# Difference between revisions of "Classical Bohr Atom"

Line 83: | Line 83: | ||

Getting back to the energy: | Getting back to the energy: | ||

$$\begin{aligned}\Delta E&\sim{e\hbar\over2m_ec}{Ze\over a_0^2}\alpha\\ | $$\begin{aligned}\Delta E&\sim{e\hbar\over2m_ec}{Ze\over a_0^2}\alpha\\ | ||

− | &\sim Z^ | + | &\sim Z^3\alpha^2\cdot Ryd\\ \end{aligned}$$ |

\item Hyperfine Structure:\par | \item Hyperfine Structure:\par |

## Revision as of 10:53, 26 September 2017

### Short Topical Videos

- Radiation and Matter, to Order of Magnitude (Aaron Parsons)
- Bohr Hydrogen Model 1: Radius (Quantum Chemistry)
- Bohr Hydrogen Model 2: Energy (Quantum Chemistry)
- Rotational Spectroscopy Example (Quantum Chemistry)

### Reference Material

## 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:

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 .