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 can be written as equal to the average value
plus a deviation from that average .
Then
Of course, a similar expression holds for , so the ground state energy is
Now the first square root above is a measure of the uncertainty in
. If is always zero, then is always
its average value, without any uncertainty. Similarly, the second
square root above is a measure of the uncertainty in . The
Heisenberg uncertainty principle can be made quantitative as, chapter
4.5.3,
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 must be . That then means that the expectation kinetic energy must be the expectation potential energy. And the two must be the very minimum allowed by the Heisenberg relation; otherwise there is still that inequality.