1. Wavefunctions: These questions pertain to a particle described by the following wavefunction: W(x) = = -{cc-1-67 = for x 0 Where C is a real, positive constant (i.e. CER,C > 0) (a) Determine the position (i.e. the value of x) at which y(x) reaches its max. amplitude. - Express your answer as both a natural logarithm, and a decimal with two sigfigs. (b) (Not marked) Make a rough sketch of this wavefunction for your reference. (c) State with brief justification whether this function is well-behaved. Explicitly address continuity, integrability, and single-valued-ness in your answer. (Ed. Note that this function has a discontinuous derivative at x=0, so it couldn't be a valid solution of the TISE unless the relevant potential function had an infinite step at this location as well.) (d) Determine the value of C such that this wavefunction is normalized (e) What are the units (i.e. the physical dimensions) of the constant, C? (f) Calculate the average position of the particle: (x) = S 4* ydx Express your answer as a fraction and a decimal to three sigfigs. - Hint: Note that sota zep dx = p, if p > 0 is a constant (g) Is (x) the same as the value you found in part a)? - Briefly discuss your answer, commenting on the relation between a wavefunction and its probability density, as well as the skew of this wavefunction. =