From the main menu or the button bar, choose ‘New Graph’ and name it Graph 1. Choose ‘Parabolic Well’ from the ‘Pre-Defined’ item on the ‘Potential’ menu. This sets the potential function to y=sqr(x). Set the number of energy levels to 6 and evaluate the graph by pressing the evaluate button. Your potential well should now resemble the following.
Right-click on ‘Graph 1’ to reveal the pop-up menu. Choose ‘Show Eigenvalues’ from the menu to reveal the following form showing the quantized energy levels at which the Schrodinger equation satisfactorily meets the boundary conditions.
Notice that the energy levels are evenly spaced. This situation is known as a harmonic oscillator potential.
Right click on ‘Graph 1’ and select ‘Spectrum’. A graph showing the spectral lines for vibrational transitions within the potential is created. Notice that the spectral lines occur on top of each other at one point. This shows the high probability of transitions between adjacent energy levels. Since these levels are evenly spaced, the spectral lines are of equal wavelenght.
Right click on ‘Graph 1’ and select ‘Transition Matrix’. A new form appears revealing the values of <A|x|B> for the transitions. Notice that transitions between adjacent energy levels are more probable—that is, they have larger values of <A|x|B>.
Now, create another graph and name it ‘Graph 2’. For this graph, choose ‘Parser’ from the ‘Potential’ menu and enter the function ‘y=sqr(x-0.5)’. This will create an identical potential function shifted a half unit to the right. Evaluate this potential for 6 energy levels.
From the ‘Graph’ menu, select ‘New Transition Matrix’. A dialog box will appear prompting you to select two graphs to calculate the transition matrix for. Click once in the graph window of each graph. A new form will appear showing the transition matrix between these two potentials.
Notice that shifting the potential increased the probability of nearly all transitions occuring. Now, choose ‘New Spectrum’ from the ‘Graph’ menu. Again, click once in the graph window of each graph. A new graph will appear showing the transition spectrum between the two potentials.