Questions tagged [linear-algebra]
To be used for linear algebra, and closely related disciplines such as tensor algebras and maybe clifford algebras.
1,078
questions
0
votes
1
answer
24
views
Why does the eigenvector I calculated for 𝑆𝑦 in the spin-1 case give an incorrect probability for measuring ℏ? [closed]
I've been working on a problem involving a spin-1 particle in state:
$ |\chi _{z}^{0}>=\begin{pmatrix}
0 \\
1\\
0
\end{pmatrix}$
Which corresponds to the $m_{z}=0$ state of the spin-1 particle. ...
- 11
6
votes
2
answers
634
views
How can I calculate derivative of eigenstates numerically?
I want to calculate $\langle n | \partial_{k_x} n \rangle$ where $| n \rangle \equiv | u_{n,\mathbf{k}} \rangle $ is the $n$-th Bloch eigenstate of a $6\times6$ Hamiltonian $H\equiv H(\mathbf{k})$. ...
- 1,417
0
votes
0
answers
54
views
Are projectors observable quantities in QM? [duplicate]
Given a certain quantum state $|\Psi\rangle$, then we can project any other quantum state $|\Phi\rangle$ on the first one by using the projector: $P_\Psi\equiv |\Psi\rangle\langle\Psi|$ in such a way ...
- 1,836
9
votes
1
answer
2k
views
"There is a bra for every ket, but there is not a ket for every bra"
This statement is an excerpt from Quantum Mechanics (Cohen-Tannoudji), but I don't quite understand why it holds: given all our kets live in a certain Hilbert space $\mathcal H$, then all the bras ...
- 1,836
0
votes
0
answers
15
views
Parity eigenstates in massive $\mathcal{N}=1$ multiplet
In $\mathcal{N}=1$ SUSY we have two scalar states in the massive chiral multiplet. These are $\left|\Omega\right>$ and $\left|\Omega'\right> = \bar{Q}_1 \bar{Q}_2\left|\Omega\right>$, which ...
- 1,925
0
votes
1
answer
78
views
Goldstein Chapter 6 Question
I have a question about a potential error in the $3^{\mathrm{rd}}$ edition of Goldstein's Classical Mechanics. In their exposition in Chapter 6 of small oscillations, the authors obtain the usual ...
- 279
0
votes
0
answers
41
views
Number of non-zero one- and two-body integrals in second quantization Hamiltonian
The electronic Hamiltonian of a molecule with Born-Oppenheimer approximation in second quantization is given by
$$\hat{\mathcal{H}}_{el} = \sum_{p,q} h_{pq} \hat{c}_{p}^{\dagger} \hat{c}_{q} + \frac{...
- 9
1
vote
1
answer
54
views
Finding inner product value in QM given change of basis
Given the linear transformation
$x\rightarrow x'$
$y\rightarrow y'$
$z\rightarrow z'$
where $\{x', y', z'\}$ are linear combinations of $\{x,y,z\}$
How can I find the value
$\langle x',y',z'|x,y,z\...
- 17
2
votes
0
answers
38
views
How do people calculate eigenstate amplitudes for finite lattices
I'm given the hamiltonian of a generalized Hatano-Nelson chain,
$$ H_{lr} = \sum_{n=1}^{N} (t_l c_n^\dagger c_{n+l} + t_{-r} c_n^\dagger c_{n-r})$$
How would one normally go about finding the spatial ...
- 343
0
votes
3
answers
72
views
Position Eigenstate and Parity Operator
I would like to ask for help with a question that I can't solve.
I know that in quantum mechanics an abstract state $|x\rangle$ (position) is defined up to a phase $e^{i\theta}|x\rangle$.
Now we know ...
user423183
0
votes
0
answers
74
views
Doubt in QFT lecture 4 by Osborne: Circulant matrices
Professor Osborne mentioned here that Hamiltonian can be written in form of circulant matrix so we can use a Fourier transform to diagonalize it.
I just wanted to read a bit more about it but I can't ...
- 21
4
votes
3
answers
438
views
Is It Possible to Assign Meaningful Amplitudes to Properties in Quantum Mechanics?
In informal discussions of quantum mechanics, it is common to talk about the amplitude for a state to have a certain property. For example, when discussing the double-slit experiment, Feynman states ...
- 714
0
votes
1
answer
79
views
How does the eigenvector interpretation of Lorentz transformations generalise to higher dimensions?
When I came across this YouTube video, I immediately realised how Lorentz transform corresponds to a linear transformation where the vectors representing the speed of light ($\vec{c}$ and $-\vec{c}$) ...
- 1
0
votes
0
answers
32
views
Eigenvalue under Lorentz metric
I'm currently reading Hawking and Ellis and came across a statement regarding eigenvalues of a matrix under the Lorentz metric, represented by $diag\{1,1,1,-1\}$. A matrix $T$ is defined as follows:
\...
- 21
0
votes
1
answer
71
views
Base change using bra-ket notation
If I have two orthonormal bases (of a vector space over $\mathbb{C}$) $A=\{|a_{1}\rangle, |a_{2}\rangle\}$ and $B=\{|b_{1}\rangle, |b_{2}\rangle\}$, the change of base matrix from $A$ to $B$ is given ...
- 3
1
vote
0
answers
47
views
Non-linear Eigenvalue problem with an ODE to solve for the Linear Stability Theory of a Boundary Layer
I am working on the Linear Stability Theory (LST) for analysing the stability of a boundary layer in fluid mechanics. I have conducted CFD simulations and extracted the base flow data. To form the LST ...
- 11
1
vote
0
answers
67
views
Associated Hermite polynomials
In quantum mechanics, one of the most common problems to solve is the quantum harmonic oscillator.
\begin{equation}
\mathcal{H}=\frac{1}{2}p^2+\frac{1}{2}x^2
\end{equation}
The most elegant way to ...
-1
votes
1
answer
49
views
What is the dimension considered in the Schmidt Decomposition?
In the Schmidt decomposition, is the dimension considered of each Hilbert space the complex or real one? Meaning the complex dimension of $\mathbb C^2$ has dimension $2$, not $4$. If so when you ...
- 139
0
votes
0
answers
64
views
Einstein's notion of "covariant"
In his The Meaning of Relativity, pg. $11-12$, Einstein explains the notion of "covariant" along the following lines:
Consider a point $\mathbf x$ on a straight line $\mathbf x -\mathbf A=\...
- 421
1
vote
2
answers
101
views
What is the correct way to think of position?
How accurate would it be to think of position (along some axis) as the component of radius vector. Example:
$$ \textbf{r} = x \hat{\textbf{i}} + y \hat{\textbf{j}} $$
And if that is correct, we could ...
0
votes
0
answers
65
views
Split Pauli Four-vector as quadratic terms of spinors
If I have the Pauli Four-vector $$x_{\mu}\sigma^{\mu} = \left(\begin{array}{cc}
t+z & x-i y \\
x+i y & t-z
\end{array}\right)$$ with $\sigma^0$ as Identity Matrix. Is there some way to write ...
0
votes
3
answers
107
views
Can I use any linearly independent, orthogonal, eigenkets as starting basis to construct $S_x$, $S_y$ and $S_z$? [closed]
I know how to construct $S_z$ using $|\uparrow\rangle$=$\left(\begin{matrix}1\\0\end{matrix}\right)$ and $|\downarrow\rangle$=$\left(\begin{matrix}0\\1\end{matrix}\right)$ as starting basis. And I can ...
- 1
-1
votes
1
answer
94
views
Why is the tensor product not a multilinear application? [closed]
I was studying multilinear algebra and I wanted to have a good understanding for relativity. I started studying multilinear applications, which are kind of like a natural extension of linear ...
- 67
0
votes
2
answers
72
views
What is the advantage of using spherical tensor over cartesian tensor?
I am trying to train a machine-learning model to forecast the polarizability of atoms within a molecule. Typically, the tensor is characterised as a Cartesian rank-2 tensor, like this:
$$\alpha= \...
- 101
3
votes
1
answer
148
views
Rotation of spherical harmonics
I have a question about the rotation of spherical harmonics. In Wikipedia it is mentioned that if we make a rotation in 3D space: $R\vec{r}=\vec{r}'$,then the Spherical Harmonics can be written as a ...
0
votes
1
answer
101
views
How to use the definition of a rank-$2$ tensor for this kind of examples?
Suppose that, a rank-$2$ tensor transforms as
\begin{align}
T'^{ij}=\frac{\partial x'^i}{\partial x^k}\frac{\partial x'^k}{\partial x^l}T^{kl}.
\end{align}
How to use this criterion to investigate if ...
- 195
-2
votes
1
answer
128
views
Multiplication of $\mathrm{U}(3)$ matrices [closed]
On this page of this paper:
I am unable to understand how they multiplied the $3\times 3$ $\mathrm{U}(3)$ matrix with $T_{3,2}$, which is a $2\times 2$ matrix, in Eqs. (26) to (28). Can anyone please ...
- 115
0
votes
3
answers
114
views
Is that right that about trace of the two density matrices multiply to each other? [closed]
We have:
$$\mathrm{Tr}\,(\rho \rho^{\prime})=1$$
then, is it right to say $\rho=\rho^{\prime}$?
- 27
0
votes
2
answers
70
views
Is the tensor product injective on pure quantum state vectors?
I am reading lecture notes on quantum information/computing, and the tensor product of two pure qubit states $|b_1\rangle\otimes |b_2\rangle\in\mathbb{C}^{2\times2}$ was introduced as the "...
- 1
0
votes
0
answers
33
views
Permanent operation's result
N-body fermionic systems are constructed by Slater determinant, and it is equal to Vandermonde polynomial. Are there any special polynomial for the permanent which is used to construct N-body ...
- 31
0
votes
1
answer
87
views
Decomposition of $(x \pm i y) \, Y_{l m}$ and $z \, Y_{l m}$ on spherical harmonics
Using the various algebraic properties of the associated Legendre polynomials $P_l^m(u)$ and of the spherical harmonics $Y_{l m}(\theta, \varphi)$, I was able to decompose the following expressions, ...
- 7,562
0
votes
1
answer
83
views
Calculating Eigenkets of Perturbed Matrix for Second-Order Correction
Q: Find the eigenvalues of the 3x3 symmetric matrix $H$ using perturbation theory where all of the elements on the diagonal of $H$ are an order greater than the elements not on the diagonal.
We can ...
1
vote
2
answers
317
views
Problem with logarithm of tensor product of matrices
In the book of From Classical to Quantum Shannon Theory, in exercise 11.8.1, there is a property of logarithm of a tensor products of two matrices, defined as follows:
$$\log ( A \otimes B) = \log(A) \...
- 29
0
votes
1
answer
119
views
Rotation of Pauli Vectors with $SU(2)$ reproduces the $SO(3)$ matrix. but do all $SU(2)$ matrices reproduces $SO(3)$?
So we can write the $SU(2)$ matrices multiplication as this.
$$\begin{bmatrix}\alpha&\beta\\-\beta^*&\alpha^*\end{bmatrix}\begin{bmatrix}z&x-iy\\x+iy&-z\end{bmatrix}\begin{bmatrix}\...
- 15
0
votes
1
answer
75
views
Why does applying Ladder operators change the eigenfunction?
When applying a ladder operator to a spherical harmonic function, it spits out the function with a lower or higher magnetic quantum number.
My question is how does this abide by the classical ...
- 3
0
votes
1
answer
88
views
Regarding Energy Eigenstate and Position Eigenstate
I am solving problem 14.4. (a) of Schwartz's Quantum Field Theory and the Standard Model. It is related to the simple harmonic oscillator in quantum mechanics. It asks the eigenstate of the position ...
- 79
0
votes
1
answer
95
views
Operators algebra for quantum mechanics [closed]
I am taking my first quantum mechanics course and I am a bit lost in operators algebra. These are the main questions I have:
Why can we write this kind of equations? $$ Ô \psi = o\psi $$
What I mean ...
0
votes
1
answer
130
views
Rotating a system
brekely physics book chapter 2 page 30 , a question about rotating a system by $ \frac{\pi}{2} $ around the z axis clockwise direction and writing vectors according to the new axis after rotation
...
- 31
1
vote
0
answers
28
views
Intuition behind dampening as a rotation in the complex plane
I'm currently building my intuition behind the meaning of the variables within the general second-order damped harmonic oscillator frequently taught in engineering:
$$
\ddot{x} + 2\zeta\omega_n\dot{x} ...
Sodapop3421
3
votes
2
answers
343
views
Question about the identity operator and the bosonic ladder operators
Consider a self-adjoint operator $B$, such that for each mode $a_1,...,a_n$ [of a quantum bosonic system with Hilbert space $\cal H$ given by the corresponding Fock space] we have $B a_i B^\dagger = ...
- 43
1
vote
1
answer
131
views
Understanding equation for eigenvalues of a Hamiltonian
I'm reading the paper Hamiltonian Truncation Study of Supersymmetric
Quantum Mechanics. I'm not understanding a claim they make about the eigenvalues of a certain Hamiltonian. In particular, how eqn 3....
- 213
0
votes
0
answers
32
views
Scattering matrix vs. unitary transformations
In quantum optics, the input/output bosonic modes at a beam splitter transform according to the scattering matrix
$$
\begin{pmatrix}
a_1 \\
a_2 \\
\end{pmatrix} = \dfrac{1}{\sqrt{...
- 1,221
2
votes
0
answers
120
views
What is the difference between a Lindbladian and a Liouvillian in open quantum systems?
As far as I know, when we try to write the Lindbladian equation in a generalized nice operator basis we get Liouvillian. Is this correct?
What are the differences between them?
0
votes
1
answer
46
views
Pauli matrix exponentials [closed]
Just a short query to confirm my understanding.
Given the Pauli-X operator $\hat{X}$ and it's eigenstates $|+\rangle:=\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and $|-\rangle:=\frac{1}{\sqrt{2}}(|0\...
- 297
1
vote
1
answer
89
views
Exponential of the metric tensor
Exponential of an arbitrary matrix can be written as
$$e^A = \displaystyle\sum_{n=0}^\infty \dfrac{A^n}{n!}$$
In Einstein notation, how this expression will look like?
In Einstein notation, what ...
- 768
1
vote
2
answers
455
views
Determinant of Rank-2 Tensor using Levi-Civita notation
In my Professor's notes on Special Relativity, the determinant of a rank-two tensor $[T]$ (a $4\times 4$ matrix, basically) is given using the Levi Civita Symbol as: $$T=-\epsilon_{\mu\nu\rho\lambda}T^...
- 462
2
votes
1
answer
151
views
Landau levels in symmetric gauge, what is the constraint on the quantum numbers?
After solving the Schrödinger equation for the charged particle in a constant and homogeneous magnetic field, using the symmetric gauge $\vec{A} = \frac{B}{2} (-y, x, 0)$, we could find the Landau ...
- 7,562
2
votes
3
answers
173
views
Why do QM books point out that $S^2$ commutes with $S_x$, $S_y$, and $S_z$?
The spin angular momentum magnitude squared operator:
$$S^2=S_x^2+S_y^2+S_z^2=\frac{3\hbar^2}{4}
\begin{pmatrix}1&0\\0&1\end{pmatrix}$$
Obviously $S^2$ commutes with everything, so why do QM ...
- 21
2
votes
2
answers
151
views
What can we say about the eigendecomposition of quantum channels?
It is known that quantum channels, being CPTP maps, map density operators to density operators. And thus, they can be seen as superoperators. Similar to operators, where eigenstates and eigenvalues ...
- 173
0
votes
2
answers
114
views
Einstein Summation Convention Confusion
My textbook:
The second bit confuses me. I asked a question on this site yesterday (Moment of Inertia tensor confusion) which involved the moment of inertia tensor and the term $$r_{i}r_{j}$$
The ...
- 75