For a particle in the lowest energy level of a one dimensional box 1 Angstrons wide, calculate the probability of finding the particle between 0.40A and 0.60A
http://itl.chem.ufl.edu/4412_aa/partinbox.html
find the prob from .4A to .6A, evaluate the integral
Pr= INT (2/L sin^2 (PIx/L) dx from.4L to .6L
Pr= 2/L INT sin^2 (PI x/L) dx
= 2/L (x/2 - L/4PI * sin 2PI x/L)
then evaluate over the limits. check that integral http://integral-table.com/
To calculate the probability of finding a particle in a specific range within a one-dimensional box, we need to use the wave function and calculate the integral of the wave function squared over that range.
The wave function for a particle in the lowest energy level of a one-dimensional box is given by:
ψ(x) = √(2/L) * sin(nπx/L)
Where L is the width of the box (in this case 1 Ångström) and n is the quantum number (for the lowest energy level, n=1).
To find the probability of finding the particle between 0.40 Å and 0.60 Å, we need to calculate the definite integral of the squared wave function over that range:
P = ∫[0.40, 0.60] |ψ(x)|^2 dx
Since the wave function is real, we can simplify it by squaring only the amplitude:
|ψ(x)|^2 = [√(2/L) * sin(πx/L)]^2 = (2/L) * sin^2(πx/L)
Now let's substitute the values into the integral:
P = ∫[0.40, 0.60] (2/L) * sin^2(πx/L) dx
Calculating this integral will give us the probability of finding the particle in that specific range.