Consider the semi-infinite square well given by V(x)=-Vo<0 for 0<=x<=a and V(x)=0 for x>a. There is an infinite barrier at x=0. A particle with mass m is in a bound state in this potential energy E<=0.
a) solve the Schroedinger Eq to derive phi(x) for x=>0. Use the appropriate boundary conditions and normalized the wave function so that the final answer does not contain any arbitrary constants.
b) Show that the allowed energy levels E must satisfy the equation
tan((√2m(E+Vo)/h)a)+√-(E+Vo)/E= 0
c) The equation in part (b) can not be solved analytically to give the allowed energy levels but simple solutions exist in certain special cases. Determine the conditions on Vo and a so that a bound state exists with E=0.