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3answers
59 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
41 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
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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 ...
-1
votes
1answer
114 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 ...
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2answers
75 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
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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
96 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
570 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
48 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
190 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
116 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
93 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
59 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
88 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
414 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
55 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
548 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
202 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
487 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 ...
27
votes
8answers
4k views

Classical mechanics without coordinates book

I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think ...
6
votes
1answer
305 views

Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
3
votes
3answers
452 views

What's the difference between “numerical methods” & “mathematical analysis” as said by Feynman in his lectures?

While reading his lectures, I came to these lines: On the basis of Newton's second law of motion,which gives the relation between the acceleration of any body & the force acting on it,any ...
6
votes
2answers
316 views

Infinite dimensional vector spaces vs. the dual space

I just happened across this over on Math Overflow. It references the following theorem from linear algebra: A vector space has the same dimension as its dual if and only if it is finite ...
1
vote
4answers
267 views

Why drops form spheres?

Consider a drop of water floating in an inertial frame in STP air (e.g., the ISS). Intuitively, the equilibrium shape of the drop is a sphere. How would one prove that? Is it equivalent to showing ...
2
votes
1answer
98 views

Relationship between Connection and Material Derivative

Suppose $D\subset \Bbb R^3$ contains a fluid and that $f : D\times \mathbb{R}\to \mathbb{R}$ is a time dependent function defined on the fluid region. In that case, the material derivative is defined ...
6
votes
5answers
881 views

What is the meaning of following expression $C=\frac{\delta Q}{dT}$ mathematically?

Our professor raised the following question during our lecture in Statistical Physics (even so it's related to Thermodynamics): Many text books (even Wikipedia) writes wrong expressions (from ...
2
votes
3answers
169 views

Spectral properties in Solid state physics

So assume we have a periodic 1d Schrödinger operator $$- f'' + V(x) f(x)= \lambda f(x)$$ and we want $V$ to be periodic. Now if we assume that we are on a finite interval and that we have periodic ...
6
votes
2answers
164 views

Akin to gauge field, why GR's lagrangian is not $R_{abcd}R^{abcd}$? What's the mathematical or physical meaning of $R_{abcd}R^{abcd}$?

For gauge field theory, the Lagrangian of the gauge field is $$\mathcal{L}=-\frac{1}{4}\mathrm{tr}(\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu})=-\frac{1}{8}F_{a\ \mu\nu}F^{a \ \mu\nu}$$ The field ...
16
votes
1answer
936 views

Intuitive meaning of Hilbert Space formalism

I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points: The observables are given by self-adjoint operators on the ...
5
votes
4answers
280 views

Kähler and complex manifolds

I was wondering if anyone knows any good references concerning Kähler manifolds and complex manifolds? I am studying supergravity theories and for the simplest $\mathcal{N}=1$ supergravity we will get ...
6
votes
2answers
179 views

Does the Hermitian operator $H=-\frac{d^2}{dx^2}$ have imaginary eigenvalues?

In quantum mechanics, Hermitian operators play a very important role because they possess real eigenvalues. Considering $-\frac{d^2}{dx^2}$, it is a Hermitian operator (Actually it's the simplest ...
4
votes
2answers
238 views

Can Minkowski spacetime be redefined as a non-flat riemannian manifold?

Minkowski space time is defined in terms of a flat pseudo-Riemannian manifold. I have wondered if it can be redefined as Riamannian manifold and in the case what type of curvature would there appear. ...
7
votes
4answers
939 views

Connection between Poisson Brackets and Symplectic Form

Jose and Saletan say the matrix elements of the Poisson Brackets (PB) in the $ {q,p} $ basis are the same as those of the inverse of the symplectic matrix $ \Omega^{-1} $, whereas the matrix elements ...
0
votes
0answers
25 views

Free groups and their appearance, emergence and applications in physics

While trying to develop my knowledge of group theory in physics from a more formal point of view, I noticed an entity called a free group. I'm aware that it is extremely important in pure mathematical ...
1
vote
0answers
58 views

Contradiction in classical analysis of the hamiltonian $\mathcal{H}=xp$?

I am writing an essay on the Berry Keating article proposing to use the $\mathcal{H}=xp$ hamiltonian to get a correspondence between the nontrivial riemann zeros and the eigenvalues of an Hermitian ...
1
vote
2answers
127 views

Why is quantum mechancis is not content with symmetric operators, but wants self-adjoint operators?

A symmetric operator has only real eigenvalues and different eigenvectors corresponding to different eigenvalues are orthogonal. These are exactly what we want for a physical observable. I think ...
1
vote
1answer
182 views

Discrete vs Continuous spectra of operators [duplicate]

Why is it that if an operator $Q$ has a discrete spectra, that the eigenfunctions are all in Hilbert space? Why is it that if the spectrum is continuous we automatically know that the eigenfunctions ...
0
votes
1answer
367 views

Expansion in solid spherical harmonics on the lattice

I'm interested in calculating scattering processes (e.g. Coulomb scattering of an electron beam by a single ion) in the context of lattice quantum field theory, and wonder if there is something like ...
1
vote
2answers
331 views

Hilbert space and Hamiltonians

Assume a system described by a Hamiltonian H, and assume that the eigenstates of H, $φ_i$(r) are integrable in absolute square. We say that these states belong to a Hilbert space (they can even form a ...
6
votes
3answers
1k views

A question on the existence of Dirac points in graphene?

As we know, there are two distinct Dirac points for the free electrons in graphene. Which means that the energy spectrum of the 2$\times$2 Hermitian matrix $H(k_x,k_y)$ has two degenerate points $K$ ...
0
votes
0answers
40 views

Symplectic Structure without predefined Hamiltonian

Here there is a link which has helped me understanding the relationship between symplectic geometry and classical mechanincs. In short, the symplectic form transforms the derivative of the ...
3
votes
0answers
69 views

Is the metric-induced topology relevant at all in a (psuedo) Riemannian manifold? [duplicate]

A (pseudo) Riemannian manifold is a tuple: $$(M,g)$$ where $M$ is a smooth manifold (in particular, a topological space with an atlas) and $g$ is a (pseudo) Riemannian metric tensor. It is apparent ...
3
votes
1answer
94 views

The index of a Dirac operator and its physical meaning

I recently read Witten's paper from the 1980s and he often uses the notion of the index of a Dirac operator in K-theory. What is the meaning of the index of a Dirac operator? What exactly is the ...
1
vote
1answer
57 views

What is the relation between renormalization and self-adjoint extension?

What is the relation between renormalization and self-adjoint extension? It seems that a renormalization scheme can be rigorously treated mathematically using the self-adjoint extension theory. Is ...
0
votes
0answers
58 views

debates on “infinity” in contemporary physics?

During the seventeenth century many philosopher-scientists, such as Leibniz, were preoccupied with the philosophy and mathematics of infinity. Famous problem were the "labyrinth of the continuum," ...
2
votes
1answer
89 views

The relationship between the structure of spacetime and the existence of spinor field?

We all know that the existence of spinor fields implies that spacetime must be time-orientable. Thus that spacetime is time-orientable is a necessary condition for existence of spinor fields. Geroch, ...
2
votes
0answers
68 views

Intuition behind $U(1)$-gauge model of Electrodynamics in a general spacetime

As the article Electrodynamics in general spacetime greatly explains, the $U(1)$-gauge theory is a good base for working in non-simply connected spaces. But I wonder whether there is a deep reason to ...