Chemistry 3730 assignment 12 solutions

1. We have to define the wavefunction, and in terms of the underlying parameters of the problem:
   

   > psi0 := x -> (beta/Pi)^(1/4)*exp(-beta*x^2/2);

   

   > beta := omega0*mu/hbar;

   

   > omega0:=sqrt(k/mu);

   

   > assume(k>0); assume(mu>0); assume(hbar>0);

   
   

   To confirm that is a solution, apply the Hamiltonian to this function:
   

   > -(hbar^2/(2*mu))*diff(psi0(x),x$2)+1/2*k*x^2* psi0(x);

   

   > simplify("/psi0(x));

   

   
   

   Note that this is .

   Now, to demonstrate that the wavefunction is normalized,
   

   > int(psi0(x)^2,x=-infinity..infinity);

   

   
   

2. Apply to :
   

   > 1/sqrt(2*mu)*(-I*hbar*diff(psi0(x),x)+I*omega0*mu*x*psi0(x));

   

   
   

   The only part that matters is the part that depends on x. The rest are constants and we need to normalize this wavefunction anyway.
   

   > psi1:=N1*x*exp(-beta*x^2/2);

   

   
   

   To normalize:
   

   > int(psi1^2,x=-infinity..infinity);

   

   > simplify(");

   

   > N1:=sqrt(2*k^(3/4)*mu^(3/4)/(hbar^(3/2)*sqrt(Pi)));

   

   
   

   Verification:
   

   > combine(simplify(int(psi1^2,x=-infinity..infinity)));

   

   
   

   Now, to show that this is a solution of Schrodinger's equation:
   

   > -hbar^2/(2*mu)*diff(psi1,x$2)+1/2*k*x^2*psi1;

   

   > simplify("/psi1);

   

   
   

   This is , as expected.
3. 
   

   > 1/sqrt(2*mu)*(-I*hbar*diff(psi1,x)-I*omega0*mu*x*psi1);

   

   > simplify(");

   

   
   

   Give or take normalization, this is .
4. 
   

   > 1/sqrt(2*mu)*(-I*hbar*diff(psi0(x),x)-I*omega0*mu*x*psi0(x));

   

   
   

   There are no harmonic oscillator wavefunctions with energies lower than .