Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 1.17.
Here’s another example of calculating the uncertainty principle. We have a wave function defined as
The constant is determined by normalization in the usual way:
The expectation value of is from the symmetry of the wave function. The expectation value of is
[We can’t calculate in this case, because we know the value of only at one specific time (), so we don’t have enough information to calculate its derivative.]
The remaining statistics are (the integrals are all just integrals of polynomials, so nothing complicated):
Thus the uncertainty principle is satisfied in this case.