# Questions tagged [quantum-mechanics]

For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales.

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### Prove eigenvectors are orthogonal. [closed]

Assuming the eigenvalues of a Hermitian matrix are non-degenerate, prove that the corresponding eigenvectors are orthogonal.
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### Proving that the limit of a given test function is a valid delta function

I am a physics student trying to gain a better mathematical understanding of the theory of distributions and namely the definition of the Dirac-Delta function. I understand that the defining properly ...
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### How to represent elements from $\bar{E}=E/Z(E)$ in the form $(a|b)$

From https://arxiv.org/abs/quant-ph/9608006 Background The group $E$ of tensor products $\pm w_{1} \otimes \dots \otimes w_{n}$ and $\pm i w_{1} \otimes \dots \otimes w_{n}$, where each $w_{j}$ is one ...
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### Transform a differential equation into Hamiltonian form

I am currently doing the following exercise in the book Modern Geometry - Methods and Applications Part I by Dubrovin, Fomenko, and Novikov. Exercise 33.4.1: Consider the differential equation \begin{...
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### Having difficult with this integral [closed]

∫e^[−i(ax−bx^2)]dx I have not idea how to do this integral
1 vote
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### Example of statistical data with the property of being contextual that is generated by quantum mechanics

Definition for the specifics of the question as well as an example of contextuality in quantum mechanics I have a set of measurements acting on a 2 qubit state for whom the statistics of the ...
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### Is there any mathematically rigorous definition of deriving a matrix valued function with respect to one of its matrix argument?

On my way of satudying Heisenberg matrix mechanics, I get blocked by formulas engaging derivations with respect to a matrix arguments. My question is the following : Is there any mathematically ...
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### Need help with an analytic solution of an integral of a particle within an infinite potential well

While working on a homework problem related to an infinite potential well, I encountered the following integral: $$\int_0^L \sin\left(\frac{n\pi}{L}x\right) \sqrt{x(L-x)}\mathrm{d}x$$ This integral ...
1 vote
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### Spectrum of a sum of self-adjoint operators

This is a "sequel" to that question where I explain why I need the spectrum of an operator given as the sum of a convolution and a function multiplication. Here, I am considering the ...
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### Tricky "Divergent" Integral: Correction to Groundstate

I am trying to rederive the results presented in the paper, in particular equation (30). That is, I am trying to compute the correction to the ground-state energy of a dipolar condensate due to beyond-...
1 vote
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### Operators and KdV equation

Consider Given the linear differential operators $$L=-\partial_x^2+u(x, t), \quad A=4 \partial_x^3-3 u \partial_x-3 \partial_x u,$$ considered as acting on a vector space of functions of $x$ verify ...
1 vote
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### Trouble understanding Berry's phase derivation

Following the the wikipedia article about the adiabatic theorem, and Sakurai's Modern QM, we start with the definition of the geometric phase that we get when doing a loop with a parameter R which ...
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### John von Neumann theorem on self adjoint extentions

Let $H$ be a Hilbert space and $A:D(A)\subset H \rightarrow H$ be symmetric and closed. Assume $A$ has a selfadjoint extention $B$. Then the Cayley transform of $A$ has also a unitary extention i.e. ...
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### Fourier Transform Duals and Multi-Variable Chain Rule

EDIT: I believe that I have come to the conclusion that my original idea was fundamentally misguided, and there is no reason to expect that such a process is possible in general. There is much more ...
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### Proving limit exists and is positive for smooth function involving integral

Let $\mu$ be a function from $\mathbb R_+\rightarrow\mathbb R_+$ in $C^\infty$ with $\Upsilon:=\sup\{|p|:\mu(|p|^2)>0\}$. Suppose that \lim_{|p|\rightarrow \Upsilon}\frac{\mu(|p|^2)}{(\Upsilon-|p|...
Hi I'm studying quantum mechanics with Stephen Gasiorowicz's book. It says for the $n=2$ states of the hydrogen atom, the $l=0$ state has even parity and the $l=1$ state has odd parity. As far as I ...
Is the following proposition true? Proposition. For any $a,k\in \mathbb{R}$, \begin{equation} \int_a^{\infty} dx e^{ikx} = 2\pi \delta(k). \end{equation} (End) I think it is true based on the ...