Chemistry 3730 assignment 3 solutions

1. Start by defining the wavefunction:
   

   > psi := (n,L,x) -> sqrt(2/L)*sin(n*Pi*x/L);

   

   
   

   We will eventually need the following assumptions:
   

   > assume(L,positive); assume(hbar,positive);

   
   To verify that the Heisenberg uncertainty principle holds, we need to compute and , which in turn require that we compute , , and . Some of these averages are trivial so I just typed them in.
   

   > average_x := L/2;

   

   > average_x2 := int(psi(2,L,x)^2*x^2,x=0..L);

   

   > delta_x := sqrt(average_x2-average_x^2);

   

   > average_p := 0;

   

   
   

   We can use the answer to question 2 to cut down our work here, or we can compute the integral. I chose to do the former.
   

   > average_p2 := 4*h^2/(4*L^2);

   

   
   

   Convert the h to hbar so that all of the calculations use the same version of Planck's constant:
   

   > h:=2*Pi*hbar;

   

   > delta_p:= sqrt(average_p2-average_p^2);

   

   
   

   Now verify the uncertainty principle:
   

   > evalf(delta_p*delta_x);

   

   > simplify(");

   

   
   

   This is much bigger than so the uncertainty principle holds for this state.
2. To answer such a question, ask yourself what you know:
   • We know the energy of the particle in a box.
   • The energy is all kinetic.
   • The kinetic energy is .
   Therefore, for the particle in a box,