All Questions
Tagged with perturbation-theory integration
23
questions
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proving complex Integral relation from perturbation theory MQ
Someone can help me to prove this identity, it comes from a normalization in MQ. From perturbation theory time dependent we have
$$H(t)=H_0+H’(t)$$
$$|Ψ>=c_a(t)e^{-iE_at/\hbar}|Ψ_a>+c_b(t)e^{-...
3
votes
1
answer
100
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Asymptotic expansion of $\int_{\xi}^{\infty}\frac{e^{-\alpha t}}{t}dt$
Question: Provided $\xi \ll 1$, find the first three terms of the asymptotic expansion of the integral $$I(\xi) = \int_{\xi}^{\infty}\frac{e^{-\alpha t}}{t}dt.$$
My approach: Assuming $\alpha>0$ ...
- 3,965
1
vote
1
answer
115
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Asymptotic approximation of an integral using splitting range
Question: Evaluate the first two terms as $\epsilon\to 0$ of $$\mathcal{N}(z)=\int_z^1\frac{dx}{\sqrt{x^3+\epsilon}}$$ where $0\le z <1$.
My approach: First, let us calculate $\mathcal{N}(z=0)$. It ...
- 3,965
1
vote
1
answer
91
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If $\int \bar \Psi \frac{1}{ r_1 - r_2} \Psi d\tau= \frac{20\pi k}{a^5}$, what is $\int \bar \Psi \left( \frac{1}{r_1 - r_2} -E_1 \right) \Psi d\tau$?
Suppose that
$$\Psi = ke^{ar_1}e^{ar_2}$$
and it is known that
$$ \left(E_0 + \frac{1}{\lvert r_1 - r_2 \rvert}\right)\Psi = E\Psi$$
for some constants $E, E_0$.
Assuming that $E$ can be expressed as ...
- 802
1
vote
2
answers
131
views
How to integrate $\int \frac{e^{-x}e^{-y}}{\lvert \vec x - \vec y \rvert} dx dy$ for the helium atom?
What methods are used to integrate
$$\int \int \frac{e^{-x}e^{-y}}{\lvert \vec x - \vec y \rvert} dx dy$$
which comes up in perturbation theory calculations such as for the helium atom (8.2.6). The ...
- 802
2
votes
0
answers
87
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Are there methods for capturing transcendentally small terms in perturbation theory?
By "transcendentally small terms" I mean corrections to a perturbation that are smaller than all polynomial orders. They are generally exponential in form. I would have linked to a Wikipedia ...
- 2,412
0
votes
1
answer
48
views
Evaluation of Integral
How do I evaluate the integral $\int_{0}^{\infty}e^{-t/\tau} e^{i\omega t }dt$
Where $\tau$ is a constant
Akshai Christy Jacob
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29
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Expanding a function containing a logistic term inside an integral
I am working my way through Mark Homes Introduction to Perturbation Methods by myself. I am having trouble solving problem 1.8(c).
I believe that I'm supposed to split the integral by introducing a ...
- 61
1
vote
0
answers
95
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Asymp. Integrals
I am trying to get a two-terms asymptotic expansion {as $c\to 0 $ } for the following integral
$$\int_{0}^{c} \sqrt{\frac{c^2-\zeta^2}{1-\zeta^2}} d\zeta$$
I have tried so far to substitute $\zeta=c\ ...
- 139
1
vote
1
answer
229
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Perturbation Integral
How to show that for $f(x,m)$ = $\frac{mx^2}{2} + \frac{x^4}{4}$,
the integral $Z(\lambda)$ = $\frac{1}{\sqrt \lambda}$$\int_{-\infty}^{\infty} dz e^{-f(z)/\lambda}$ is $\sqrt{\frac{m}{2\lambda}}$$e^...
- 195
2
votes
1
answer
59
views
Endpoint Perturbation Theory
So, suppose we want to evaluate the integral
$$\int_{a}^{b+\epsilon c}f(x)\, dx$$
where $f:\mathbb{R}\to\mathbb{R}$ is assumed to be smooth and regular in the integration region $[a,b+\epsilon c]\...
- 334
1
vote
1
answer
55
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Asymptotics of $C(x) = \int_0^x \cos(0.5\pi t^2) \ dt.$
Consider the Fresnel integral
$$C(x) = \int_0^x \cos(0.5\pi t^2) \ dt.$$
I've calculated that, as $x\to 0$, $C(x) \sim x$ and, as $x\to\infty$, $C(x) \sim 0.5 + \frac{\sin(0.5\pi x^2)}{\pi x}$. ...
- 1,295
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1
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39
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Approximation scheme for integrating small differentials?
Let's say I have some differential $dy/dt$. I want to calculate the definite integral over some interval $t = 0-T$ i.e.
$$ y(T) = \int_0^{T} \frac{dy}{dt} dt$$
My question is if $dy/dt$ is very ...
- 508
1
vote
1
answer
517
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Why is it so important in the method of steepest descent that we follow the steepest descent contour that passes through a saddle point?
In the method of steepest descent, we approximate to leading order an integral of the form $$I(x) = \int_C f(t) \exp(x \phi(t)) dt$$ as $x \to \infty$ by observing that if we write $t = \xi + i \eta$ ...
1
vote
1
answer
264
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Applying the method of steepest descent to $ \int_{-i \infty}^{i \infty} \frac{\exp{x (t^2 - 2t)}}{t - a}\,dt$ as $x \to \infty$
Consider the integral $I(x) = \int_{-i \infty}^{i \infty} \dfrac{\exp{x (t^2 - 2t)}}{t - a} dt = \exp(-x) \int_{-i \infty}^{i \infty} \dfrac{\exp{x (t - 1)^2}}{t - a} dt$.
I would like to find the ...