Gaussian integration in infinitesimal limit

jror
Messages
2
Reaction score
0

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!
 
Physics news on Phys.org
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.
 
Last edited:
Mr-R said:
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:
$$\int_x^{x+dx} f(t)\,dt = f(x)\,dx$$
 
  • Like
Likes Mr-R
Cheers vela. Will edit it.
 
Thread 'Need help understanding this figure on energy levels'
This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
Thread 'Understanding how to "tack on" the time wiggle factor'
The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
Back
Top