Hydrogen Energy Levels

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Abstract:

This paper expands on "Ground state of the Hydrogen Atom"[1] where we use a fixed grid and fixed timesteps, similar to frames in a movie, to model the Hydrogen molecule. Electrons and protons exert no force on each other if they are within a Bohr radius (53 picometers) of each other. The assumption that there is no attraction between a proton and an electron when the electron is inside a radius of 53 picometers around the proton leads to an understanding of the forces that bind molecules together. 

In addition to binding forces, any interpretation of quantum mechanics must predict the discrete photon emissions one observes from hydrogen molecules.  Specifically, the  Lyman, Balmer and Paschen series of emissions are shown as changes in the local energy environment that may or may not cause an electron to emit a photon as it passes from one level to another.  All hydrogen energy levels are derived in this manner. Fine and Hyperfine detail are shown for the 10.2 and the 12.09 evolt levels.  Lamb Shift detail is shown for the red (1.89 evolt) photon.

Photon Emission

Here we define the photon as neither a particle nor a wave, but rather as a massless very skinny, very long string of energy.  The relationship between a photons energy and its length is controlled by a relative of  Plancks constant ħ, where: length = ħ / energy (length of the photon is usually in nanometers and the energy of the photon is in evolts).  Low energy photons like heat are very long, high energy photons like ultraviolet rays are very short.  Photons travel at the speed of light c = 3.0 x 10⁸ meters/second.  Photons can be curled around an electron or captured by an electron, resulting in the kinetic energy of the photon being transfered to the electron and often in the case of a bound electron, directly to the entire molecule.  Photons are emitted from an electron if the electron passes to a lower energy level where that photon can no longer be held by the electron and it pops off like a long string from a yoyo.

If the electron moves from one energy level to a lower energy level, and if it has enough spin energy to begin with, it will emit a photon equivalent in energy to the difference in the two energy level locations.

Figure 1. (a) illustrates the before and after of a free electron dropping into an empty proton shell (0-1 transition) and emitting a 10.2ev photon.  It also illustrates a free electron dropping or flying into an occupied proton shell (0-2 transition) resulting in a photon emission of 12.1 evolts.

Figure 1. (b) shows a bound electron entering the shell of a second bound electron resulting in a green photon emission of 2.56 evolts from one electron (1-3 transition) and a red 1.89 evolt photon emission from the other electron (1-2 transition).  Figure 1. (c) shows a heat photon of 0.97 evolts (2-3 transition) being emitted. 

Location of hydrogen energy levels

The total energy of a free electron is 511,000 Evolts. Proton shading represents different energy levels.

To find the energy level that an electron will find itself, follow these steps.

1. if the electron is in a shell, add 1;
2. add 1 for each additional electron that is in that same shell;
3. repeat 1. and 2. for each shell the electron is in.

The final number tells you the energy level the electron is in.

Figure 2 illustrates the first six energy levels and their construction.

Depending on the density of the gas, electrons have large or small chances of making specific transitions.  At room temperature and light pressure, 0-1, 1-2 or 0-2 transitions are seen in great abundance.  At higher pressures, you start to see higher transitions like 6-7, 7-8, 6-8 type transitions with no 0-1 transitions as large numbers of protons are overlapping each other.[3]

Fine and Hyperfine detail

Figure 3. (a) illustrates the fine structure for hydrogens first energy level (10.2 evolts).  4.5x10^-5 ev depends on the matching of the spin of the energy of the electron with the spin of the energy of the proton when the electron enters the proton. If the spin of the electron has to be flipped then less energy will be emitted in the photon.

Figure 3. (b) illustrates the fine structure for hydrogens second energy level (12.09 evolts).  7.3x10^-7 evolts is required to match the spin of the energy of the electron with the spin of the energy of the proton when the electron enters the proton at this energy level.

 
Figure 3. (c) shows hyperfine structure for the first energy level (10.2 evolts).  The hyperfine structure deals with the interaction of the magnetic orientation of the electron with the magnetic orientation of the proton. Depending on the relative orientation there is an energy difference of 5.9x10^-6 evolts.

Both the fine structure and the hyperfine structure depend on the energy level the electron drops into. Fine and Hyperfine spin are independent of each other.

In summary, for the first or 10.2 evolt energy level (1s1/2), fine and hyperfine structure are listed in the traditional wavelength, energy and frequency.

1. Fine structure 2.7cm, 4.5x10^-5ev, 10900mhz
2. Hyperfine structure 21cm, 5.9x10^-6ev, 1420mhz

Lamb shift

Figure 4. shows the slight difference in the 10.2 evolt energy level depending on its construction. Notice the three different ways a red photon can be produced. The slight difference in energies is required to flip the electron in different configurations.

Actual measured red photon emissions and intensities from the nist.gov database:[2]

 Int.   Precise energy level shift
 180  1.889422ev (6562.8518)
  90   1.889463ev (6562.7110)
  30   1.889459ev (6562.7248)

Without an external electric or magnetic field, the Zeeman and Stark effects are not felt.