In a previous post, we showed the Bohr atom as represented at the Georgia State University Physics Department’s website in the following way:

Is it possible to derive the total energy of electrons in the Bohr model from Schrodinger’s wavefunctions?

In general, we can calculate the total energy, , of the electron as the sum of its Coulombic potential, and kinetic energy, :

With, , we can calculate the average total energy, , using:

**So what is the form of ?**

The average potential energy, , is calculated using Coulomb’s law:

The average kinetic energy, , can be determined if the electrons travel around the nucleus in a circular fashion.*

From Newton’s second law: and for circular motion

while . Therefore, , so

This means that we have our average total energy for the circulating electron: , and

From here, it is a simple matter of applying the integral we learned earlier: to the and to derive the total energy of the electrons in these Bohr orbits:

**The Energy of the 1s electron**

**The Energy of the 2s electron**

Looking at the energy differences between these two energy levels, the can be calculated and compared to the energy levels observed in the hydrogen spectrum (Lymann series):

A wavelength of 121.566 nm corresponds to an energy of: . Not bad agreement! Bohr’s energy levels inversely depend upon the square of the principle quantum number, n. The model is derived from the consideration of a lone electron (of a hydrogen atom or a helium anion) and therefore didn’t work well to describe the spectral lines of the many-electron atoms. This was something that vexed him. As Max Tegmark and John Archibald Wheeler describe in their wonderful “100 Years of Quantum Mysteries”: “Back in Copenhagen, Bohr got a letter from Rutherford telling him he had to publish his results. Bohr wrote back that nobody would believe him unless he explained the spectra of all the elements. Rutherford replied: Bohr, you explain hydrogen and you explain helium, and everyone will believe all the rest.”

In a later post, at some point, it would be interesting to see if we could derive the *general* square dependence of the Bohr radius on the principle quantum number, n. This has been done for the specific case above, but for a general derivation, it would likely be necessary to determine how the principle quantum number arises from the wave equation. I have never taken a formal quantum course so need to do some reading before I can post about this. Also, it would be fun to try to derive the hydrogen wavefunctions too!

Overall, I think this wraps up my experiments with the overlap between quantum and multivariable calculus and physics for some time. I had a blast and I hope you got to share in the fun too!

*Note: I wonder if we could calculate another version of the kinetic energy using a deBroglie relationship: , treating the electron as a wave.

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