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.
1,257
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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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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
$$\mathbf{w}=\frac{\partial \omega}{\partial \mathbf{...
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What is the most appropriate mathematical theory for electrical circuits?
What exactly are electrical circuits as mathematical objects?
It seems quite intuitive to me, that they are geometric realization of some graph with some additional structure.
Another thing I notice ...
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What does it mean to differentiate a scalar with respect to a vector?
I am reading the special relativity lecture notes that I got from a professor of mine. It says that the Lagrangian is
$$L = \frac{1}{2}m|\dot{\boldsymbol{x}}|^2 - V(\boldsymbol{x}) \tag{1}$$
The notes ...
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Is math really fundamental or would it not matter if physics did not depend on math? [closed]
Can you disprove this please because it's been bugging me and I don't know much about physics at all as I'm only in 10th grade. It's one of those weird thoughts but I would like confirmation to keep ...
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Greens Theorem for periodic functions
Ashcroft and Mermin supply the following proof of their equations (I.1/2), which get used often in computing integrals over the first Brillouin zone (in computing current densities etc.).
I find ...
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Expanding state variables and state functions of a thermodynamic system
In this Wikipedia article under the section "Heat capacities of a homogeneous system undergoing different thermodynamic processes" there is on line that says:
$$
\delta Q=dU+pdV=\bigg(\frac{\...
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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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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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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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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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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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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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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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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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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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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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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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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
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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)...
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3
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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 ...
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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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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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2
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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
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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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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
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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 ...
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1
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295
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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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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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Which summation should be chosen for a divergent series arising from the expression of relative mass in order to always preserve the same mass? [closed]
In all physical theories, the appearance of infinity is generally regarded as a sign that the theory is either incorrect or being applied outside its applicable domain, necessitating the search for ...
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Schur's Lemma in Zee's Group Theory book for reducible representations
Main question
Schur's lemma says:
$$D(g) A = A D(g) \Rightarrow A = \lambda I\tag{1}$$
if $D$ is irreducible. How can I use this to show that if $D$ is reducible and if $SDS^{-1}$ is a direct sum of ...
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Deriving Azimuth Algorithm: Limitations of Accelerometer and Gyro Sensor Setup
I have a navigation device for which I am trying to derive an azimuth algorithm. I am getting stuck with the derivation, as I'm pretty sure it is impossible given the hardware I have access to. I'm ...
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What does an antenna actually measure? The magnitude of the electric field, or the real component of the complex phasor representation?
I'm in the process of writing a physics simulation code, involving some antenna modeling. In the process of doing so, I've realized that I'm not so confident in my understanding of what an antenna ...
2
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1
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If two models generate identical predictions, must one be mathematically reducible to the other?
If two models always generate identical predictions of measurable observables given identical inputs of measurements over a given domain, must one be mathematically reducible to the other over that ...
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Concept of generators in group theory
I am in the process of learning group theory, and I am a little confused about the concept of generators. As I understand it, groups have associated "generators" that can produce every ...
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1
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Reformulating problem into form of Ising Hamiltonian
The Ising Hamiltonian has the following form:
$$H= -\sum_{j<k}J_{j,j+1}\sigma_j\sigma_{j+1}-\sum_{j} h_j\sigma_j + \varepsilon,$$
Where $\sigma$ are the spins that take values of $\pm$ 1
I have a ...
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1
answer
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How to compute this form factor? [closed]
I would really appreciate if someone could help me about computing the form factor $$F(q^2)=\int_0^{\infty}\rho(r)e^{-iq\cdot r}d^3r \quad, $$ when $$\rho (r)=\begin{cases}\rho_0& r<R\\0& r&...
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What are the applications of fractional convolutions to physics - if any?
There are multiple applications of the convolution operation to physics. For instance, it comes about in calculations in fluorescence spectroscopy, acoustics, and computational fluid dynamics.
The ...
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3
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What is wrong with this analogy finding equality between two formulae of average velocity?
I have seen several questions on this confusion. Most are related with the issue of using variable acceleration. So here is an example where I am using a constant acceleration, but it still seems to ...
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Why are quantum harmonic oscillators everywhere?
My question is not referring to the fact that the harmonic oscillator (HO) is employed in a lot of physics models. For example, this question asks about the universality of HO's, and the answers focus ...
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Prove a Hermitian operator satisfies a property
Assuming $A$ as hermitian operator, then $\langle A \phi|\phi \rangle= \langle \phi|A\phi \rangle$ holds. I need to show that this is equivalent to $\langle A\psi|\psi' \rangle = \langle \psi|A\psi' \...
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The eigenvectors obtained in the diagonalization in the paper "Two Soluble Models of an Antiferromagnetic Chain" by Lieb, Schultz and Mattis [closed]
The diagonalization involves few transformations, that transforms the anisotropic XY model in a matrix eigenvalue equation as $(A-B)(A+B)\phi_k= \lambda^2_k \phi_k$ where the matrix $(A-B)(A+B)$ has a ...
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3
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What theorem is behind writing an operator in matrix form as outer products?
Consider a Hamiltonian of a two state system that follows, for the two eigenstates:
$$H|\phi_1\rangle = E_1|\phi_1\rangle \ ; \ H|\phi_2\rangle = E_2|\phi_2\rangle $$
Its matrix can be represented as:
...
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0
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Solving an equation with Laplace operator for a specific solution in spherical and polar coordinates
I am trying to solve an eigenvalue problem related to Laplace operator. I want to find a specific solution that only depends on the radial coordinate $r$ to the equation
$$\nabla^2 \psi = -k^2 \psi$$
...
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0
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72
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More mathematically formal textbook on general relativity [duplicate]
I was going through the lectures of F. Schuller in the International Winter School on Gravity and Light 2015, and I finally understand thing due to the differential geometry chartless formalism. But I ...
5
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0
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Validity of Lorenz gauge in non-Abelian gauge theory
I understand that this is a long shot, especially because it's such a niche question but: has it been mathematically proven that (under sufficient smoothness conditions, etc.) any field configuration ...
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3
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Temperature is discrete but not countable? [closed]
So I was reading a a question and top comment on math stack exchange that didn't make sense to me.
you can measure the temperature of something, but you can't count it. Incidentally, I claim ...
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Confusion regarding working around with the mathematics of operators acting on the whole tensor product state
In 'Quantum Information by Stephen Barnett' (Page 95), we have:
$$
P(m, l)=\left|{ }_a\langle l|\otimes\langle m|\hat{U}| \psi\rangle \otimes| A\rangle_a\right|^2=\left\langle\psi\left|\hat{\pi}_{m l}\...
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votes
2
answers
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Why isn't there a discontinuity in the derivative of the wavefunction with a finite square well?
Consider a finite square well with depth $-V_0$ over $[0,L]$. Let's say we want to study the continuity of the derivative of $\psi$ around the point $0$. Then, we can use $(\epsilon>0)$:
$$\psi'_{\...
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votes
1
answer
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How to find the density operator of two joint systems given the density operator of the individual systems?
If I have two systems A and B and I want to find the density operator of the joint systems $\rho_{AB}$, is it just the tensor product of the density matrix of the individual systems, $\rho_{AB} = \...
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