In this question we are going to implement the technique used to find the eigen states of the harmonic oscillator different potential. consider a particle in the potential to a v (z) vo (e-20r-2e) where vo and a are real and positive constants. a) make a qualitative sketch of the potential v (x). what is the condition on the energy e for the particle to be in a bound state? b) define the dimensionless quantities 2 2mvo v2m v -2me -ar ah ah ah and show that the schrodinger equation can be written as 1 d (2) (2) v+ 8 + 1/2 + dz2 dz 0() e) explain why the variable z is only defined in the interval (0, o). find the boundary conditions on show that for z0o the asymptotic solution is ()~exp (-/2) while for z0 if is (s) ~ 2'. d) e) based on (d) plug in a solution of the form ()e/2f (2) and obtain the following equation for f () _pf(2) + (2s +1- : ) 4e) + vf (2) = 0 d: d22 f) solve the equation for f (2) by plugging in a taylor series 1)-σc.2 and find a recursion equation for the coefficients c. show that v (z) will satisfy the boundary conditions only if the series is cut of at a finite n. use this to find the eigen energies. g) find the wave function of the ground state o (z) up to a normalization constant. h) how many bound states does the system have? under what conditions are there no bound states? hint: express s in terms on and remember that, by definition, s > 0.