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0answers
14 views

Random walk of a polymer in an annular space - filling in some steps

In Muthukumar, M., 2003. Polymer escape through a nanopore, J. Chem. Phys., 118, 5174–5184, the author solves the diffusion equation for the probability $P(\vec{r},\vec{r_0},N)$ that the ends of a ...
1
vote
1answer
91 views

Question on quotient groups and SLOCC

I have a math-physics question, which is based on an interest in SLOCC systems for black hole entanglement. The Cartan decomposition of a group $G$ such that $H = G/K$ is such that the derivation or ...
5
votes
0answers
58 views

Is there an algorithm to diagonalize a matrix using gauge transformations

I have two matrices $U(\lambda, x,t)$ and $V(\lambda, x,t)$, where $\lambda$ is a parameter, which belong to the $sl(2)$ algebra, and satisfy the zero-curvature equation $$ \partial_t U - \partial_x V ...
0
votes
1answer
49 views

Velocity from the cumulative distribution function of the Boltzmann distribution

I want to get a Boltzmann distribution of the $v_x$, $v_y$ and $v_z$ velocity components (please, notice that the distribution is one-dimensional). To do so, I need the cumulative distribution ...
1
vote
2answers
37 views

Rotation matrix for aligning x-axis in an arbitrary direction

I want to align the x-axis of my coordinate system, with an arbitrary direction in space $\hat{n}$. About which axis should I rotate? Ceratinty rotation about x-axis or $\hat{n}$-axis will not serve ...
0
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0answers
22 views

Strain Flow in Fluid Mechanics

I am working on hydrodynamic instabilities and was looking for non axisymmetric vortex models that are not Stuart or Taylor green vortices. What I came across was multipolar vortices where a uniform ...
1
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0answers
53 views

How to calculate the contour integration with branch point? [closed]

The question come from a Mutusbara Sum like this $${ \sum _{ { z=i\omega }_{ n } } { \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } }$$ it equal a contour integral around Imaginary ...
7
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0answers
113 views

How to properly use Perturbation Theory in classical systems?

Context: If we consider a particle in upwards motion near the Earth's surface and acted by a quadratic drag we get the non-linear eom: $$\frac{dv}{dt}=-g-\frac{b}{m}v^2.$$ We can solve it ...
0
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0answers
19 views

How to make good approximation for a sum of squared expression? [migrated]

In both expression, n is integer and nmax is the maximum n and can be very large. How to use nmax to approximately and analytically to express these two expressiones? Are there any analytical ...
1
vote
2answers
55 views

Is there any general theorem which specifies conditions where the critical solution of an action is unique (for given boundary conditions)? [duplicate]

Consider a classical mechanical system with generalized coordinates $q_i$, $i \in \{1,\dots\,n\}$. And Lagrangian $L$. Given a path $\gamma$ (with coordinates $\gamma_i$) and two times $t_1$ and $t_2$ ...
9
votes
1answer
621 views

Non-separable solutions of the Schroedinger equation

I'm studying an undergraduate Quantum Mechanics course and I have some doubts about the solution of the Schroedinger equation by the separation of variables method. If we suppose that the solutions ...
3
votes
1answer
57 views

Wave speed of a hanging rope

Let us consider a homogeneous rope hanging from the ceiling. I will call the vertical direction $x$ and the horizontal displacement $y$. When we apply the second Newton's Law to a portion of mass ...
1
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1answer
50 views

Anomalies and determinant bundle curvature

I heard that anomalies and curvature of determinant bundle are related. Namely, curvature of determinant bundle is related to Chern-Simons form (which are involved in description of gauge anomalies). ...
-3
votes
0answers
81 views

What's the value of $\int f(x)\delta(x-a) dx$ if $a$ is not in the domain of integration? [migrated]

A problem occurs when I was solving an exersice of perturbative kind. The delta function has the fundamental property that \begin{align} \int_{-\infty}^{\infty}f(x)\delta(x-a)dx=f(a) \end{align} ...
2
votes
0answers
55 views

(Causal) Set notation round brackets vs square brackets? [closed]

In many (quite old) papers & books I have been reading recently in the causal theory of general relativity (e.g. On the structure of causal spaces, Kronheimer & Penrose, 1967) I find sets ...
2
votes
0answers
41 views

What is the charge density for line and surface charges?

In electrostatics it is common to see line, surface and volumetric charges being described differently. A line distribution is a function defined on the line, a surface charge distribution is a ...
0
votes
1answer
45 views

Thermofluid mechanics inclined plane

As shown in the attacked image, a tank has an inclined wall at an angle of 450 to the horizontal. On this wall, there is a 1m square door that is hinged at A and has a simple latch at B. The distance ...
0
votes
0answers
20 views

Construct recurrence relation for the temporal evolution of a Master equation

Say that we have a system evolving over discrete timesteps. The quantity we are interested is X and is given by a distribution $P_X$. This distribution is evolving temporally, and we have a ...
1
vote
1answer
38 views

Homoclinic orbit and a particle in a double well

The physical set-up is a classical particle in a parabolic double well: Physically, a particle with reasonable amount of potential energy would be able to roll down the slope of the well, roll past ...
1
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0answers
29 views

Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal or how to marginalize a dynamical system? Clearly my question looks at the same time fairly ...
1
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0answers
70 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems?

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
0
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0answers
29 views

Simplify $K$-matrix

2+1D Abelian topologically ordered states are believed to be described by multicomponent $U(1)$ Chern-Simons theories, with Lagrangian \begin{equation} ...
1
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1answer
70 views

When and where to check the formal definition of a manifold

In most texts on GR we are first introduced to a formal and rigorous definition of a manifold. We then learn the point that in GR "any coordinate system" might be used for the 4D spacetime metric. ...
0
votes
2answers
48 views

What's meant by: “All observers agree on the combination of basis vectors and components for a tensor”?

There's a Youtube video given by Dr Dan Fleisch on Tensors, where he states at 11:22: You may be wondering, what is it about the combination of components and basis vectors that makes tensors so ...
3
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1answer
40 views

Generators of a certain symmetry in Quantum Mechanics

In Classical Mechanics to describe symmetries like translations and rotations we use diffeomorphisms on the configuration manifold. In Quantum Mechanics we use unitary operators in state space. We ...
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0answers
44 views

Extending projection operator to infinite-dimensional case

Hi I have a basic question regarding bra-ket notation. Given that $\{|e_n \rangle \}$ is a discrete orthonormal basis, $$\langle e_m | e_n \rangle = \delta_{mn}$$ then $$\sum_{n}|e_n \rangle \langle ...
0
votes
1answer
46 views

Allowed Wave Functions of System

Given a single-particle system with Hamiltonian $H$, what constraints can be put on the wave function at a particular point in time $\psi(x)$? Of course $\psi(x)$ must obey boundary conditions given ...
1
vote
0answers
36 views

Lippmann-Schwinger equation and time dependence

Consider the Lippmann-Schwinger equation (LSE) $$ |\psi\rangle = |\phi\rangle + \hat{G}_0(\epsilon) \hat{V} |\psi\rangle \tag{1}$$ where $\hat{G}_0(\epsilon) = \frac{1}{\epsilon - \hat{H}_0 + ...
4
votes
0answers
58 views

Interpreting the Fourier transform of a Gibbs measure

Recall that a Gibbs measure gives a probability distribution on states $x$ of the form $$ p_X(x) = \frac{1}{Z(\beta)}\exp(-\beta E(x)) $$ As I understand, the function $E$ is interpreted as the ...
2
votes
0answers
38 views

Contour integral of the retarded Klein Gordon propagator

I've been trying to prove by hand the Peskin's formula for the retarded propagator of the Klein Gordon equation, that is, $$\int_{x^0 > y^0} \frac{d^4p}{(2\pi)^4} \frac{-e^{-ip(x-y)}}{i(p^2 - ...
-1
votes
1answer
55 views

Cross-differentiation to derive the maxwell relation $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$ [closed]

How can I use $T=\left(\frac{\partial E}{\partial S}\right)_V$ and $P=-\left(\frac{\partial E}{\partial V}\right)_S$ to derive $$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial ...
0
votes
3answers
68 views

A relationship between entropy and temperature

I tried deriving a formula relating entropy (not change in entropy, but entropy itself) to temperature. I’ve only seen two equations really relating to entropy thus far, and only one of them includes ...
1
vote
1answer
61 views

QM sytem with eigenvalues of the form $f(m*n)$ and prime number gap spectrum

Depending on the dimension and the symmetry and form of the potential, the energy eigenvalues of a quantum mechanical system have different functional forms. Eg. The particle in the 1D-box gives rise ...
1
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0answers
37 views

Are physical functions always differentiable [duplicate]

I know that physicist usually don't really think too much about differentiabillity of functions. Usually there are at most finite many points where functions aren't differentiable and if there are ...
3
votes
1answer
66 views

Equivariant cohomology and Mayer-Vietoris sequence

I'm reading this article upon topological field theory and I'm a bit confused about the way he compute equivariant cohomology of $S^2$ wrt $\mathrm{U}(1)$, i.e. $H^\bullet_{S^1}(S^2)$. You can find ...
2
votes
1answer
87 views
+50

Potential of an axisymmetric disc with constant rotation velocity

I am having trouble understanding why the form of the 3D potential for a disc with a constant rotation velocity for circular orbits of stars within the disc \begin{equation} v(R) = v_0, \tag{1} ...
1
vote
1answer
46 views

Book to study Dirac delta function from a physics point of view [duplicate]

I am a beginning physics graduate student. I am often bewildered by the strange properties of the Dirac delta function such as: $\delta (a x)= \frac{1}{a} \delta (x)$ The derivative of $\delta (x)$ ...
1
vote
0answers
36 views

Sufficient condition for square integrability [duplicate]

The necessary condition for $\int\limits_{-\infty}^{+\infty}|\psi(x)|^2dx$ to be integrable is that $\psi(x)\rightarrow 0$ as $x\rightarrow\pm\infty$. But this is not the sufficient condition. For ...
6
votes
2answers
195 views

Implementing Category Theory in General Relativity

I was thinking if it may be possible to implement category theory in general relativity. I don't mean writing simply in terms of categories, but actual fundamental ideas (i.e. physics of the theory ...
1
vote
0answers
39 views

Understanding the mathematical terms in Physics [closed]

I understand well only when I get a physical picture of the concept. Is it good? or should I change.
4
votes
2answers
157 views

When does causal separation imply no spacelike separation?

(See here for notation.) In Minkowski space, if $p\prec q$, then there is no spacelike curve $c:[0,1]\to \mathbb{R}^{n-1,1}$ with $c(0)=p$ and $c(1)=q$. This is obvious from a spacetime diagram. Here ...
3
votes
1answer
36 views

Pointwise temperature and Thermodynamic Temperature

When I studied Thermodynamics the definition of temperature I've learned (based on the Gibbs' postulates) was: $$T = \dfrac{\partial U}{\partial S}.$$ This quantity has all the properties of the ...
4
votes
0answers
204 views

How does one determine if a spacetime is globally hyperbolic?

A spacetime $M$ is said to be globally hyperbolic if it is strongly causal and if the sets $J^+(p)\cap J^-(q)$, for all $p,q\in M$, are compact. (For more information, see the Wiki article on causal ...
2
votes
0answers
43 views

Seeking masters degree advice [closed]

I am graduating this semester with a bachelors in physics. My goal is to do theory in my graduate level course work. The problem I am in now is that I missed the deadline to take the physics GRE (I'm ...
0
votes
1answer
43 views

Assumptions in physics for Helmholtz decomposition

A version of the Helmholtz theorem says that, under opportune assumptions on the vector field $\boldsymbol{F}:\mathbb{R}^3\to\mathbb{R}^3$ and on $V\subset\mathbb{R}^3$ the following identity holds: ...
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0answers
44 views

About the formula of pendulum simple

for the modulation and the simulation of a pendulum simple , I'm Find this formula : a(t) = a0 * sin ( sqrt(g/l) * t * Pi/2 ) - [ k/(mll) * cos ( sqrt(g/l) * t * Pi/2 ) * t ) ] ...
1
vote
1answer
84 views

Spectral functions in quantum mechanics

I'm a math student and a totally newcomer to quantum mechanics and I'm trying to teach myself this subject by studying Faddeev and Yakubovski's Lectures on Quantum Mechanics for Mathematics Students. ...
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0answers
50 views

What are some resources for Algebraic quantum mechanics for Physicists

I am interested in the GNS construction and other stuff. I am aware of Valter Moretti's book but I want something that is more inclined towards physicists.
3
votes
1answer
87 views

Is it possible to integrate a function over a null hypersurface?

For $f$ a compactly supported function on a spacetime $(\mathcal{M},g)$, one may define its integral over $\mathcal{M}$ by $$f\longmapsto \int_\mathcal{M}\star f,$$ where $\star$ is the Hodge star of ...
3
votes
3answers
89 views

Examples of Bernoulli Numbers, Euler-Mascheroni Integration, and the $\zeta(n)$ in physics [closed]

In Arfken's Mathematical Methods for Physicists, there is a subsection of the "Infinite Series" chapter which covers the Bernoulli numbers, Euler-Mascheroni integration (or summation), and the ...