Tagged Questions

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. Mathematical physics is the mathematically rigorous study of the foundations of physics, and the application of advanced ...

5k views

Why does calculus of variations work?

How does it make sense to vary the position and the velocity independently? Edit: Velocity is the derivative of position, so how can you treat them as independent variables? Doesn't every physics ...
25k views

Best books for mathematical background?

What are the best textbooks to read for the mathematical background you need for modern physics, such as, string theory? Some subjects off the top of my head that probably need covering: ...
2k views

Rigged Hilbert space and QM

Are there any comprehensive texts that discuss QM using the notion of rigged Hilbert spaces? It would be nice if there were a text that went through the standard QM examples using this structure.
4k views

Why $\displaystyle i\hbar\frac{\partial}{\partial t}$ can not be considered as the Hamiltonian operator?

In the time dependent Schrodinger equation $\displaystyle, H\Psi = i\hbar\frac{\partial}{\partial t}\Psi$ , the Hamiltonian operator is given by $\displaystyle H = -\frac{\hbar^2}{2m}\nabla^2+V$ ...
1k views

Reason for the discreteness arising in quantum mechanics?

What is the most essential reason that actually leads to the quantization. I am reading the book on quantum mechanics by Griffiths. The quanta in the infinite potential well for e.g. arise due to the ...
650 views

Are all scattering states un-normalizable?

I am an undergraduate studying quantum physics with the book of Griffiths. in 1-D problems, it said a free particle has un-normalizable states but normalizable states can be obtained by sum up the ...
1k views

Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?

These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
2k views

In quantum mechanics, given certain energy spectrum can one generate the corresponding potential?

A typical problem in quantum mechanics is to calculate the spectrum that corresponds to a given potential. Is there a one to one correspondence between the potential and its spectrum? If the ...
3k views

Path integral formulation of quantum mechanics

I'm a mathematics student with not much background in physics. I'm interested in learning about the path integral formulation of quantum mechanics. Can anyone suggest me some books on this topic with ...
10k views

Don't understand the integral over the square of the Dirac delta function

In Griffiths' Intro to QM [1] he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being $$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right)$$ (cf. last formula on ...
4k views

The Role of Rigor

The purpose of this question is to ask about the role of mathematical rigor in physics. In order to formulate a question that can be answered, and not just discussed, I divided this large issue into ...
5k 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, ...
5k views

Mathematically-oriented Treatment of General Relativity

Can someone suggest a textbook that treats general relativity from a rigorous mathematical perspective? Ideally, such a book would Prove all theorems used. Use modern "mathematical notation" as ...
5k views

Is there something similar to Noether's theorem for discrete symmetries?

Noether's theorem states that, for every continuous symmetry of an action, there exists a conserved quantity, e.g. energy conservation for time invariance, charge conservation for $U(1)$. Is there any ...
3k views

Is the converse of Noether's first theorem true: Every conservation law has a symmetry?

Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Is the converse true: Any conservation law of a physical ...
1k views

Discreteness of set of energy eigenvalues

Given some potential $V$, we have the eigenvalue problem $$-\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi$$ with the boundary condition $$\lim_{|x|\rightarrow \infty} \psi(x) = 0$$ If we ...
2k views

Rigorous underpinnings of infinitesimals in physics

Just as background, I should say I am a mathematics grad student who is trying to learn some physics. I've been reading "The Theoretical Minimum" by Susskind and Hrabovsky and on page 134, they ...
4k views

A book on quantum mechanics supported by the high-level mathematics

I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators ...
The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = ... 7answers 3k views +100 Reading list in topological QFT I'm interested in learning about topological QFT including Chern Simons theory, Jones polynomial, Donaldson theory and Floer homology - basically the kind of things Witten worked on in the 80s. I'm ... 8answers 5k views Classical mechanics without coordinates book I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think ... 4answers 3k views Introduction to string theory I am in the last year of MSc. and would like to read string theory. I have the Zwiebach Book, but along with it what other advanced book can be followed, which can be a complimentary to Zwiebach. I ... 3answers 563 views Why we use$L_2$Space In QM? I asked this question for many people/professors without getting a sufficient answer, why in QM Lebesgue spaces of second degree are assumed to be the one that corresponds to the Hilbert vector space ... 6answers 3k views Formalizing Quantum Field Theory I'm wondering about current efforts to provide mathematical foundations and more solid definition for quantum field theories. I am aware of such efforts in the context of the simpler topological or ... 2answers 878 views Lie bracket for Lie algebra of$SO(n,m)$How does one show that the bracket of elements in the Lie algebra of$SO(n,m)$is given by $$[J_{ab},J_{cd}] ~=~ i(\eta_{ad} J_{bc} + \eta_{bc} J_{ad} - \eta_{ac} J_{bd} - \eta_{bd}J_{ac}),$$ ... 1answer 617 views Hilbert space of a free particle: Countable or Uncountable? This is obviously a follow on question to the Phys.SE post Hilbert space of harmonic oscillator: Countable vs uncountable? So I thought that the Hilbert space of a bound electron is countable, but ... 4answers 7k views Mathematical background for Quantum Mechanics [duplicate] What are some good sources to learn the mathematical background of Quantum Mechanics? I am talking functional analysis, operator theory etc etc... 2answers 457 views What restrictions on time boundary conditions does it have to use Fourier transform to solve wave equation? The wave equation can be solved using Fourier transform, by assuming a solution of the form of $$\mathbf{E}(x,y,z,t)~=~\mathbf{E}(x,y,z)e^{j\omega t}$$ and then reducing the equation to the Helmholtz ... 7answers 8k views Number theory in Physics [closed] As a Graduate Mathematics student, my interest lies in Number theory. I am curious to know if Number theory has any connections or applications to physics. I have never even heard of any applications ... 3answers 2k views What is the relation between renormalization in physics and divergent series in mathematics? The theory of Divergent Series was developed by Hardy and other mathematicians in the first half of the past century, giving rigorous methods of summation to get unique and consistent results from ... 4answers 2k views If all conserved quantities of a system are known, can they be explained by symmetries? If a system has$N$degrees of freedom (DOF) and therefore$N$independent1 conserved quantities integrals of motion, can continuous symmetries with a total of$N$parameters be found that deliver ... 7answers 14k views Recommendations for Statistical Mechanics book I learned thermodynamics and the basics of statistical mechanics but I'd like to sit through a good advanced book/books. Mainly I just want it to be thorough and to include all the math. And of course,... 5answers 1k views Complex integration by shifting the contour In section 12.11 of Jackson's Classical Electrodynamics, he evaluates an integral involved in the Green function solution to the 4-potential wave equation. Here it is: $$\int_{-\infty}^\infty dk_0 \... 5answers 4k views Haag's theorem and practical QFT computations There exists this famous Haag's theorem which basically states that the interaction picture in QFT cannot exist. Yet, everyone uses it to calculate almost everything in QFT and it works beautifully. ... 8answers 5k views Crash course on algebraic geometry with view to applications in physics Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an ... 10answers 4k views Physics for mathematicians How and from where does a mathematician learn physics from a mathematical stand point? I am reading the book by Spivak Elementary Mechanics from a mathematicians view point. The first couple of pages ... 1answer 581 views On a trick to derive the Noether current Suppose, in whatever dimension and theory, the action S is invariant for a global symmetry with a continuous parameter \epsilon. The trick to get the Noether current consists in making the ... 5answers 2k views Laplacian of 1/r^2 (context: electromagnetism and poisson equation) We know that a point charge q located at the origin r=0 produces a potential \sim \frac{q}{r}, and this is consistent with the fact that the Laplacian of \frac{q}{r} is$$\nabla^2\frac{q}{r}=... 3answers 2k views Can we have discontinuous wavefunctions in the Infinite Square well? The energy eigenstates of the infinite square well problem look like the Fourier basis of L2 on the interval of the well. So then we should be able to for example make square waves that are an ... 7answers 3k views Quantum mechanics on a manifold In quantum mechanics the state of a free particle in three dimensional space is$L^2(\mathbb R^3)$, more accurately the projective space of that Hilbert space. Here I am ignoring internal degrees of ... 3answers 2k views A “Hermitian” operator with imaginary eigenvalues Let $${\bf H}=\hat{x}^3\hat{p}+\hat{p}\hat{x}^3$$ where$\hat{p}=-id/dx$. Clearly${\bf H}^{\dagger}={\bf H}$, because${\bf H}={\bf T} + {\bf T}^{\dagger}$, where${\bf T}=\hat{x}^3\hat{p}$. In this ... 7answers 2k views Tensor Operators Motivation. I was recently reviewing the section 3.10 in Sakurai's quantum mechanics in which he discusses tensor operators, and I was left desiring a more mathematically general/precise discussion. ... 3answers 1k views Representing forces as one-forms First of all, sorry if any of those things are silly or nonsense, I'm just trying to understand better how the concepts of forms, exterior derivative and so on can be used in physics. This question ... 3answers 2k views Book covering differential geometry and topology required for physics and applications I am a physics undergrad, and interested to learn Topology so far as it has use in Physics. Currently I am trying to study Topological solitons but bogged down by some topological concepts. I am not ... 4answers 3k views Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)? All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before. Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ... 2answers 403 views Is there a 1-1 correspondence between symmetry and group theory? The professor in my class of mathematical physics introduces the definition of groups and said that group theory is the mathematics of symmetry. He gave also some examples of groups such as the set ... 11answers 12k views Quantum Field Theory from a mathematical point of view I'm a student of mathematics with not much background in physics. I'm interested in learning Quantum field theory from a mathematical point of view. Are there any good books or other reference ... 10answers 6k views Applications of Algebraic Topology to physics I have always wondered about applications of Algebraic Topology to Physics, seeing as am I studying algebraic topology and physics is cool and pretty. My initial thoughts would be that since most ... 5answers 5k views Trace of a commutator is zero - but what about the commutator of$x$and$p\$?
Operators can be cyclically interchanged inside a trace: $${\rm Tr} (AB)~=~{\rm Tr} (BA).$$ This means the trace of a commutator of any two operators is zero: $${\rm Tr} ([A,B])~=~0.$$ But what about ...