Questions tagged [mathematics]
DO NOT USE THIS TAG just because your question involves math! If your question is on simplification of a mathematical expression, please ask it at math.stackexchange.com The mathematics tag covers non-applied pure mathematical disciplines that are traditionally not part of the mathematical physics curriculum, such as, e.g., number theory, category theory, algebraic geometry, general topology, algebraic topology, etc.
1,184
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Coupled Resonators with sinusoidal coupling [closed]
I have two coupled resonators with complex resonance frequencies of $\beta_1$ and $\beta_2$. The coupling between them is varying sinusoidal with time. The time-evolution of their resonant mode field (...
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2
votes
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46
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Characteristic classes and index theorems for physicists
Since characteristic classes and index theorems are occasionally used in the QFT (for example, when discussing instantons or quantum anomalies), I want to learn more about them. Is there any good ...
0
votes
1
answer
51
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The Weird Interpretation for Contravariant and Covariant Vector
I have seen the answer for related topics, and it makes sense to me for the trivial contranvariant expression for a vector,
$$\pmb{v} = v^i\hat{e}_i\tag{1}$$
and it is said that if the base $\hat{e}_i$...
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votes
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33
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Finding numerical solution of differential equation [closed]
I investigated my system and encountered differential equation such as
$y''(x) = e^{y(x)}-e^{-2y(x)}-e^{y(x)-y(x+d)}$,
where $d$ is a constant. I can write python script rest of the thing, but I ...
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0
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21
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Can this problem be sloved by calculation in coordinate system [closed]
Point M is the shared point of line AB ande the circle,and you need to calculate the acceleration of piont M at the moment that is shown on the picture
1
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0
answers
43
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Why there is only one modulus for a Klein bottle?
In Polchinski Page 207, there is a claim that only one modulus for a Klein bottle, I tried to understand this claim, Here's what I have tried:
As I understand, modulus means the geometries that are ...
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votes
0
answers
54
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How do we know the geometry of our physical world is made from real numbers and not rational numbers? [duplicate]
If I draw a line on a paper from point a to point b, how do we know that each point on the line exists in the real space, and not the rational space? How do we know if I randomly draw a dot, it won't ...
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2
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105
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What does it mean to transform a tensor?
My electrodynamics lecture only defines a tensor (in $R^3$) of rank $n$ as a "set of quantities $T_{i_1i_2...i_n}$, $i_1, ... i_n = 1,2,3$ that transforms under rotation as follows:
\begin{...
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1
answer
84
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Scalar Product Calculation and Identical Particles in Quantum Mechanics
In the book "Nolting, Theoretical Physics Part 5/2" (German), on Page 264, Formula 8.80, the author introduces second quantization in the case of identical particles. One considers the ...
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How to obtain approximation solution for this differential equations? [migrated]
I am looking for approximation solution for this differential equations
\begin{align*}
& \ddot{x}=-2\,\Omega\,\cos \left( \phi \right) {\dot z}+2\,\Omega\,\sin \left(
\phi \right) {\dot y}\\
&...
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0
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Is the character table of $S_3$ unique? [migrated]
I'm trying to construct the character table of $S_3$ group.
$n_c$
class
$1$
$\bar{1}$
$2$
1
$I$
1
1
2
3
$(12),(23),(13)$
1
a
b
2
$(123),(132)$
1
c
d
As a brute force method, I imposed every ...
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58
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Reference for mathematics of quantum mechanics with infinite degrees of freedom?
I am looking for a book, or lecture notes or even courses available on YouTube where there is a good and detailed discussion on the mathematical aspects of Quantum Mechanics with infinite degrees of ...
3
votes
1
answer
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Do matrices with this property appear in physics?
First I should mention that my background is in Mathematics, but I am looking for a motivating example in physics. I apologize in advance if my question does not meet the standards of this site.
...
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votes
2
answers
98
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Eigenvalues of an time-ordered exponential operator
Let's consider a simple 1-qubit time-dependent Hamiltonian: $$H(t) = \delta(t) \sigma_x + \sigma_z \ ,$$
where $\delta(t)$ is some time-continuous (real-valued) function.
Evolving $H(t)$ continuously ...
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1
vote
1
answer
61
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Lagrangian total time derivative - continues second-order differential
In the lagrangian, adding total time derivative doesn't change equation of motion.
$$L' = L + \frac{d}{dt}f(q,t).$$
After playing with it, I realize that this is only true if the $f(q,t)$ function has ...
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1
vote
0
answers
26
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After equation (1.6) the author said that "The sequence of integration and limiting process in Eq. (1.6) must not be interchanged" but he interchanged [migrated]
In the attached screenshot, after equation (1.6) the author stated clearly that the sequence of integration and limiting process in Eq. (1.6) must not be interchanged, which imply to me that if we did ...
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votes
1
answer
49
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Matrix representation of squeezing operator
In quantum mechanics we know that every operator can be represented by a matrix.Being a beginner of quantum optics, my question is does there exist a matrix for squeezing operator also? If does, can ...
0
votes
1
answer
40
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Is the ${u_{vs}}=0$ when $P$ is a constant?
In Callen's book Thermodynamics and I (second edition) p.124
To corroborate equation 4.22 show that $${\left( {\frac{{\partial P}}{{\partial s}}} \right)_T} = - {\left( {\frac{{\partial T}}{{\...
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votes
1
answer
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Doubt about Huang's proof of Equipartition Theorem in Microcanonical Ensemble
I'm trying to understand the proof of the Generalized Equipartition Theorem in the Microcanonical Ensemble. The proof that appears in Wikipedia is the same that can be found in Kerson Huang's ...
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2
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0
answers
99
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A question regarding Hermite polynomials and exponential operators $\exp[e^{x^2/2}p(\frac{d}{dx})e^{-x^2/2}]f(x)$
Is it possible to express $$\exp\left[\mathrm{e}^{x^2/2}p\left(\frac{d}{dx}\right)\mathrm{e}^{-x^2/2}\right] f(x)$$ as an integral transform or something similar? $p(x)$ is a polynomial.
$$\exp\left[\...
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2
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185
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Why does separating variables in the spherically-symmetric TISE work?
When solving the one-particle Schrodinger equation for a spherically symmetric potential, we use the substitution $$\psi(r,\theta,\phi) = R(r)Y_l^m(\theta,\phi)$$ in order to solve. However, I am ...
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0
votes
1
answer
33
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Group velocity vector in spherical coordinates
I am trying to understand a derivation from the textbook Radiation Processes in Plasmas by G. Bekefi (p. 14).
Start with the group velocity vector
$$\mathbf{w}=\frac{\partial \omega}{\partial \mathbf{...
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32
votes
9
answers
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What is the most appropriate mathematical theory for electrical circuits?
What exactly are electrical circuits as mathematical objects?
It seems quite intuitive to me, that they are geometric realization of some graph with some additional structure.
Another thing I notice ...
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1
vote
1
answer
98
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What does it mean to differentiate a scalar with respect to a vector?
I am reading the special relativity lecture notes that I got from a professor of mine. It says that the Lagrangian is
$$L = \frac{1}{2}m|\dot{\boldsymbol{x}}|^2 - V(\boldsymbol{x}) \tag{1}$$
The notes ...
- 1,160
0
votes
1
answer
99
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Is math really fundamental or would it not matter if physics did not depend on math? [closed]
Can you disprove this please because it's been bugging me and I don't know much about physics at all as I'm only in 10th grade. It's one of those weird thoughts but I would like confirmation to keep ...
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0
answers
37
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Greens Theorem for periodic functions
Ashcroft and Mermin supply the following proof of their equations (I.1/2), which get used often in computing integrals over the first Brillouin zone (in computing current densities etc.).
I find ...
- 1,135
2
votes
1
answer
37
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Expanding state variables and state functions of a thermodynamic system
In this Wikipedia article under the section "Heat capacities of a homogeneous system undergoing different thermodynamic processes" there is on line that says:
$$
\delta Q=dU+pdV=\bigg(\frac{\...
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0
answers
28
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Are elements of different invariant subspaces of a self-adjoint set orthogonal?
I know that self-adjoint operators have orthogonal eigenspaces, but how does that generalize to the orthogonality of invariant subsapces?
I am reading Fonda's Symmetry Principles in Quantum Physics ...
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2
votes
2
answers
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Question about the time-ordered exponential operator
I learned that a unitary matrix generated by time-dependent Hamiltonians is written down as
\begin{equation}
U(t) = \mathcal{T}\exp\Big(-i\int_0^t H(t') dt' \Big),\tag{1}
\end{equation} where $\...
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1
vote
0
answers
63
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Computing the eigenvalue of an operator in second quantization [closed]
At the moment I am reading a specific paper: "Effective pair interaction between impurity particles induced by a dense Fermi gas" by David Mitrouskas and Peter Pickl.
I am wondering about a ...
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10
votes
2
answers
860
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Distributions "more singular than a Dirac delta" must have negativity
I am looking at properties of the Glauber P-functions, which are distributions (in the sense of a dirac delta) on the complex plane, normalized so that $\int_{\mathbb{C}} d^2 \alpha P(\alpha) = 1$. On ...
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1
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1
answer
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A standing wave may be expressed as superposition traveling waves - is the converse, traveling as superposition of standing, also true?
So I consider the wave equation:
\begin{align*}
\frac{\partial^2 u}{\partial t^2} = c^2 \left(\frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\...
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2
votes
1
answer
91
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Derivation of Leibniz Rule for Exterior Derivative
I was reading Sean Carrol's GR book, when on page 85 he introduces the Leibniz rule analogue for exterior derivatives:
$$\text d(\omega\wedge\eta) = (\text d\omega)\wedge\eta + (-1)^p\omega\wedge(\...
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0
answers
44
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How does convex splitting method work?
I'm an undergraduate physics student and I'm simulating some partial differential equations using finite element method. For non-linear equations I found a method called linear convex splitting ...
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2
votes
2
answers
399
views
What is the cardinality of intervals in space, and what is the cardinality of intervals in spacetime?
The interval $|(0, 1)| = |\mathbb{R}|$. I naively thought that one could treat intervals in space in kind, i.e., that the cardinality of any interval in space has the cardinality of the continuum. You'...
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1
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0
answers
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Ladder Operations and Rodriguez formula [closed]
During my attempt to prove the Rodriguez Formula for Hermite Polynomials by using the Ladder Operators,
$H_n(x) = (-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}$, I arrived to the formula $H_n(x) = e^{x^2/2}(...
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14
votes
4
answers
3k
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Can electric field be discontinuous?
"This is because of abrupt discontinuity of fields"
I have read this or similar sentences in many papers. I am bit puzzled. How and under what conditions electric field can be discontinous? ...
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0
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37
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About the form of the cost functional for quantum optimal control theory
In quantum optimal control theory, the cost functional is often defined as (e.g, see Eq.(9) in here, as well as many other solid references such as this):
$$J = \langle \psi(T) \rvert O \lvert \psi(T) ...
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votes
1
answer
24
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Diagonal block of a capacitance matrix is also a capacitance matrix
I am trying to conclude the answer to the following question (which I posed on the mathematical branch of the community):
https://math.stackexchange.com/questions/4739174/linear-system-ax-y-with-...
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0
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Determine time taken for light to travel between points through circular, sub-wavelength aperture
Background
I'm working on numerically modeling some physical phenomenon using electromagnetics simulation software. I have a particular physical setup modeled, wherein I've noticed some non-intuitive ...
2
votes
0
answers
57
views
Green's identity for arbitrary differential operators
If we have a scalar field $\psi$ that satisfies an equation $\nabla^\mu \nabla_\mu \psi = \rho$ where $\rho$ is some known source we can use Green's identity to express it as
\begin{equation}
\psi (x)...
- 4,550
3
votes
3
answers
112
views
Can 2 vectors in dimensions greater than 3 be found on a plane? [closed]
I was thinking about how, given any 2 linearly independent vectors in a 3D cartesian coordinate system, one can always find another 3D cartesian coordinate system (or should I say frame) where the 2 ...
- 155
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votes
0
answers
47
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Non-differentiable solution of the Brachistochrone problem
Is there a solution to the brachistochrone problem where the solution is non-differentiable everywhere (angular point)?
The Euler-Lagrange method fails if the first or second derivative of the ...
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vote
1
answer
91
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Jackson's Electrodynamics: Green's function prefactor
In Ch. 6 of Jackson's Classical Electrodynamics 3rd ed., the Helmholtz equation Green's function is written as satisfying the following inhomogeneous equation (Eqn. 6.36):
$$ (\nabla^2 + k^2)G(\mathbf{...
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1
vote
2
answers
56
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Expressing infinitesimal physical quantities
In physics class, my teacher demonstrated that in polar coordinates, an infinitesimal area involving radial length dr and infinitesimal angle dθ is equal to rdr dθ, since the area is roughly a square ...
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0
answers
33
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The right domain for Hamiltonians
This question came to me today, and I am now intrigued about it. For a system described by a Lagrangian $L$, the associated Hamiltonian is its Legendre transform. Suppose we consider a given ...
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0
answers
290
views
Nancy Cartwright's distinction [closed]
Nancy Cartwright introduced an interesting distinction in the context of her study of the history of the evolution of our understanding of superconductivity. She emphasized the distinction between ...
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0
votes
0
answers
60
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Reference request - classical field theory and mathematics
I am looking for references (books, lecture notes etc) on mathematical classical field theory. By that, I mean classical field theory under a rigorous point of view. However, I am more interested in ...
0
votes
1
answer
91
views
Is sum of two operators hermitian? [closed]
Given a three dimensional Hilbert space with the three basis vectors $|1\rangle, |2\rangle, |3\rangle,$ and two state vectors, $|\psi_1 \rangle = a|1\rangle -b |2\rangle +c|3\rangle, |\psi_2 \rangle = ...
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0
votes
1
answer
93
views
Lie algebra with $\sim \!N^3$ generators [closed]
Is there a Lie algebra whose number of generators scales as $N^3$, or in general $N^p$ with $p$ an arbitrary positive integer?
All the familiar examples, such as $\mathrm{U}(N)$ or $\mathrm{SU}(N)$ or ...
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