Discuss the below:
Semi-infinite square well
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
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.
tan((√2m(E+Vo)a)/h) +-√(E+Vo)/E =0