It is well known that incorrect information is often propagated over the internet. Physicists are not immune to this, and I would like to encourage my colleagues to take the time to check the information they post.

As an example, an internet search for the hydrogenic wave functions turned up many variants for the n =3, l =1 radial wave function. Two of these are

R31r=4816Za03/26Zra0Z2r2a02eZr/3a0,
(1)
R31r=423Z3a03/2Zra01Zr6a0eZr/3a0,
(2)

where the variables have their common meaning. What is interesting is that the second of these is exactly 3 times larger than the first. Hence, (at least) one of these is incorrect.

While it is possible to check the normalization integral by hand, it is even more straightforward to check the normalization numerically, using

0r2R312dr=1n=0Nn·Δr2R31n·Δr2Δr.
(3)

The numerical calculation can be done using virtually any programming language, or even a spreadsheet, and does not require a high level of precision. Equation (2) is found to be too large by a factor of 3. It is difficult to identify the original source of this particular error, since posted materials often lack citations, however the same error can be found in print form from at least 50 years ago. It is likely a simple typo that has continued to be propagated forward.

Unfortunately, this is not the only re-posted error I have run across. Numerical techniques provide a simple and expedient way to check correctness. In addition to normalization, simple numerical techniques can be useful to check, or at least spot check, summations, solutions to equations including differential equations, and many other mathematical results. Please check results before posting them.