Particle in a 1D-Box

Expectation values
<> =

Normalization

The normalized wave function for a one dimensional particle in the box is given by:

Some needed substitutions

Check the normalization

Averages

Calculate the average position, <    >

The particle on the average wil be found in the middle of the box

Calculate the average of the square of the position ,  <>

Now we calculate the average momentum, < >

The average momentum is zero.  This makes sense since standing waves are a combination of a wave moving to the right and a wave moving to the left

Finally we calculate the average square of the momentum, <  >

Variance and Standard Deviation

With the previous calculations we can find the variances

From the variances we can find the standard deviations

Uncertainty Principle

Now we can check the Uncertainty Principle

Rewriting the final result and its minimum value, n = 1.

the values is greater than 0.5000 hb.

Therefore, the Schrodinger Equation includes the Uncertainty Principal