Classical Bohr Atom

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Order of Magnitude Interaction of Radiation and Matter

It turns out that you can tell a cute little story about the energy levels in hydrogen-like atoms using a simple Bohr model of the hydrogen atom and classical physics. You just need a couple small doses of quantum mechanics in the beginning to get you started on the right track.

Energy Levels

  • Electronic Transitions:
Sketch of Rydberg- and Lyman- for hydrogen

We’ll start with the Bohr atom, and follow a classical derivation/estimation path. The only quantum mechanics that we need to inject is that angular momentum comes in units of . So let’s begin by quantizing the angular momentum associated with the electron’s orbit around the nucleus. We’ll define the radius of the electron’s orbit as , and the velocity that the electron travels at will be . This gives us the following expression for the angular momentum for units of :

If we balance the force required to keep the in a circular orbit with the electric force:

A Rydberg is the energy required to ionize an H atom from the ground state. It is . We can estimate it by integrating the electric force from to , but in reality, there is another factor of 2:

  • Fine Structure:

Fine Structure comes from the interaction of the magnetic moment of the with the a caused by the Lorentz-transformed Coulomb field of the proton (generated by the ’s motion). The energy of a dipole interaction is , so we’d expect that:

To estimate B: , and ( is the fine structure constant):

Estimating : The of a dipole goes as , so , where is the classical radius. Estimating that the rest mass energy of the should be about the electrostatic potential energy left in the , :

We can estimate by reverting to Maxwell’s equation:

Again, we estimate that , and because the electron spin is quantized, . After some algebra, we get that:

(Bohr magneton for )

(Bohr magneton for nucleus) Getting back to the energy:

  • Hyperfine Structure:

Instead of using the Bohr magneton, we use the intrinsic magnetic moment (spin) of the nucleus:

Thus:

Note that Fine and Hyperfine are magnetic dipole transitions. level transitions are electric dipole transitions. Magnetic dipole transitions are generally weaker.

  • Vibrational Transitions in Molecules:

Our general technique with vibrational transitions is to model them as harmonic oscillators. Thus, they should have the characteristic harmonic energy series:

For a harmonic oscillator, . We estimate that since the force for a spring is , and that force should be about the Coulomb force on ’s. If we say that atoms stretch with respect to each other about a Bohr radius:

where A is the atomic mass # of our atoms.

  • Rotational Transitions in Molecules:

The thing to remember is that angular momentum comes in units of .