The given qualitative explanation of the ground state of the harmonic oscillator in terms of the uncertainty principle is questionable. In particular, position, linear momentum, potential energy, and kinetic energy are uncertain for the ground state. This note gives a solid argument, but it uses some advanced ideas discussed in chapter 4.4 and 4.5.3.
As explained more fully in chapter 4.4, the
expectation value
of the kinetic energy is defined as
the average value expected for kinetic energy measurements.
Similarly, the expectation value of the potential energy is defined as
the average value expected for potential energy measurements.
From the precise form of expectation values in quantum mechanics, it
follows that total energy must be the sum of the kinetic and potential
energy expectation values. For the harmonic oscillator ground state,
that gives
Now any value of
Of course, a similar expression holds for
Now the first square root above is a measure of the uncertainty in
So the minimum value of the final two terms in the expression
(1) for the ground state energy is the complete ground state energy.
Therefore, in order that the right hand side in (1) does not exceed
the left hand side, the first two terms must be zero. So the average
particle momentum and position are both zero. In addition, for the
estimates of the final two terms, equalities are needed, not
inequalities. That means that