Gaussian integration in infinitesimal limit

[jror]

Homework Statement

Given the wave function of a particle $\Psi(x,0) = \left(\frac{2b}{\pi}\right)^{1/4}e^{-bx^2}$, what is the probability of finding the particle between 0 and $\Delta x$, where $\Delta x$ can be assumed to be infinitesimal.

Homework Equations

The Attempt at a Solution

I proceed as I normally would when trying to obtain the probability of finding a particle within a certain interval, by calculating the integral $\int_a^b |\Psi(x,0)|^2 dx$, where the limits here are $a=0$ and $b=\Delta x$. I am stuck in trying to calculate the Gaussian with these limits. I know the answer in the infinite limit, but for abritrary limits one usually has to deal with error functions. What is the trick here with setting the upper limit to some assumed infinitesimal number? Would appreciate a hint!

 

[Mr-R]

I think you just have to expand the wave function to first order in $\Delta x$ and then $$P(\Delta x)=|\Psi(\Delta x)|^2 \Delta x$$
Imagine the area under an infinitesimal interval, in this limit you can substitute multiplication for the integral. I am not entirely sure though.
See the post after this one.

 

[vela]

Imagine the area under an infinitesimal interval, in this limit the integral can be substituted for a normal multiplication. I am not entirely sure though.

You worded that backwards. You can substitute multiplication for the integral:
$$\int_x^{x+dx} f(t)\,dt = f(x)\,dx$$

 

[Mr-R]

Cheers vela. Will edit it.