Home work Physical and Theoretical Chemistry IV

- Molecular Spectroscopy -

Discussion  Fr 25.4.2003 at 12:15,  HR30.1 (Hagenring 30)



 

Home work 1
 
 

Exercise 1:  Perturbation theory

An electron is captured in a one-dimensional box of length a=20 Angström (2 nm). What is the Schrödinger equation within that box (V = 0 for 0 ≤ x £ a; V = ¥ elsewhere). What is the wavefunction Yn and the energy eigenvalues En? (don't calculate it; just look in a textbook (or in the internet). You need it as a guide for the following problem:)
 

  1. Now you perturb the electron in the box by positioning all into a charged capacitor (10kV) where the distance of  the two parallel plates is 1 cm (the electric field points toward x). The perturbation operator has the simple form H' = αx, i.e. for x = 0 the perturbation is 0. Calculate α (if you don't succeed, continue with the letter α).
  2. To begin with don't perform a calculation, but guess what shift in energy you would expect for the lowest energy level.
  3. Calculate the shift in energy in a first approximation using perturbation theory. What is the shift in energy for the lowest energy level? Can you observe this shift by spectroscopic techniques? Why, resp. why not?
You will encounter the integral oa x sin2(nπ x/a) dx. The result is a2/4. (If you like math: what is the appropriate technique to calculate this integral?)
 
 

Exercise 2:  Variational method

Think of the one-dimensional harmonic oscillator: V(z) = (k/2)·z2. Of course, you do know the wavefunction for the oscillator. But let's assume you don't. Try the following ansatz for the wavefunction of the ground state (a gaussian function with λas parameter):

Yansatz = A exp[- λ·z2]

  1. Optimize the wavefunction using the variational method. Normalize first Yansatz (i.e. determine A according to -∞ò¥|Yansatz |² dz = 1).
  2. What is the energy of the ground state? Compare your result with the exact solution.