Physics Friday 58

In a previous Friday post, I demonstrated one method of determining the energy eigenvalues for the one-dimensional quantum harmonic oscillator. In particular, we took the (time-independent) Schrödinger equation , and by defining dimensionless parameters , , and then attempting a solution of the form , we derived the differential equation , where the series solution has recursion relation . Then, the requirement that the wavefunction be normalizable requires that the series solution terminate after finitely many terms, requiring that , for n a non-negative integer.

Now, let us consider the ground state case: n=0. Then , and the series recursion relation becomes , so that , and the terminating solution is that u(ξ) is a constant (a0). Then , and so
.
Normalizing this wavefunction,

and
.
So the ground state wavefunction of the 1-dimensional quantum harmonic oscillator is a gaussian.

Now, let us examine the uncertainties of position and momentum. The Heisenberg Uncertainty Principle tells us that (see here). For our wavefunction, we see

and

as in both cases the integrand is odd (see here).
Thus

Using and , we see that
, and so
.
Similarly,
.
Now,
,
so

Taking the product,
,
which is exactly the lower limit allowed by the uncertainty principle. Thus, the non-zero value of the ground-state energy (zero-point energy) can be seen as being a result of the Heisenberg Uncertainty Principle.

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