Recent content by junt

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...

Thanks. But I think it is safe to say that your argument still leads to pole, which is linear in ##x_2##. Thanks though for replying

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...

Hi All, What are the main differences between statistical and dynamics properties in physics? Could you please explain the difference for problems in both classical and quantum mechanics. For instance, path integral molecular dynamics is supposed to give statistical properties of a quantum...

Hi All, $$\int{\exp((x_2-x_1)^2+k_1x_1+k_2x_2)dx_1dx_2}$$ I can perform the integration of the integral above easily by changing the variable $$u=x_2+x_1\\ v=x_2-x_1$$ Of course first computing the Jacobian, and integrating over ##u## and ##v## I am wondering how you perform the change of...

Is square root of delta function a delta function again? $$\int_{-\infty}^\infty f(x) \sqrt{\delta(x-a)} dx$$ How is this integral evaluated?

I am quite new here, and was wondering if anybody knows how this 2D integral is evaluated. $$ \int_{-\infty}^\infty \int_{-\infty}^\infty \delta(k_1 x-k_2y)\,dx\,dy$$Any help is greatly appreciated! Thanks!