Gaussian integration in infinitesimal limit

In summary, the conversation discusses how to calculate the probability of finding a particle between 0 and an infinitesimal interval, given the wave function of the particle. The suggested approach is to expand the wave function to first order in the interval and then substitute multiplication for the integral.
  • #1
jror
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0

Homework Statement


Given the wave function of a particle [itex] \Psi(x,0) = \left(\frac{2b}{\pi}\right)^{1/4}e^{-bx^2} [/itex], what is the probability of finding the particle between 0 and [itex] \Delta x [/itex], where [itex] \Delta x [/itex] 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!
 
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  • #2
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.
 
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  • #3
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:
$$\int_x^{x+dx} f(t)\,dt = f(x)\,dx$$
 
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Likes Mr-R
  • #4
Cheers vela. Will edit it.
 

FAQ: Gaussian integration in infinitesimal limit

What is Gaussian integration in infinitesimal limit?

Gaussian integration in infinitesimal limit is a mathematical method used to approximate the integral of a function over an infinite range by breaking it down into smaller, more manageable parts. It involves using the Gaussian distribution, also known as the bell curve, to model the behavior of the function and then summing up these small parts to get an approximation of the integral.

Why is Gaussian integration in infinitesimal limit useful?

Gaussian integration in infinitesimal limit is useful because it allows us to solve integrals that would otherwise be impossible to solve analytically. It also provides a more accurate approximation compared to other numerical integration methods, especially for functions with rapidly changing behavior over a wide range.

What is the difference between Gaussian integration in infinitesimal limit and other numerical integration methods?

The main difference between Gaussian integration in infinitesimal limit and other numerical integration methods is in the way they approximate the function. Gaussian integration uses the Gaussian distribution to model the function, while other methods such as the trapezoidal rule or Simpson's rule use a series of straight lines or curves to approximate the function.

What are the limitations of Gaussian integration in infinitesimal limit?

One of the main limitations of Gaussian integration in infinitesimal limit is that it can only be applied to functions that are continuous and have a finite integral. It also requires knowledge of the function's behavior over the entire range, which may not always be available. Additionally, the accuracy of the approximation may decrease significantly if the function has sharp peaks or rapidly changing behavior.

Can Gaussian integration in infinitesimal limit be used for multidimensional integrals?

Yes, Gaussian integration in infinitesimal limit can be extended to solve multidimensional integrals. This is known as multivariate Gaussian integration and involves using multiple Gaussian distributions to model the behavior of the function in different dimensions. However, the computational complexity increases significantly for higher dimensions, making it more challenging to apply in practice.

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