Discussion  Fr 18.11.2005 at 12:15,  HR 30.1

Home work 3
 
 

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 Y_n and the energy eigenvalues E_n? You need it as a guide for the following problem:
 

1. Now you perturb the electron in the box by positioning it into a charged capacitor (10kV) where the distance between two parallel plates is 1 cm (the electric field is parallel to x axis). The perturbation operator has a 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 energy shift you would expect for the lowest energy level.
3. Calculate the energy shift in the first approximation using the perturbation theory. What is the energy shift for the lowest energy level? Can you observe this shift by spectroscopic techniques? Hint: you may have to calculate a matrix element integral using integration by parts, or just taking the integral value from mathematical handbooks.  

Exercise 2:  Variational method

Think of the one-dimensional harmonic oscillator: V(z) = (^k/₂)·z². Of course, you know the wavefunction for the oscillator. But let's assume you do not. Try the following approximate expression for the ground state wavefunction:

which is the Gaussian function with l as parameter.

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

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