Input := 

<<Quantum`QuickReIm`
<<Algebra`Trigonometry`
<<Graphics`Animation`

Definition of constants

Input := 

m=1;
k=1;
hbar=1;
beta=Sqrt[m*k/hbar^2];
y=Sqrt[beta]x;

Wave functions and energy of a harmonic oscilator

Input := 

psi[n_,x_] := Sqrt[(1/(2^n*n!))*Sqrt[beta/Pi]] *
				HermiteH[n, y] * Exp[- y^2 /2]
e[n_]:=(n + 1/2)*hbar*Sqrt[k/m]

Now we can choose two quantum states, n1 and n2:

Input := 

n1=0;
n2=1;

Definition of a time-dependent wave function

Input := 

PSI[x_,t_] := a(Exp[-I e[n1] t/hbar] psi[n1,x] + 
				Exp[-I e[n2] t/hbar] psi[n2,x])

Absolute square root of the time-dependent wave function

Input := 

AbsSqrPsi[x_,t_]=Conjugate[PSI[x,t]]*PSI[x,t]\
					//ComplexToTrig//TrigReduce//Simplify

Output =

 2         2
a  (1 + 2 x  + 2 Sqrt[2] x Cos[t])
----------------------------------
             2
            x
           E   Sqrt[Pi]

Normalization

Input := 

norm=Integrate[AbsSqrPsi[x,t], {x, -Infinity, Infinity}]

General::intinit: 
   Loading integration packages -- please wait.

Output =

   2
2 a

Input := 

a=a/.Solve[norm == 1, a]

Output =

      1         1
{-(-------), -------}
   Sqrt[2]   Sqrt[2]

Input := 

a=a[[2]]

Output =

   1
-------
Sqrt[2]

Input := 

Plot[AbsSqrPsi[x,2Pi], {x,-3,3}]

Output =

-Graphics-

Animation:

The next command will generate a set of plots. To do actual animation select graphics and go to the Graph menu and choose Animate Selected Graphics. To stop animation use control buttons which will appear at the bottom of the window.
Note: It may take some time before Mathematica will respond to pressing any of these buttons!

Input := 

Animate[Plot[AbsSqrPsi[x,t], {x,-3,3}, Axes->False],
						 {t,0,2 Pi,2 Pi/20}]

For the HTML version we already did perform the animation. It is shown first.

Here are the separate frames.