2. Bohr Model¶

This note explains another failure of classical physics, and derives the energy levels of electrons in the hydrogen atom.

Brief History¶

In classical radiation theory, Thomson and Rutherford found that electrons must revolve around the nucleus to avoid falling into it.

Maxwell's EM theory supposed that if an electron revolves with frequency $f$, then it must emit radiation with frequency $f$.

However! As energy is radiated, electrons should fall into a lighter orbit.

In 1913, Bohr "solves" the mystery and publishes "On the Constitution of Atoms and Molecules".

Bohr proposes two things: 1. Classical radiation theory does not apply to electrons 2. Only certain orbits are stable, and are separated by $hf = \Delta E$.

Derivation of Bohr Model¶

In this derivation, we will discoverthe allowed radii that an electron can be in, as well as the energy at a specific level, and how to find the energy emitted upon a change in level.

We start with three presumptions: 1. Circular orbit 2. Only certain orbits are stable 3. Jumping from one orbit to another absors or emits $hf$.

Begin with circular motion:

\[ a_c = \frac{v^2}{R} \]

We have an attractive electrostatic force with two charges of $e$.

\[ \frac{F_c}{m} = \frac{ke^2}{mR^2} \]

I want to get some form of angular momentum, so I multiply both sides by $m^2R^3$.

\[ \frac{v^2}{R} = \frac{ke^2}{mR^2} \]

\[ m^2v^2R^2 = mke^2R \]

\[ L^2 = mke^2R \]

Let's solve for $R$:

\[ R = \frac{L^2}{mke^2} \]

Now we use the idea that angular momentum is quantized:

\[ mvr = n\hbar \]

Which can be justified using the idea of de Broglie that all matter is a wave with

\[ \lambda = \frac{h}{p} \]

\[ f = \frac{E}{h} \]

Stable states are thus standing waves which satisfy periodic boundary conditions. Thus,

\[ 2\pi R = n\lambda = \frac{nh}{mv} \]

And $mvR = n\hbar = L$. Substitute this into our equation for $R$.

\[ R_n = \frac{n^2\hbar^2}{mke^2} \]

This shows that radii is quantized.

Now we will look at the situation in terms of energy.

\[ U = qV = -\frac{ke^2}{R} \]

\[ K = \frac{1}{2}mv^2 \]

Thus, the total energy is

\[ E = \frac{1}{2}mv^2 - \frac{ke^2}{R} \]

Note that we originally stated

\[ m^2v^2R^2 = mke^2R \]

Let's solve for $mv^2$:

\[ mv^2 = \frac{ke^2}{R} \]

Substitute this into our energy equation

$$ E = \frac{1}{2}\frac{ke^2}{R} - \frac{ke^2}{R} = -\frac{ke^2}{2R} $$ Let's substitute our quantized radius equation to the energy equation:

\[ E = -\frac{ke^2}{2}\frac{mke^2}{n^2\hbar^2} \]

\[ E_n = -\frac{1}{n^2} \left(\frac{mk^2e^4}{2\hbar^2}\right) \]

Maybe it could be a bit better to separate the charges of the nucleus and the electron charge, then it would be

\[ E_n = -\frac{1}{n^2}\left(\frac{mk^2q_N^2e^2}{2\hbar^2}\right) \]

Showing that energy is discrete as well.

We can describe energy changes by finding the difference in energy orbits:

\[ \Delta E = \frac{mk^2q_N^2e^2}{2\hbar^2}\left(\frac{1}{n_i^2} - \frac{1}{n_f^2}\right) \]

Emission will happen if the electron falls to a lower orbit, and absorption will happen if the elctron is excited to a higher orbit.

What is Missing in Bohr's Model¶

Several things: 1. Failed to predict intensity of spectral lines 2. Limited success for multi-electron atoms 3. Failed to produce time dynamics 4. Ignores wave nature/penomenon 5. No general quantization scheme

Therefore, we need Quantum Mechanics as developed by Heisenberg, Schrodinger, Dirac, ...