Final answer:
The normalization constant a for the given wave function is √3 nm-3/2. The probability of finding the particle between x = 0.240 nm and x = 0.590 nm is 0.089 nm^2. The expectation value of the electron's position is 0.25 nm. The electron has an equal probability of being to the left or to the right at x = 0.500 nm.
Step-by-step explanation:
(a) To find the normalization constant, we need to normalize the wave function, which means ensuring that the probability of finding the electron over all space is equal to 1. The normalization condition is given by:
∫|ψ(x)|² dx = 1
where ψ(x) is the wave function and the integral is taken over all space. In this case, the normalization constant a can be determined by solving the integral:
∫(ax)² dx = ∫a²x² dx = a²/3 * x³ |01.00 nm = 1
0.333a² = 1
a² = 3
a = √3
(b) The probability of finding the particle between x = 0.240 nm and x = 0.590 nm is given by:
∫|ψ(x)|² dx
= ∫(√3x)² dx
= 3 * ∫x² dx = 3 * x³/3 |0.240 nm0.590 nm
=(0.590 nm)³ - (0.240 nm)³
(c) The expectation value of the electron's position is given by:
E(x) = ∫x * |ψ(x)|² dx
= ∫x * (√3x)² dx
= 3 * ∫x³ dx = 3 * x⁴/4 |01.00 nm
=3 * (1.00 nm)⁴/4
(d) The electron has an equal probability of being to the left or to the right of a certain position when the wave function is symmetrical about that position. In this case, the probability is equal when the wave function is equal to zero, which occurs at x = 0.500 nm.