ...
3
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5answers
230 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
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1answer
73 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
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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}$$
Now the matrix is ...
2
votes
1answer
70 views
Eigenvalues of a mean correlation matrix (integral over correlation matrices with arbitrary density)
Consider a stationary dynamic system with state $s(t)$ and correlation structure described by $C_{ij}(\tau)=\mathbb{E}[(s_i(t+\tau)-\bar{s_i})(s_j(t)-\bar{s_j})]$. Given an arbitrary density function ...
0
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1answer
137 views
Jacobi's matric formulation for tensors
Hi I am trying to derive the E field equation and am stuck using the Jacobi formula, is this correct: $$\delta \det g_{\mu \nu} = Tr(ADJ(g_{\mu\nu}\delta ...
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1answer
130 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
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1answer
170 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
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1answer
72 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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4answers
205 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 = ...
1
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2answers
374 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 ...
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4answers
601 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 ...
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2answers
151 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 ...
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2answers
695 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 ...
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2answers
160 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
56 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:
$$T = T^{\mu\nu}_{\rho} e_\mu \otimes e_\nu ...
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2answers
127 views
Will a one year undergraduate course of Linear Algebra be enough for QM? [duplicate]
Possible Duplicate:
Linear Algebra for Quantum Physics
Can you get all/most of the knowledge you need of Linear Algebra for QM in a one year course? I know for certain my course also ...
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5answers
446 views
Linear Algebra for Quantum Physics
A week ago I asked people on this forum what mathematical background was needed for understanding Quantum Physics, and most of you mentioned Linear Algebra, so I decided to conduct a self-study of ...
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2answers
337 views
Metric tensor and its inverse
Is it always allowed to represent the metric tensor $g_{\mu \nu}$ in General Relativity as a $4\times 4$ matrix?
If the last one is represented for example with a $4\times 4$ matrix ...
1
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1answer
192 views
Tensors: relations between physics and linear algebra
In continuum mechanics we use finite deformation tensors to exprime deformations in a point. The 9 components of the tensor (in reality 6 because of its symmetry) are defined as
$$
...
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2answers
141 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?
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5answers
589 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 ...
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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 ...
7
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3answers
561 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 ...
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0answers
28 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 $\sigma(H)$ ...
3
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1answer
81 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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1answer
146 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 ...
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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
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3answers
133 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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3answers
176 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 ...
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1answer
360 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
...
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3answers
120 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 ...
5
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2answers
190 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 ...
2
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2answers
302 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 ...
2
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3answers
126 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 ?
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4answers
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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 ...
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3answers
552 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 ...
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5answers
482 views
Operator vs linear transformation
One of the postulates of quantum mechanics is that every physical observable corresponds to a Hermitian operator $H$, that the possible outcomes of the measurements are eigenvalues of the operator, ...
1
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1answer
445 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 ...
0
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0answers
174 views
Equations that govern SPICE models and how to perform the same simulations in MATLAB [closed]
SPICE is a great tool, but I want to be able to write my own code in MATLAB such that I can get faster results.
To start with, I have a simple but specific requirement. Once I understand and can ...
1
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3answers
124 views
Question about orthonormal decompositions over unitary operators
I'm teaching myself quantum information theory using Nielson and Chuang's "Quantum Computation and Quantum Information" and I'm at a point in the book where the formalism is starting to make my eyes ...
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3answers
485 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
...
