QM harmonic oscillator - integrating over a gaussian?

[tarkin]

Homework Statement

[/B]
For the first excited state of a Q.H.O., what is the probability of finding the particle in -0.2 < x < 0.2

Homework Equations

Wavefunction for first excited state: Ψ= (√2) y e^-y2/2

where:

The Attempt at a Solution

To find the probability, I tried the integral of : |Ψ|²

but this gives the integral of gaussian. From what I've read, the integral of a gaussian can only be solved from -infinity to infinity. So how can I find it from -0.2 to 0.2?

 

[kuruman]

Look up error function, then use a canned algorithm such as on a spreadsheet to find its value.

 

[TSny]

Are you allowed to use a calculator that can do numerical integration?

https://www.zweigmedia.com/RealWorld/integral/integral.html

 

[Delta2]

Use that $e^x=\sum\limits_{n=0}^{\infty}{\frac{x^n}{n!}}$ so that $e^{-\frac{x^2}{2}}=\sum\limits_{n=0}^{\infty}{(-1)^n\frac{x^{2n}}{2^nn!}}$.

So you can integrate like it is a polynomial with infinite terms. You can choose up to which term of n to keep but I think for your value of x between 0.2 and -0.2 , the first three or four terms of integration are enough. You ll probably have to use a computer program or at least a calculator if you choose a very high value for n like the first 10 terms or more.