Final answer:
The probability of finding a particle between two points in a quantum box is calculated by integrating the squared wave function for the given energy level between those points. We examine the integration for ground (n=1) and first excited (n=2) states between 0.65L and 0.67L.
Step-by-step explanation:
To calculate the probability that a particle will be found between 0.65L and 0.67L in a box of length L, we use the quantum mechanical concept of a particle in a one-dimensional box. The wave function for a particle in a box is given by:
Ψn(x) = √(2/L) sin(nπx/L),
where 'n' is the quantum number representing the energy level, and 'L' is the length of the box. The probability density is the square of the wave function, Ψn(x)₂.
The probability of finding the particle between two points, a and b, is obtained by integrating the probability density between these points:
P(a ≤ x ≤ b) = ∫ab Ψn(x)2 dx.
For n = 1 (ground state) and n = 2 (first excited state), we calculate:
n = 1: Integrate Ψ₁(x)₂ from 0.65L to 0.67L.
n = 2: Integrate Ψ₂(x)₂ from 0.65L to 0.67L.
To perform these integrations, one must apply the specific integral of sin2(nπx/L) from 0.65L to 0.67L. This typically requires the use of a mathematical software or a table of integrals since the integrand includes a sine function squared.