The Hydrogen Atom

First, a look at what inspired the planetary type of model of the atom: In 1911-1913, Ernest Rutherford and collaborators came up with one of the greatest combination theoretical and experimental breakthroughs in history.

They bombarded a gold foil with alpha particles, then detected where the a's came out. Most of them went almost straight through, but a small percentage of them came approximately straight back. Rutherford said "It was almost as incredible as if you had fired a 15-inch shell at a piece of tissue paper and it came back and hit you." You see, the prevailing theory was that the positive charge was spread out within the entire volume of the atom. The positive part (now called the nucleus) was known to have almost all of the mass, since electrons, the negatives, have very little mass. To make a long story short, Rutherford's analysis of the data showed that the positive part is concentrated in a tiny portion of the atom. So it was reasonable to assume that electrons went around it the way planets go around the sun.

Another series of experiments that had been going on for many years was finding the wavelengths emitted by glowing gases. It was found that they didn't emit a continuous smear of wavelengths, but a discrete set. Like the bugle compared to a slide trombone-- the bugle has a discrete set of notes.

Niels Bohr was looking for a planetary model for the H atom, and in 1913 he found that if he assumed that the electron could only have angular momentum = a multiple of h/2p (h is Planck's constant, 6.63x10^-34 J× s), then a circular orbit model gave results which agreed with the wavelengths in the spectrum of a glowing hydrogen gas. Angular momentum is mvr, so mvr=nh/2p (n=1,2,3,…)

This made no sense. He probably figured that since h has units of angular momentum, it would be reasonable to look for a model in which the electron angular momentum is related to h in a simple way, then he played around with the data until he found the above result. This was incomplete as a theory of physics, just as Kepler's laws on the solar system were; a sound theory of physics is not just an equation that works. (Newton laws later made sense out of Kepler's laws.)

It was the theory of Louis de Broglie, about 10 years later, that finally made sense of the Bohr model. De Broglie figured that if light, the wave, acts like a particle, why doesn't the electron, a particle, act like a wave?

The photon: E = hf = hc/l , so mc² = hc/l , or l = h/mc.
The electron: l = h/mv, by analogy. (By the way, v is proportional to the square root of the kinetic energy, and it is common to measure energy in electron-volts, eV. If you do the numbers you can show that the electron’s wavelength
is l = 1.23/(E)^1/2 nm∙(eV)^1/2. In other words, l in nanometers is 1.23 divided by the square root of the kinetic energy in electron-volts.)

Then he assumed that the permissible orbits were those in which the electron is a standing wave, so the circumference is a multiple of the wavelength. The sketch below shows an example, with the circumference = 4l .

2p r = nh/mv, so

1. mvr = nh/2p.

The electron, charge -e, is attracted to the nucleus (a proton), charge e, with force ke²/r², where k=1/4pe_o = 8.99 x 10⁹ Nm²/C². With circular motion we have

2. ke²/r² = mv²/r (now cancel one factor of r)

Solve equation 1 for v and plug into equation 2 and we have

3. ke²/r = n²h²/(4p ²mr²).

Solve equation 3 for r and we find

4. r = n²h²/(4p ²mke²).

The total energy E is kinetic plus electrical potential energy. The latter is
-ke²/r, where the zero of PE is at r = ¥, and after canceling an r in equation 2, then dividing both sides by 2, we see that KE is ke²/2r. Adding this to the PE, we find

5. E = -ke²/2r.

Combining this with equation 4, we have

6. E_n = -2p ²k²e⁴m/n²h².

(Everything is constant except n, so it is known as E_n, but it is the same as the previous E.)

Plug in the numbers (in SI units,
k = 8.99x10⁹, e=1.60x10^-19, m=9.11x10^-31, h=6.63x10^-34), then convert to electron-volts (1 eV=1.6x10^-19J) and we have

E_n = -13.6/n² eV

In a normal H atom, the electron is in the lowest energy state (the "ground" state),
n = 1. You can zap it with light, heat, electric current or whatever and get the electron excited up to a higher level. Then it will drop down, maybe in one jump or in stages, emitting a photon each step of the way. For example if it is excited to n=3, it could jump down from 3 to 1 and emit one photon, or it could jump down to 2 and then from 2 to 1, emitting a photon for each jump.

Take the 3 to 2 jump, for example. The loss of energy of the electron, E₃ - E₂, is the energy of the emitted photon, hc/l . It is useful to know that
hc is 1240 eV× nm. So -13.6(1/9 - 1/4) = 1240/l . Solving for l , get
l = 656 nm. This is visible, a red light. (It turns out that the photon from H is visible if and only if the electron jumps down to n = 2. This is an accident of nature due to the range of wavelengths that human eyes can detect.)

When you jump off a chair, you emit gravity waves, but they are not detectable. Any accelerating mass emits gravity waves just as any accelerating charge emits electromagnetic waves. Einstein’s theory of general relativity predicts gravity waves, and if you can detect them, you might get a Nobel Prize.

Now take a quantum jump to a main physics page:

mechanics
fluids, heat, electricity and magnetism
vibrations and waves
quantum
index

Comments, questions: fredrick.gram at tri-c.edu, but remove “at” and use @.