Questions tagged [linear-algebra]
To be used for linear algebra, and closely related disciplines such as tensor algebras and maybe clifford algebras.
1,128 questions
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Is the symmetry of a system self-adjoint?
Say we have the statement;
An observable $A$ is a symmetry of the system if $[A,\hat H]=0$ where the Hamiltonian doesn't explicitly depend on time.
We now know that $A\hat H = \hat H A$
does this ...
0
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0
answers
112
views
Vector transformations that preserve norms but changes inner products between different vectors
Unitary transformations conserve the inner product structure of a set of vectors, they only change the direction of the vectors, i.e. rotate them all in the same way. A unitary transformation $U$ can ...
- 418
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votes
2
answers
238
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How to obtain the general explicit form of the vector state and wavefunction in the case of a continuous degenerate spectrum
$\newcommand{\ket}[1]{|#1\rangle}$
I'm trying to figure out what the general form of the vector state (and wave function) look like in the case of a continuous spectrum with (for now) discrete ...
3
votes
0
answers
107
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Computing the trace of powers of a transfer matrix when we can diagonalize its components
Suppose that we have two operators $A$ and $B$ which satisfy $[A,B]\neq0, A^{T}=A, B^{T}=B$. I'm going to keep these operators vague to be concise, but I have precise definitions of $A$ and $B$ in ...
- 941
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Eigenvectors of Bosonic Bogoliubov transform
I have a Bosonic Hamiltonian of the form
\begin{align}
H = - \sum_{k} \Big[
& A (\, a_k^\dagger a_k
+ \, b_k^\dagger b_k )
+ \frac{B}{2} \left( a_k a_{-k} + a_k^\dagger a_{-k}^\dagger
+ b_k b_{-...
- 1
7
votes
7
answers
916
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Vector spaces and how vectors represented by different units are related
To preface, I have an extensive academic and professional background in physics and math. I have done enough math where I am very comfortable solving problems and understanding what calculations need ...
- 303
1
vote
1
answer
242
views
Isotropic tensor function
I am studying turbulence and I came across the concept of isotropic tensors, that is tensor that are invariant under rotations and translations. After googling for a while the thing I found is that, ...
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124
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Tensor products and simultaneous eigenstates
In A Modern Approach to Quantum Mechanics, Townsend writes:
One of the most evident features of the position-space representations
(9.117), (9.127), and (9.128) of the angular momentum operators is
...
- 1,119
2
votes
0
answers
140
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About 'good basis' in time independent degenerate perturbation theory
In Sakurai's modern quantum mechanics (3rd edition), he tries to explicitly show the eigenvalue equation that shows
"we should use the linear combinations of the degenerate unperturbed kets that ...
- 41
3
votes
2
answers
204
views
Doubt About Euler Angles and Matrix Order Multiplication
I'm asking this question here because the doubt comes from trying to understand a physical problem (kinematics of rotations), but this question would easily fit the MathExchange site also.
I was ...
- 2,069
4
votes
1
answer
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Non-equilibrium thermodynamics in an eigenvector basis
I am currently studying non-equilibrium thermodynamics in the linear regime, and I am wondering why we define the constitutive equations (i.e: the relationship between flux and force) in a basis that ...
- 2,638
3
votes
2
answers
410
views
The inverse of a specific metric tensor [closed]
I am studying general relativity and here is a problem I encountered:
Suppose
$$
\mathrm{d}s^2=-M^2(\mathrm{d}t-M_i\mathrm{d}x^i)(\mathrm{d}t-M_j\mathrm{d}x^j)+g_{ij}\mathrm{d}x^i\mathrm{d}x^j
$$
or ...
- 333
2
votes
0
answers
205
views
Basis and inner product of the Dirac (Clifford) algebra
The Dirac algebra in 4D spacetime is composed of four $4\times 4$ gamma matrices $\{\gamma^\mu\}=\{\gamma^0,\gamma^1,\gamma^2,\gamma^3\}$ satisfying the following anticommutation relation:
$$\{\gamma^\...
- 769
5
votes
1
answer
382
views
The product of the square root of Pauli matrices
When we deal with the Dirac equation and its solutions, we often define the following quantities:
$$
\sigma^\mu\equiv(\mathbf{1},\vec{\sigma}) \ \ \ \text{and} \ \ \ \bar\sigma^\mu\equiv(\mathbf{1},-\...
- 769
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votes
0
answers
102
views
How to use the symmetry to get the eigenstates in twisted bilayer graphene (TBG)?
In tripod model of twisted bilayer graphene (TBG), the Hamiltonian with $ \mathbf{k} = 0$ is
\begin{align}
H = \begin{pmatrix}
0 & T_{1} & T_{2} & T_{3} \\
T_{1} & -\mathbf{q}_{1} ...
- 101
3
votes
1
answer
109
views
Operational meaning of spectral radius of quantum channels
It is known from Chap.6 of Wolf's lecture notes that CPTP maps or quantum channels have spectral radius 1. That is the absolute value of the eigenvalues is at most 1, corresponding to the asymptotic ...
- 205
0
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0
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123
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Probability of measuring $S_z$ when the particle has an arbitrary spin direction
Suppose we have to find the eigen values of spin operator $\hat{S}$ along a unit vector $\hat{n}$ that lies in XZ plane and then we are to find the probability measuring $S_z$ with $+\frac{\hbar}{2}$ ...
- 175
1
vote
2
answers
130
views
Canonical commutation relation between raising/lowering operators not satisfied [duplicate]
We know that the canonical commutation relation between the raising and lowering operator $\hat a_I$, $\hat a^\dagger_J$ should result in the identity $\delta_{IJ}$:
$$[\hat a_I, \hat a^\dagger_J] = \...
- 283
3
votes
2
answers
190
views
Why is the Kochen-Specker Contextuality proof difficult to do continuously?
I am an underclassmen in University, doing a little study on Kochen-Specker's and Bell's Theorem. I have not myself taken a Quantum class yet, though I am aware of some constructions in it, such as ...
- 31
0
votes
1
answer
165
views
Uniqueness of basis in QM
While reading Chapter 1 of Principles of Quantum Mechanics by R. Shankar, I got confused in the topic of "Simultaneous Diagonalisation of two Hermitian Operators". There the author claims ...
- 9,156
3
votes
1
answer
140
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Why do we require Hilbert spaces describing quantum systems to have a countable orthonormal basis [duplicate]
It is postulated that a quantum system is (in part) represented by a separable Hilbert space. It seems that the actual (equivalent to separability) property of interest is that the Hilbert space has a ...
- 4,103
6
votes
3
answers
543
views
How can I make the notion that a basis is fixed in time with respect to itself more precise?
Let $A = (O, (e_x,e_y,e_z))$ be the standard frame in $\mathbb R^3$, such that $O = (0,0,0)$ and $(e_x,e_y,e_z)$ is the standard basis. Additionally, let $B = (O, (b_x,b_y,b_z))$ be a moving frame in $...
- 199
1
vote
1
answer
141
views
Decomposing density matrix on non-orthogonal and incomplete set of states
I've recently come across the following conclusion in a quantum optics paper. The paper studies the dissipative dynamics of a two-qubit system and presents the following approximation for the steady-...
- 11
0
votes
1
answer
220
views
About the connection between distance in Hilbert space and fidelity
I have the feeling that there is a direct relationship between the distance in Hilbert space and the fidelity (similarity) between quantum states.
short version : the distance between two quantum ...
- 150
1
vote
1
answer
129
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Potential Textbook Typo in Calculation of Lorentz Transformation
In the 3rd edition of Goldstein's Classical Mechanics (pp. 283-284), the authors consider three inertial frames $S_{1}$, $S_{2}$, and $S_{3}$ where the Lorentz transformation from $S_{1}$ to $S_{2}$ ...
- 335