Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Chapter 9, Exercises 9.4.1 – 9.4.2.

Here we’ll look at a couple of calculations relevant to the application of the uncertainty principle to the hydrogen atom. When calculating uncertainties, we need to find the average values of various quantities. First, we’ll look at an average in the case of the harmonic oscillator.

The harmonic oscillator eigenstates are

where is the th Hermite polynomial. For we have

so

For this state, we can calculate the average

where we evaluated the Gaussian integral in the second line.

We can compare this to as follows:

Thus and have the same order of magnitude, although they are not equal.

In three dimensions, we consider the ground state of hydrogen

where is the Bohr radius

with and being the mass and charge of the electron. The wave function is normalized as we can see by doing the integral (in 3 dimensions):

We can use the formula (given in Shankar’s Appendix 2)

We get

as required.

For a spherically symmetric wave function centred at ,

with identical relations for and . Since

Thus

We can also find

Thus both and are of the same order of magnitude as .

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