Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.12.
Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 7.4, Exercise 7.4.2.
In the study of the harmonic oscillator, we can express and in terms of the raising and lowering operators:
We now have
The reason this is zero is that, as we saw when working out the normalization of the stationary states,
and since the wave functions are orthogonal, we get
for the same reason.
For the mean squares:
In going from the first to the second line, we’ve thrown out terms where we integrate two orthogonal functions. For example,
We have used the relations above and the fact that is normalized to get the third line.
The uncertainty principle then becomes
and the kinetic energy is
which is half the total energy, as it should be.