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:)
-
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).
-
To begin with don't perform a calculation, but guess what shift in energy
you would expect for the lowest energy level.
-
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 o∫a
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]
-
Optimize the wavefunction using the variational method. Normalize first
Yansatz (i.e. determine A according
to
-∞ò¥|Yansatz
|² dz = 1).
-
What is the energy of the ground state? Compare your result with the exact
solution.