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Questions tagged [mathematical-physics]

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 mathematical methods to problems in physics. Examples include partial differential equations (PDEs), variational calculus, functional analysis, and potential theory.

Mathematical physics is the application of mathematics to problems in physics and mathematical methods suitable for such applications, e.g., partial differential equations (PDEs), functional analysis, variational calculus, and potential theory. It also includes the study of problems inspired by physics within a mathematically rigorous framework, such as rigorous derivation of an atomic energy spectrum, rigorous construction of a quantum field theoretic model, and rigorous description of a phase transition.

Do not use just because your question involves math!

See also Wikipedia's article on mathematical physics.

2344 questions
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Ring theory in physics

Surely group theory is a very handy tool in the problems dealing with symmetry. But is there any application for ring theory in physics? If not, what's this that makes rings not applicable in physics problems?
iii
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Intuitively Re-Deriving Equations of Mathematical Physics

Using the intuitive interpretation of the Laplacian $\vec{\nabla}^2$ as the difference between the average value of a field in the neighbourhood of a point & the value of the field at that point, one can pretty easily & quickly derive the form of…
bolbteppa
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Examples where an ill-behaved function leads to surprising results?

In mathematical derivations of physical identities, it is often more or less implicitly assumed that functions are well behaved. One example are the Maxwell identities in thermodynamics which assume that the order of partial derivatives of the…
Lagerbaer
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Integral representation of Thomas-Fermi Equation

The Thomas-Fermi equation with dimensionless variables is identified as; $$ \frac{d^2\phi}{dx^2} = \frac{\phi^{3/2}}{x^{1/2}} $$ with the boundary conditions as $$ \phi(0) = 1 \\ \phi(\infty) = 0. $$ There are many series approximation solutions…
renormalizedQuanta
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Can the math for physics be expressed without any uncountable sets at all?

I am wondering about this and have wondered about it for a short while. Usually physics is modeled using things based off of the Real Number Line $\mathbb{R}$, which is uncountable. (E.g. we may use powers of $\mathbb{R}$, we may use $\mathbb{C}$,…
The_Sympathizer
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Natural systems that test the primality of a number?

There might be none. But I was thinking of links between number theory and physics, and this would seem like an example that would definitely solidify that link. Are there any known natural systems, or physical systems in general (possibly…
Justin L.
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$\nabla ^2\psi$ equals $\psi -$ average value of $\psi$ at neighboring points

Let $\psi (x,y,z)$ be a scalar field. I found the following statement in Morse & Feshbach Methods of Theoretical Physics: The limiting value of the difference between $\psi$ at a point and the average value of $\psi$ at neighboring points is…
a06e
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How to prove Gegenbauer's addition theorem?

I asked this at: https://math.stackexchange.com/questions/210153/, but didn't get any reply, so I am trying here, since I actually need this in physics anyway. How can one prove the following identity: $$ V_k(r_1, r_2) = {2k+1\over 2 r_1…
Ondřej Čertík
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Are there any systems in physics which can only be formulated as an integral equation?

My question is are there any systems in physics that can only be formulated as an integral equation? Or do all integral equations have an equivalent differential equation?
Anode
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Homogenous vector bundles

In the definition of homogenous vector bundles, an equivalence class is defined. Briefly: G is a lie group and H a (lie) subgroup. Define $$ \rho : H \rightarrow GL(V) $$ where V is a vector space. The equivalence class is defined as $$ (g_1,…
Tbh
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Normalizing the free particle wave function

One way to normalize the free particle wave function "is to replace the the boundary condition $\psi(\pm{\frac{a}{2}}) = 0$ [for the infinite well] by periodic boundary conditions expressed in the form $\psi(x)=\psi(x+a)$" -- Quantum Physics,…
H Taylor
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Why are $\pi$ and $e$ used in so many physical formulae?

It's a fact that in many physical formulae $\pi$ (or even multiples of it as far as I can see) and $e$ show up. But why would that be so? Is because the two are "connected" in the well-known formula $e^{i\pi}+1=0$, though this only shows the two in…
Deschele Schilder
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From Number Theory to Physics.

I have asked a question here: I want to see an example which is related to (integral) quadratic forms or theta series. @Kiryl Pesotski answered me in some comments as following: For example, you may want to compute the partition function…
Davood
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Discrete sum over a gaussian function

I have a sum of the form $$\sum_{n,m=-N}^N e^{-\alpha (n-m)^2}$$ where $α>0$ is some constant, and I don't mind if the limit $N\rightarrow\infty$ is taken. I know there is a possibility of exchanging the "variable of summation" by introducing some…
Koby Yavilberg
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Scale Factors via Ellipsoidal Coordinate System Scale Factors

In Morse & Feshbach (P512 - 514) they show how 10 different orthogonal coordinate systems (mentioned on this page) are derivable from the confocal ellipsoidal coordinate system $(\eta,\mu,\nu)$ by trivial little substitutions, derivable in the sense…
bolbteppa
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