The Central Potential

1. Introduction and Instructions

In this exercise you will investigate the behavior of potential energy functions that depend only on the radial distance from the force center. Section 2 is an introduction to the Mathematica  notebooks that we will be using throughout the semester and demonstrates how one defines functions, plots, prints, and makes 'movies'. In Section 3 you will explore some of the behavior of these central potentials analytically and graphically.  Include a purpose, derivations, calculations, figures, and a discussion in your report on Section 3.

2. Mathematica Notebooks and Commands

Mathematica  documents are known as notebooks and are used to manage your work session.  Each notebook in turn consists of several "cells" containing different parts of the session. Below you should see several subsection titles for the different parts of this exercise. To open any section or cell place the pointer over the small box located at the bottom of the bracket immediately to the right of each section title. You should see the pointer change shape. When this occurs double-click the mouse by pressing down momentarily on the mouse button twice in succession and then releasing. Now open section 2.1 and follow the instructions there.

2.1 Mathematical Syntax and Test Functions

The lines below display definitions of the functions from your assignment that you are to plot in their Mathematica  format. Please note several things. The software interprets a blank space or the character "*" as multiplication (each character is used randomly in the definitions below). It is worth noting here that it is best to use the "*" character to represent multiplication instead of a blank space. Countless generations of Mathematica users have been deceived when debugging their code by the presence or absence of the space representing multiplication. It is better to be explicit so your code will be easier to read, interpret, and debug. Notice also the need for the trailing underscore, "x_", when defining the function and the use of ":=". To actually define these functions in Mathematica  place the pointer anywhere in the cell (note cell bracket on the right) and click by pressing down briefly on the mouse button. The editing cursor should now appear and will be located somewhere within the cell. Now  press "Enter" (not "Return") on the keyboard. Notice how the bracket of the cell changes shape. This means Mathematica   is executing the commands you have given it. When it is done the bracket will return to its original shape. Now open section 2.2 and follow the instuctions there.

psi4[x_] := A Cos[w*t] Sin[k x + B]
psi2[x_] := A*Cos[k*(x - c*t)]
psi3[x_] := A Exp[- 0.02*(x + c*t)^2] Cos[k*(x + c*t)]

2.2 Plotting Commands

The lines below display some of the commands needed to plot the functions defines above. First one must assign values to the constants in the problem. Place the pointer in the cell below and click. After the editing cursor appears hit "Enter".

A = 1.0
k = 1.0
B = 3.14
c = 1.0
w = 1.0
t = 41.5

In the cell below is the command to plot the first function defined above (psi1[x]). Several options of the "Plot" command have been used to illustrate their function and to produce a clear, well-labelled graph. The Mathematica  book by Wolfram contains detailed descriptions of the meaning of each option.To evaluate the commands in the cell place the pointer anywhere in the cell and click. After the editing cursor appears hit "Enter". Try different values for the numbers to see how they affect the plot. You can edit them by placing the pointer just to the right of the number you want to change and clicking. The editing cursor will appear and you can use the delete key to remove the original values and then enter the ones you desire with the keyboard.

Plot[psi4[x], {x,0,10}, 
       Frame->True,
       PlotRange->{-1,1},
       FrameLabel->{"x",
                    "psi4",
                    "Fig 4c. t = 41.5",
                    " "},
       RotateLabel->False]

2.3 Printing

To print a portion of your work highlight the brackets on the right of the notebook. This is done by placing the pointer over the top bracket first. The pointer will change shape. Now press and hold the mouse button down and drag the pointer through the brackets you want to print (this is known as clicking and dragging). The brackets will be highlighted if this is done correctly. Release the mouse button and go to the "File" menu at the top of the screen. Press the mouse button down and you will see a number of options. Holding the mouse button down drag the pointer down to "Print Selection". When the mouse is correctly positioned the text for this option will be highlighted. Now release the botton. You will be presented with a dialog box. Place the pointer in the box that says "PRINT" and click the mouse button. Your output will soon appear at the laser printer.

2.4 Making Movies

If you really want to demonstrate your mastery of the Mathematica  package, use the following commands to create a 'movie' of a function developing in time. After the 'MoviePlot' command is executed select all the different frames by clicking on the bracket that encloses the movie frames. Then, choose the command 'Animate Selected Graphics' from the 'Graph menu above.

(* The command below reads in some more Mathematica 
   commands from a file on the hard disk. It has to 
   be executed only once. *)
   
<<Graphics`Animation`

(* This last command defines a function, f[x,time], 
   that is used to make the movie. *)
   
f[x_,time_] := Sin[x] + 0.2*Sin[5.0(x - 0.2*time)]
Animate[Plot[f[x,time],
          {x,-10,10},
          Frame->True,
          PlotRange->{-1.4,1.4},
          FrameLabel->{"x","f(x)"}],
          {time,0,40*Pi,2*Pi/100}
         ]

3. Some Properties of a Central Potential

A particle of mass m  moves in a "central potential," V(r), where r denotes the radial displacement of the particle from the fixed origin.

a.  What is the vector force on the particle? Recall the components of the gradiant operator in spherical coordinates.

b.  Show that the angular momentum L of the particle about the origin is constant. (Hint: Calculate the time derivative of L = r x p and recall that p = m dr/dt.)

c.  Show the energy of the particle may be written as 

                                   E = pr2/2m + L2/2mr2 + V(r)
                                   
where pr is the radial component of the momentum.

d.  For convenience one can treat the last two parts of the expression for the energy as an "effective" potential where Veff = L2/2mr2 + V(r). Consider the electromagnetic case where
V(r) = -Z1Z2e2/r and Z1 and Z2 are the number of electronic charges of the two interacting particles. Make a sketch of Veff versus radial displacement r for an electron orbiting a proton. Use the sketch to establish the conditions for circular motion. What are the conditions for a bound state? What are the conditions for an unbound state? Make a clear choice of constants and be careful with your units. You should use units of angstroms and electron volts.