Quantum Mechanics Notes

The z component in the two one one mode changes as a function of z, unlike the one one zero mode where it's constant because it's cosine of zero ( $\cos(0)$ ). In the two one one mode, arrows flip from front to back.
In the two one one mode with $\cos(\omega t)$ , all arrows flip due to this factor. The diagram shows the state at t=0.
The two one one mode has a sine term $\sin(2 \pi x / l)$ , indicating it's the second harmonic in the x direction. There is a zero in the middle, with arrows getting larger, then zero, then going away from us.
There's also a y dependence $\sin(\pi y / l)$ , which is zero at y=0 and increases to y=l, representing the first harmonic in the y direction.

Experiments by Balmer and Lyman revealed that ionized atoms in a gas discharge tube emit light at specific, quantized frequencies when high voltage is applied.
This was early evidence suggesting something unique occurring within the atom because light from sources like the sun provides a continuous spectrum.

The quantum model of the atom reveals that electron energy levels are quantized, influenced by Planck and Einstein's work ( $E = nhf$ ) with electromagnetic waves and photons.
Energy levels of orbits are distance-dependent, following a one over r squared relationship but the textbook’s representation is misleading.
When an electron transitions between energy levels, it emits a photon. Although energy conservation allows multiple photons, single-photon emission is more consistent with observed clear frequencies.
This is similar to Einstein's photoelectric effect, where a photon interacts with one electron:
$E_{\text{photon}} - \text{Work Function} = \text{Kinetic Energy of Ejected Electron}$
The process typically involves a one-to-one interaction between a photon and an electron.

Spectroscopy indicates that when an electron changes energy levels, a single photon is emitted, maintaining a one-to-one process.
This contrasts with scenarios where multiple microwave photons could theoretically be emitted but aren't observed.
Photons are localized, limiting their interaction to single electrons, unlike the wave model where electromagnetic entities interact with numerous electrons, as seen in refraction.
Scientists in the 1880s found distinctive frequencies of light are emitted. An empirical equation was developed relating emitted light and quantum state:
$\frac{1}{\lambda{\text{emitted}}} = R \left( \frac{1}{n{\text{initial}}^2} - \frac{1}{n_{\text{final}}^2} \right)$
Where R is the Rydberg constant, $n{\text{initial}}$ and $n{\text{final}}$ are initial and final energy levels.
Balmer specifically studied cases where $n_{\text{final}} = 2$ .

Balmer's equation, like Snell's Law in optics ( $n1 \sin\theta1 = n2 \sin\theta2$ ), was initially empirical without a theoretical basis.
Kepler's law (period squared proportional to radius cubed) also fit planetary orbits empirically before a theoretical explanation was available.
Data collection often leads to empirical equations, with theoretical understanding potentially following.

Faraday's electrolysis experiment (m proportional to q, circa 1830s), then spectral lines experiment (1880s), then e/m experiment (1897).
Faraday's electrolysis experiment with sodium chloride demonstrated that mass (m) is proportional to charge (q).

The Millikan oil drop experiment aimed to determine the fundamental unit of charge.
By observing the motion of charged oil drops under a microscope, Millikan deduced the charge without directly observing electrons (too small to see with visible light).
Without an electric field, the oil drop reaches terminal velocity where mass $m$ times gravity $g$ equals the friction force:
$mg = cv_{\text{without}}$
With an electric field, we have:
$-mg - cv_{\text{with}} + Eq = 0$
Where $c$ is a constant.
From these equations, the charge $q$ can be calculated:
$q = \frac{c(v{\text{without}} + v{\text{with}})}{E}$
The total charge $q$ equals $n$ times $Q$ , where $Q$ is the charge of a single electron.
By analyzing multiple drops, the ratio of total charges relates to the ratio of speeds, revealing quantized charge units:
$\frac{n1}{n2} = \frac{v{\text{without,1}} + v{\text{with,1}}}{v{\text{without,2}} + v{\text{with,2}}}$
Experimental data confirms these ratios, allowing determination of the fundamental charge.

Millikan found the charge of an electron to be $1.6 \times 10^{-19}$ coulombs.
Combining this with Thomson's e/m experiment, with the value of $\frac{e}{m}$ , the mass of the electron can be calculated as approximately $9.1 \times 10^{-31}$ kilograms.

Rutherford's experiment involved firing alpha particles (helium nuclei) at a thin gold foil to understand charge distribution in a neutral atom.
Alpha particles are two protons and two neutrons.
Most alpha particles passed straight through or were slightly deflected, but some bounced back at high angles.
This suggested that the positive charge occupies only a small area compared to the size of the atom.
This experiment reveals charge distribution.

Rutherford's results indicate positive charge is heavily concentrated, with the nucleus size about $10^{-15}$ meters compared to the atom size of $10^{-10}$ meters.
This allows imagining electrons in orbit around the nucleus.
Inspired by Planck and Einstein, Bohr proposed quantized energy levels, leading to his quantum model explaining spectral lines.
Bohr's model introduces equations for the allowed values of the energy based on the quantum number. The energy of the orbit is a function of $n$ .