Questions tagged [quantum-mechanics]
For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales.
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How do I evaluate this type of integral that involve complex parameters?
In particular, I am asking about integrals shaped like this:
$$\int_{-\infty}^\infty e^{-ax^2-bx}dx$$
where $a\in\mathbb C$ with ${\rm Re}(a)>0$, and $b\in\mathbb C$. This kind of integral appears ...
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Why does it seem like two parameters $k_1$ and $k_2$ are needed to match $e^{-r}$ and $k_2\sin(k_1\,r)$ as well as their derivatives $\frac{d}{d\,r}$?
The Spherical Bessel functions that solve the Spherical Helmholtz equation in the Spherical Coordinate system come in four kinds, the Spherical Bessel Functions of the first kind, the Spherical Bessel ...
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Eigenvectors of Pauli vectors
For a Pauli vector defined by $\vec{\sigma}=\sigma_1 \hat{x_1}+\sigma_2 \hat{x_2}+\sigma_3 \hat{x_3}$, wikipedia states that $\vec{a}\cdot\vec{\sigma}$ has two eigenvector given by
$$
\psi_+ = \frac{1}...
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Simplex integral in connection with time ordered exponential
In Quantum Mechanics, one often defines the time ordered exponential like e.g. here.
Now my question is how the factor of $N!$ arises. I know the simplex volume as the following integral:
\begin{...
- 1,368
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Is this difference of surface integrals zero? $\oint_S\bar{\psi}\nabla(x\cdot\nabla\psi)\cdot n dS-\oint_S(x\cdot\nabla\psi)\nabla\bar{\psi}\cdot ndS$
This is the follow-up question of When does this integral vanish, which appears in the derivation of the quantum virial theorem? and building on this answer.
Does the following difference of surface ...
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Two notions of a pure state
Let $\mathscr{H}$ be a complex infinite-dimensional separable Hilbert space and let $\mathscr{S}$ be the state space associated to $\mathscr{H}$, which is defined as the set of all positive (hence ...
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Quantum Harmonic Oscillator Eigenvalues Confusion
In standard PDE theory, one generates eigenvalues to Sturm-Liouville problems over a finite domain. So, for a wave equation, we have an infinite number of eigenvalues $𝜆_𝑛$ for a Dirichlet problem, ...
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Spectrum of a sum of self-adjoint operators
This is a "sequel" to that question where I explain why I need the spectrum of an operator given as the sum of a convolution and a function multiplication. Here, I am considering the ...
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Integral equations for research in quantum mechanics
In my research, I have reached a point where I need to find a nonnegative function which satisfies the following properties:
It's an even function:
$$f(-\theta) = f(\theta)$$
It's normalised as:
$$...
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Is there actually any bijection between characters and global sections of the spectral presheaf?
In pages 5-6 of the article https://arxiv.org/pdf/quant-ph/9911020.pdf, the notion of a spectral presheaf is basically introduced as (in a more contemporany notation):
Definition(Valuation): A ...
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Fourier transform of a product of the Hermite polynomials
I tried to find the second-order correction to eigenenergies of the quantum harmonic oscillator perturbed by a cosine potential: $V=2A\cos(Bx)=Ae^{iBx}+Ae^{-iBx}$. But I have no clue how to calculate ...
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Poisson brackets for function of function
I have a problem which I am finding difficult to derive. I think I am missing something.
Assuming that the Poisson Bracket for two functions $(u, v)$ is defined on the canonical coordinates and ...
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Contour integration with 2 poles in Scattering theorey
I am studying Scattering theory but I am stuck at this point on evaluating this integral
$G(R)={1\over {4\pi^2 i R }}{\int_0^{\infty} } {q\over{k^2-q^2}}\Biggr(e^{iqR}-e^{-iqR} \Biggl)dq$
Where $ ...
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Disentangling and reordering operator exponentials from Lie groups
Consider a Lie algebra $\mathfrak{g}$ with elements $\{g_1, g_2,\ldots,g_N\}$, with a Lie group defined by the exponential map $\exp(g)$ for $g\in\mathfrak{g}$. Given an arbitrary general element $g=\...
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Solving a PDE arising from physics
Is there a way to find an analytic solution to the following PDE?
$i \partial _t \psi = - \gamma \partial _x ^2 \psi - c x $cos$(\omega t) \psi $,
where $\psi (x,t)$ is defined (in $x$) on the ...
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