Homework Statement
[/B]
I have the following expression:
$$S=T+V$$
$$T=\frac{m}{\tau_0+it}((x_1-x_0)^2+(x_2-x_1)^2)+\frac{m}{2(\tau_1-it)}(x_2-x_0)^2$$
$$V= \frac{(\tau_0+it)}{2}(\frac{k_0 x_0}{2}+\frac{k_0 x_2}{2}+k_0 x_1)+(\tau_1-it)(\frac{k_1 x_0}{2}+\frac{k_1 x_2}{2})$$
The main goal...
Homework Statement
Can this function be integrated analytically?
##f=\exp \left(-\frac{e^{-2 \theta } \left(a \left(b^2 \left(e^{2 \theta }-1\right)^2 L^2+16\right)-32
\sqrt{a} e^{\theta }+16 e^{2 \theta }\right)}{b L^4}\right),##
where ##a##, ##b## and ##L## are some real positive...
##x= r Cosh\theta##
##y= r Sinh\theta##
In 2D, the radius of hyperbolic circle is given by:
##\sqrt{x^2-y^2}##, which is r.
What about in 3D, 4D and higher dimensions.
In 3D, is the radius
##\sqrt{x^2-y^2-z^2}##?
Does one call them hyperbolic n-Sphere? How is the radius defined in these...
What if, I modify my argument of the delta function a little bit, which looks as follows:
##\delta(px_1^2 - q x_2^2 +r x_1-s x_2+\epsilon)##, where r, s and ##\epsilon## are again some constants.
Does your argument still hold?
In fact, I just noticed that, your arrangement of the argument...
Homework Statement
I have a 2D integral that contains a delta function:
##\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp{-((x_2-x_1)^2)+(a x_2^2+b x_1^2-c x_2+d x_1+e))}\delta(p x_1^2-q x_2^2) dx_1 dx_2##,
where ##x_1## and ##x_2## are variables, and a,b,c,d,e,p and q are some real...
Integrals like this appear when one is looking at chemical reaction rates. The exponent is basically the classical action. A and B contains space coordinates, which will be integrated after integral over ##\tau## has been performed.
Is it possible to integrate the following function analytically?
##\int_{0}^{\infty} \frac{\exp{-(\frac{A}{\tau}+B\tau+\frac{A}{\beta-\tau})}}{\sqrt{\tau(\beta-\tau)}}d\tau,##
where ##A##, ##B## and ##\beta## are real numbers. What sort of coordinate transformation makes the integral bounded...
I am trying to numerically integrate the following complicated expression:
$$\frac{-2\exp{\frac{-4m(u^2+v^2+vw+w^2+u(v+w))}{\hbar^2\beta}-\frac{\hbar\beta(16\epsilon^2-8m\epsilon(-uv+uw+vw+w^2-4(u+w)\xi...
Nope! F is just flux operator from reactant to product ##(|0><1|-|1><0|)##. This is for my case (or non-adiabatic case). However, in adiabatic case it is ##(p \delta(x-s)+\delta(x-s) p)##. But I am mainly interested in non-adiabatic case, where there are two electronic states. Basically, in...
This is a chemically inspired problem, but the path is fully quantum mechanics and a bunch of integrals.
How does one calculate fully quantum mechanical rate ($\kappa$) in the golden-rule approximation for two linear potential energy surfaces?
Attempt:
Miller (83) proposes...
That looks totally good. But when I modify my Integrand with different parameters, the same Integral doesn't seem to be Integrable under Cauchy Principal-Value. For instance:
$$2\frac{\exp{(-\frac{1}{4}(-2+u)^2-u^2)}}{|u-2|}$$
This integral doesn't converge in mathematica. Why is it so...
Hi All,
I am wondering if the function below is Integrable:
$$\frac{\exp{(-\frac{1}{2}(u-2)^2-2u^2)}}{u-2}$$
When I work it out on computer, the integral is finite from -Inf to Inf. But clearly it has a pole at u=2. Is this pole integrable? If yes, what kind of coordinate transform is...
In addition, what does +p and -p here mean. If it is +p, does it mean the area between the two hand is V1. And if it is -p, does it mean the area between the two hands is V2?
I want to understand what changing coordinate system means for hands of clock. Let's say the clock only has hour and minute hand. It can move let's say just in the upper 180 deg. of the clock (as shown in the figure). The area between the two hands is V1, and the rest is V2. Depending on the...