A Particle in a Square Well

1. Introduction

In this notebook you will explore the behavior of the solution to the Schroedinger  equation for a particle bound in a one-dimensional square well. You will use the behavior of the solution (the wave function) to determine some of the energy states of the square well.

2. Constants, Commands, and Functions

2.1 The well parameters:

This section defines some of the constants in the problem and demonstrates the Mathematica  commands necessary to perform the exercise. The parameters of the well are defined (width a and depth (Subscript[V, 0]))  and a plot is made to illustrate it. The Which function used below allows you to define a discontinuous function. You should choose your own values of a and (Subscript[V, 0]) first. To simulate an atomic nucleus pick the width a to be the order of fermis and the depth (Subscript[V, 0]) to be the order of 100 MeV. Do NOT use the values displayed here.

a  = 4.0;
V0 = 150.0;
V[x_] := Which[x < -a,0,
               -a <= x <= a, -V0,
               a < x, 0
              ]

v1=Plot[V[x], {x,-10,10},Frame->True,FrameLabel->{"x","V(x)", "Square Well Potential"," "},FrameStyle->Black,
     PlotRange->{{-10,10},{-200,40}},PlotStyle->RGBColor[1,0,0],Exclusions->None
    ]

2.2 Solving the differential equation:

The Mathematica  command NDSolve is used to find a mathematical solution to a given differential equation. The user must define the equation to be solved and the initial conditions the solution must satisfy. The initial conditions can be determined by the behavior of the solution outside the well as discussed in Section 3.1. You must pick values for the numbers in red in the NDSolve command below. Some of the constants associated with the Schrodinger equation are defined here and a plot of the solution generated by NDSolve is displayed.

(* constants and parameters *)
hbarc = 197.0;
mc2   = 931.5;
A       = -((2*mc2)/(hbarc^2));
Energy = -70;
xmin = -10;
xmax= 10;
(* numerical solution of DE. *)
sol=NDSolve[ {psi''[x] + A*(V[x] - Energy)*psi[x] == 0, psi[xmax] == 0.5, psi'[xmax] == 0.001},psi, {x,xmin,xmax}];
(* wave function normalization factor. *)
norm=(1/(Sqrt[NIntegrate[Evaluate[ psi[x]*psi[x] /. sol],{x,xmin,xmax}]]));
(* Make a plot. *)
p1=Plot[Evaluate[ norm*psi[x] /. sol],{x,xmin,xmax},
       Frame->True,FrameLabel->{"x(fm)", "ψ",StringForm["E=`` MeV",NumberForm[Energy,20]], " "},
       FrameStyle->Black,PlotStyle->RGBColor[0,0,1],PlotRange->Automatic,BaseStyle->FontSize->16, ImageSize->6*72
      ]

3. Procedure

1. Clearly state the conditions the wave function must satisfy and the requirements you can place on ψ and ψ'. What are the constraints on ψ and ψ' at a point far outside the potential well? What parameters in the differential equation are 'adjustable'? The wave function  should be zero at a point far outside of the well.  The first derivative ψ’ should also be small far outside of the well (but negative on the positive side of the well and positive on the negative side of the well).  The energy is adjustable (quantized in the square well - so only certain values are appropriate).

2. Using your own parameters for the square well, pick a trial energy and calculate the solution using the commands demonstrated in Section 2. Is your solution a valid wave function? Why or why not? If not, what can you do to find a valid wave function? Carefully outline a procedure to find a valid wave function.
    
3. Use the procedure you outlined above to find several valid wave functions and their associated energies. Obtain a value for each energy that is accurate to within 1%.

4. Analysis and Discussion

1. Are your final wave functions properly normalized? If not, how would you normalize it?

2. Will a particle in a square well always form a bound system? Does this differ from the expectations of classical physics? Cite examples and/or calculations to support your reasoning.

3. How could you experimentally determine the nature of the potential for a bound system? In other words how could distinguish between a particle in a harmonic oscillator potential, an infinite square well, or a finite square well? A well-designed figure or sketch might be useful here.

4. Discuss the symmetry of the wave functions you found for different energy states. Support your discussion with an appropriate set of figures. Your report should include a Purpose, Calculations and Analysis, and a Discussion.