Question

A particle trapped in a one-dimensional box of length L is described by the wave function ψ = ax. What is the probability that the particle is lying between L1 and L2?

Answer 1

By definition, the probability that the particle is lying between L1 and L2 is given as follows (see http://www.iitg.ac.in/asil/QM-02.pdf):

$$P = \int_{L_1}^{L_2}|\psi|^2dx$$

where

$$|\psi|^2 = (ax)^2 =a^2x^2$$

is the modulus of the wavefunction. Thus, evaluating the integral, obtain:

$$P = a^2\int_{L_1}^{L_2}x^2dx = a^2\left.\dfrac{x^3}{3}\right |_{L_1}^{L_2} = \dfrac{a^2}{3}(L_2^3-L_1^3)$$

Answer. $P = \dfrac{a^2}{3}(L_2^3-L_1^3)$.