The wave function for a particle in a box of length 3.3 a.u. is given by:
ψ(x) = Asin((nπx)/3.3)
where n is an integer corresponding to the quantum number of the particle and A is a normalization constant.
To find the value of A, we need to normalize the wave function so that the total probability of finding the particle in the box is equal to 1.
The probability of finding the particle between two points x1 and x2 is given by:
P = ∫x1x2 |ψ(x)|^2 dx
In the case of a particle in a one-dimensional box, the probability density |ψ(x)|^2 is given by:
|ψ(x)|^2 = A^2sin^2((nπx)/3.3)
To normalize the wave function, we need to ensure that:
∫0^3.3 |ψ(x)|^2 dx = 1
Using the identity ∫0π sin^2(u) du = π/2, we get:
1 = A^2 ∫0^3.3 sin^2((nπx)/3.3) dx = A^2 [3.3/2 - (1/(2nπ))sin((2nπ)/3.3)]
Solving for A, we get:
A = sqrt(2/(3.3 - (1/(nπ))sin((2nπ)/3.3)))
Therefore, the final expression for the wave function is:
ψ(x) = sqrt(2/(3.3 - (1/(nπ))sin((2nπ)/3.3))) * sin((nπx)/3.3)
where n is an integer corresponding to the quantum number of the particle.