Question

The normalized wavefunction for a particle trapped in a 3-dimensional box of dimensions 0 < x < a, 0 < y < b, 0 < z < c, is given by

ψx,y,z = (2/a)^(1/2) (2/b)^(1/2) (2/c)^(1/2) sin (n_x pi y/b) sin (n_y pi y/b) sin (n_z pi z/c)

For the case in which a = b = c = 10 Å approximate the probability that an electron will be found in a volume element whose dimensions are Δx = Δy = Δz = 0.01 Å and whose center is at x = 2 Å, y = 3 Å, z = 5 Å for the three states

nx ny nz

2 1 1

1 2 1

1 1 2

*HINT: ⌠_x⌠_y⌠_z ψ∗ψ dx dy dz is the probability. When the argument ψ∗ψ is nearly constant over xyz

the range it can be removed from the integral and the probability approximated as ψ∗ψ ΔxΔyΔz.