Solving Finite Potential Well Problem: Confusing Math & Descriptions

[bluestar]

I am working on a finite potential well problem and the math presented in a paper is simple algebra; but, the descriptions of the process has become so confusing that I can not follow.

The reference paper is:
http://www.physics.mq.edu.au/~jcresser/Phys201/LectureNotes/SchrodingerEqn.pdf

There are no page numbers but the area in question follows equation 6.56. What he is doing in this section is setting up the equations to plot figure 6.2 on the following page.

Here is the setup
• The first curve is defined as tan(.5kL) where k = sqrt(2mE/hbar^2) m = mass & E = energy eV
• The second curve is defined as -cot(.5kL) where k = sqrt(2mE/hbar^2) m = mass & E = energy eV
• The third curve is defined as f(.5kL) = sqrt((.5koL/.5kL)^2 -1)
• These functions are plotted for various values of ko = sqrt(2mV/hbar^2) thus incrementing V

1. What I don’t understand is that k is defined in terms of E and that is the value we are looking for.

The following graph 6.2 reflects the plots indicating points of intersection which represents 1, 2, 3 or 4 solutions for allowalble energy values.

2. I don’t understand why the x-axis is labeled .5kL and what does the y-axis represent?

ps, I would have used latext but I could not get it to work here.

 

[bluestar]

Ok, ok, I have figured out the graph.

The x-axis represents .5kL and this whole term is incremented from 1 thru 6 for the calculation of the 3 functions.
The .5koL element is set at specific values for the calculations of the sqrt((.5koL/.5kL)2-1) function.
Now by setting .5koL equal to 1, 2, 3.8 and 6, I was able to duplicate his/her figure 6.2.

New question:
• How did he/she know that 1, 2, 3.8 & 6 were the correct values for .5koL?
• How do you determine the value of energy from the graph values?