Bound States in a Square Well

Niels Walet, 1998-2009

Description

We investigate the bound states in a square well, of width a and depth -(Subscript[V, 0]).

Plot[If[Abs[x]<1,-1,0],{x,-2,2},PlotStyle->Hue[0.0]]

The Schrodinger equation in the three regions I: x<-a, II: -a<x<a, and III: x>a can easily be written down,
I, III -((ℏ^2)/(2m))((d^2)/(dx^2))ϕ(x)=E ϕ(x),
II:-((ℏ^2)/(2m))((d^2)/(dx^2))ϕ(x)-(Subscript[V, 0])ϕ(x)=E ϕ(x).
We shall solve these equations for -(Subscript[V, 0])<E<0. 
Argue that in that case the solution in regions I and III are exponential (real exponents), and in region II is oscillatory (sum of sines and cosines). 
Define k = (Sqrt[-((2m)/(ℏ^2))E]), κ=(Sqrt[((2m)/(ℏ^2))((Subscript[V, 0])+E)]). Is this a sensible definition?

Solution to the bound-state problem.

Just to be sure, make k,κ,(Subscript[A, 1]),(Subscript[B, 1]),(Subscript[B, 2]),(Subscript[A, 3])undefined. When they are not defined this gives some error messages...

k=.;κ=.;(Subscript[A, 1])=.;(Subscript[B, 2])=.;(Subscript[A, 3])=.;(Subscript[B, 3])=.;

Define the wave functions in regions I, II and III. We have an exponentially damped wave function in regions I and III (why?), and the full solution in region II.

(Subscript[ϕ, I])[x_]:=(Subscript[A, 1])(E^( k x))  ;
(Subscript[ϕ, II])[x_]:=(Subscript[A, 2])cos(κ x) +(Subscript[B, 2])sin(κ x) ;
(Subscript[ϕ, III])[x_]:=(Subscript[B, 3]) (E^(- k x));

Set up the four matching equations, for the functions and their derivatives at x=±a.

eqns1=(Subscript[ϕ, I])(-a)==(Subscript[ϕ, II])(-a);
eqns2=(Subscript[ϕ, II])(a)==(Subscript[ϕ, III])(a);
eqns3=(Subscript[ϕ, I])'(-a)==(Subscript[ϕ, II])'(-a);
eqns4=(Subscript[ϕ, II])'(a)==(Subscript[ϕ, III])'(a);
eqns={eqns1,eqns2,eqns3,eqns4}

Eliminate (Subscript[A, 1])and (Subscript[B, 3])from the equations.

solution=Eliminate[eqns,{(Subscript[B, 3]),(Subscript[A, 1])}]

These solutions imply (Subscript[A, 2])=0∧(Subscript[B, 2])=0!

Assigments

Make a graphical representation of these two types of solutions using Mathematica.