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2answers
157 views

Isomorphism of rigged Hilbert spaces

In connection with the statement that QM can be formulated in terms of separable complex (rigged) Hilbert spaces, the fact that all infinite dimensional separable complex Hilbert spaces are isomorphic ...
0
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0answers
50 views

Does this Hamiltonian have point spectrum?

Consider such a Hamiltonian $$ H = - \frac{1}{2} \frac{\partial^2}{\partial x^2} - F x + V(x) ,$$ with $F$ being some constant, and $V(x)= V(x+L)$ being some periodic potential. Does this ...
2
votes
0answers
38 views

What is the essential concept behind the difference in the fundamental solutions of the Stokes and Poisson equations?

The fundamental solutions, i.e., the solution with a point source, of the Poisson's equation and the Stokes equations in 3D are: $$\nabla^2 f=\delta(\boldsymbol x) \ \Longrightarrow\ G(\boldsymbol ...
5
votes
1answer
174 views

Noether's Theorem: Lie algebra, Lie groups

I've had a brief look through similar threads on this topic to see if my question has already been answered, but I didn't find quite what I was looking for, perhaps it is because I'm finding it hard ...
2
votes
2answers
132 views

Wave functions as $x$ goes to infinity

This problem emerged when I was going through some QM exercises: I've been asked to find the commutator $[A,B]$ where $A,B$ are defined as $$A\psi(x)=x\frac{\partial }{\partial x}\psi(x),$$ ...
1
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0answers
27 views

Necessity of infinitesimal steps for reversible processes

Is there a mathematical proof for why a reversible process is one that has many infinitesimal equilibrium steps, rather than fewer large steps? Maybe something along the lines of this?: ...
3
votes
0answers
92 views

Chocolate dynamics

Now I have found a possible model on how to describe chocolate when it is chewed. It has to do with geometrical transformations when a curve $\gamma$ intersects a manifold $M$. The chocolate is ...
0
votes
0answers
13 views

Are centrifugal fans subject to triboelectric effects, and if so can the increase in charge be predicted and measured?

A centrifugal blower is a device that moves gas, usually under power of an electric motor. The blower housing and impeller may be metallic, but often are made of plastic materials. My question is ...
1
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0answers
49 views

Is it possible to generalize quantum gauge theories? [closed]

I know that there are nonabelian gauge theories and their supersymmetric extensions. Mathematically, gauge theories basing on the fact that one can introduce a fiber bundle with a Connection. From ...
0
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0answers
36 views

Algebraic number theory and physics [duplicate]

I would like to ask if there are any aspects of algebraic number theory related to physics (for example in string theory or Moonshine etc). I am thinking of attending a course on algebraic number ...
2
votes
0answers
45 views

Rheological behavior of chocolate

If someone eats chocolate, the chocolate goes through the following configurations: $\chi_0:$ chocolate is solid and has a smooth Surface everywhere; the Riemann Tensor vanishes on every Point of the ...
3
votes
0answers
64 views

What is a modular tensor category / functor?

I have reads several answers here about this notion, especially regarding topological order, see e.g. this answer, but this notion sounds completely new for me. Also, I found nothing really helpful on ...
1
vote
1answer
71 views

How is this a gauge choice mathematically?

I've been reading an article about the "square cat", which is described as the system bellow Such system is a deformable body that can change $a$ and $\theta$ but has $b$ fixed. The article uses ...
1
vote
1answer
108 views

About the orthogonality of the Hamiltonian eigenstates for the the continuous energy spectrum

I would like first to describe a strange case that I encountered. $ \ \ - $ I solved the Schrodinger equation with a potential barrier (a potential well limited by a finite height wall which decrease ...
0
votes
1answer
37 views

A series of bound states covering an interval

Generally, the bound states (normalizable eigenvectors) of a Hamiltonian have discrete eigenvalues. Is it possible for the eigenvalues to cover an interval? Say, $(a,b)$? That is, for each $E \in ...
2
votes
2answers
123 views

If path integrals aren't well-defined, how can they have any physical meaning?

I am confused about a particular point about the nature of path integration. According to what I've read, what we really mean when we say functional integration is \begin{equation} ...
32
votes
9answers
4k views

Rigor in quantum field theory

Quantum field theory is a broad subject and has the reputation of using methods which are mathematically desiring. For example working with and subtracting infinities or the use of path integrals, ...
1
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0answers
36 views

Deriving general boundary conditions from first principles for elastodynamic scattering

It seems that most of the relevant books only give the linear case and the rest say something along the lines of "here are common examples of boundary conditions." What are the most general boundary ...
6
votes
3answers
295 views

Coadjoint orbits in physics

I am looking for some application of coadjoint orbits in physics. If you know some of them please let me know.
1
vote
0answers
68 views

Non Hermitian Quantum Mechanics

I was just reading about Non-Hermitian Quantum Mechanics dealing with Hamiltonians $H$ that are not Hermitian operators. Then it is unclear that we get orthonormal eigenstates. Now, I was reading a ...
5
votes
3answers
567 views

Axiomatic statistical mechanics

Ive read a few courses on statistical mechanics, and while their textual explanations and example choices differ, the flow of information from microscopy to macroscopy seems the same, and reading ...
7
votes
2answers
95 views

Is there a reason why the subset of our Hilbert space that corresponds to a particle is a vector subspace?

I'm trying to gain some intuition behind the definition that states a particle is an irreducible unitary representation of the restricted Poincare group (or more specifically, its double cover). ...
1
vote
0answers
46 views

Can we avoid singularities by embedding the manifold on a bigger space, maybe Euclidean or Riemannian?

Can singularities be avoided by embedding Riemannian manifold on bigger space, or more specifically black hole singularities can be avoided or not by embedding in any other manifold. We don't have ...
3
votes
1answer
64 views

Is there a general theorem stating why the restricted Lorentz group's exponential map is surjective?

The exponential map for the restricted Lorentz group is surjective. An outline of why is shown on the wiki page Representation Theory of the Lorentz Group. Is there a more general theorem that states ...
0
votes
1answer
49 views

Simulating of Fraunhofer Diffraction of Zigzags by FFT

I tried to study the diffraction pattern of the following zigzag grating by Matlab(FFT of this image).. And the result showed like this(please ignore the scale bar in this img) I think the ...
4
votes
0answers
65 views

Derivation of the Lippmann-Schwinger equation

I was trying to understand the derivation of the Lippmann-Schwinger equation in Sakurai's Modern Quantum Mechanics, Section 6.1. Our teacher presented a much simpler derivation, similar to that on ...
1
vote
1answer
37 views

How does the Hermiticity of an operator imply that functions have an expansion in in multiple bases?

In Shankar QM it is stated that since the $\boldsymbol K$ operator is Hermitian, vectors, which are expanded in the $\boldsymbol X$ basis with components $f(x) = \langle x | f \rangle$, must have an ...
6
votes
2answers
541 views

Schrödinger equation in position representation

We start from an abstract state vector $ \newcommand{\ket}[1]{|{#1}\rangle} \ket{\Psi}$ as a description of a state of a system and the Schrödinger equation in the following form $$ ...
2
votes
3answers
57 views

What relative masses are required for them to collide n times in this scenario?

Consider two masses, m and M, where M>m. They begin at rest on an infinite frictionless surface that is flat in one direction and sloped in the other direction. Mass m is placed a little bit up the ...
1
vote
0answers
38 views

Legendre Transformation for multiple variables

I need to show that for $F(x_1, .., x_n)$, the Legendre transformation is, $$G(s_1, ..., s_n) = \sum_{i}^{N} x_i s_i - F$$ where $$s_i = \frac{\partial F}{\partial x_i}$$ and has the property that ...
1
vote
0answers
18 views

QM scattering in a finite-sized box

Background Consider a non-relativistic particle in a one-dimensional box of length $L$ with (for definiteness) an attractive delta function at the origin: $H = \frac{P^2}{2m} -|c|\delta(x), \qquad ...
0
votes
1answer
113 views

How do we know there are dimensions above the third?

Excluding the interpretation of time being the forth dimension, what mathematical evidence is there that suggests it exists? When talking about string theory they say how there is 12 dimensions and I ...
-3
votes
2answers
69 views

How much does it cost to become a theoretical physicists? [closed]

Does becoming a theoretical physicist require a huge amount of money? Does theoretical physics require the same resources as mathematics: just papers and pencil?
1
vote
2answers
82 views

What is the wave propagated away from an impulsively excited spherical shell?

Consider a spherical shell of radius $R$ centered on the coordinate origin, and an impulsive excitation $\delta (t)$ distributed over its surface ('ie. a single layer'). Each point on the sphere’s ...
3
votes
2answers
94 views

Identifying irreps of $SU(2)$

How does one verify that, the representations of $SU(2)$ corresponding to $j=1/2$ or $j=1$ is irreducible? I think showing the irreducibility (taking the representative matrices into a block-diagonal ...
8
votes
2answers
555 views

Nonseparable Hilbert space

What kind of things can go wrong if we try to do quantum mechanics on a nonseparable Hilbert space? I have heard that usual mathematical manipulations that we take for granted will no longer hold. ...
0
votes
1answer
43 views

Applications of partial differential equations in material science [closed]

I've been asked to find a partial differential equation that has applications in material science. However we are not allowed to use the heat equation. I have found Fick's laws (basically the heat ...
7
votes
2answers
276 views

Ambiguity in number of basis vectors [duplicate]

The dimension of the Hilbert space is determined by the number of independent basis vectors. There is a infinite discrete energy eigenbasis $\{|n\rangle\}$ in the problem of particle in a box which ...
0
votes
3answers
176 views

About the postulates of quantum mechanics and self-adjointness

I am a freshman trying to understand the very basics of quantum mechanics but I met barriers at the beginning. What really matters is the postulates of quantum mechanics and their relationship with ...
3
votes
2answers
111 views

How can projection operators be expressed in form $\frac{1}{d} (I + \sum_i r_i \lambda_i)$?

How can projection operator be expressed in form $\frac{1}{d} (I + \sum_i r_i \lambda_i)$? I was reading a paper and found out that the density matrix in $d$-dimensional Hilbert Space can be ...
1
vote
1answer
90 views

Must the wavefunction be analytic?

In order to show the preservation of normalization of the wave function (in one dimension for now), one shows that the time differential is zero, which entails the following step: $$ ...
3
votes
0answers
56 views

Hodge dual and the Moyal bracket? Any link? [closed]

I have already asked this on the mathematics Stack exchange but I thought I'd try it here too! The Hodge star operator $\star$ is a linear map between $\bigwedge ^pV$ and $\bigwedge ^{n-p}V$ for an ...
3
votes
0answers
80 views

What's the physical meaning of the integral of the squared temperature in a one-dimensional rod, used to prove uniqueness of soln. in heat eqn.?

Before getting to the question, some background. Let $u(x,t)$ be the temperature in a laterally insulated rod of length $L$, at position $x$ and time $t$. The temperature satisfies the heat equation ...
6
votes
1answer
387 views

Mathematically, what is the kernel in path integral?

Mathematically, what is the kernel in path integral? At first, I thought that it is the kernel in the integral transform because when we use the (physical) kernel to transform the wave function (Eq ...
0
votes
0answers
17 views

Recommendations for Complex variable calculus [Physics Oriented]

I'm looking for a good book Complex variable calculus that give a really good mathematical background for a physicist. I already know real-variable calculus and vector calculus and I'm also ...
2
votes
1answer
53 views

Advice on Major Selection [closed]

I need a bit of an advice in deciding my major. My university allows only one major (along with minors) and I'm having a little bit of trouble in deciding what to do. Here's the thing. I want to go ...
2
votes
2answers
534 views

Quantum mechanics in a metric space rather than in a vector space, possible?

Quantum mechanics starts with wave functions living in Hilbert space. But later for Born's interpretation, the wave function need to be of unit energy (I mean total probability = 1, ...
4
votes
2answers
195 views

In what way are the Mathematical universe hypothesis and A New Kind of Science connected

The Mathematical universe hypothesis, mainly by Max Tegmark and A new Kind of Science, mainly by Stephen Wolfram both claim (as least as I understand it) that at its innermost core reality is ...
4
votes
1answer
480 views

Time-ordering in QFT

In Srednicki QFT page 37. In the derivation of LSZ reduction formula, he introduces the time-order operator $T$, so no time-dependent creation/annihilation operators are left in the transition ...
1
vote
1answer
75 views

0+0 self interacting QFT - $ e^{-\sin^2 x}$ type integral — Bessel function expansion around infinity

In a physics paper (here) I found this variant of the Bessel function of the first kind. $$ \tag{1} Z(g) ~=~ \frac{1}{\sqrt{g}} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} e^{-\frac{1}{2g} \sin^2 x} \, dx ...