Quantum Mechanics 1

Schrödinger’s equation in one dimension

The uncertainty principle

The uncertainty principle for position and momentum for particles moving in one dimension is expressed by the inequality

where the uncertainties in x and p are given respectively by 

			, 	

and the individual averages are defined by 

Now it is obvious that ÜpÝ and Üp2Ý are proportional to   and 2  respectively, so that  is proportional to .  So for the purpose of verifying the Heisenberg principle numerically it is best to write    where k is the wave number corresponding to the momentum k. Then , so that  cancels and the uncertainty principle is written in the simpler form 
		  	where		 

and		, 		.

A.	The first task is to verify that the average momentum is  zero, i.e. < k > = 0, for the eigenfunctions corresponding to a selection of potentials. (Choose the square well and the asymmetric well as typical examples, using the default options for the well parameters.) To do this you select a potential, get the computer to calculate the eigenvalues and eigenfunctions, then use the menu "Part 2: Wavefunction Properties" to examine the properties of the eigenfunctions. In particular, choose both "function 1" and "function 2" as the eigenfunction in question, and select the operator which operates on the second function as the differential operator d/dx. Then integrate the product and check that the answer is always zero. Why should this be, physically, i.e. why should the expectation value of the momentum of particles confined to potential wells be zero when in energy eigenstates? What is the result when they are not in energy eigenstates?

B.	Now extend this process by evaluating the integrals in which the operator  which operates on the second function is chosen in turn to be x, x2 and (d/dx)2 . Accepting the default values of well parameters, do this for the ground state and first two excited states for (i) the square well, (ii) the parabolic well, and (iii) the asymmetric well, and draw appropriate conclusions from your results for the values of the product Dx Dk. Make a note of any problems you encounter in terms of the number of significant figures in the answer.