A particle is described by the wave function W(x) = b(a2 - x2) for -a s x s a and W(x) 0 for x s -a and x z a, where a and b are positive real constants. (a) Using the normalization condition, find b in terms of a (b) What is the probability to find the particle at x = 0.31a in a small interval of width 0.01a? (c) What is the probability for the particle to be found between x = 0.08a and x = 0.80a ? a2/ל (a) 0.9375 (b) 0.0076 (c)

In the region 0 w, V3 (x) = 0. (a) By applying the continuity conditions atx = a, find c and d in terms of a and b. (b) Find w in terms of a and b. -

The particle is confined to a one-dimensional box between x=0 and x=2. Its wave function is A(x)=6x2/N (0≤x≤1) and A(x)=6/N (1<x≤2), where N is a normalization constant. Calculate the constant N and the average position of the particles.

For a quantum particle in a scattering state as it interacts a certain potential, the general expressions for the transmission and reflection coefficients are given by T = Jtrans Jinc R = | Jref Jinc (1) where Jinc, Jref, Jtrans are probability currents corresponding to the incident, reflected, and transmitted plane waves, respectively. (a). potential For the particle incident from the left to the symmetric finite square well -Vo; a < x < a, V(x) = 0 ; elsewhere, show that B Ꭲ ; R = A A