Tidbits of Quantum Theory

Starting my spectroscopy notes review now. Note that there will be times when I don't really know what I am talking about...apologies for any inaccuracies and errors (feel free to point them out)! And, this being the notes from an introductory course, alot of the cool stuff (e.g. the derivations) will be absent. Sorry *sadly*.

Technically, I really should start with Planck and Einstein, who are the first to postulate that matter absorbs energy in a discrete way. But, that's physics. So...might as well start with the hydrogen atomic emission spectrum.

Most first-years probably have seen a hydrogen lamp, and its very, very pink light. Split that using a prism or a diffraction grating of some sort, and well, we don't get pink. Instead, we get discrete bands of light. Balmer is the first to derive an equation for predicting the wavelength of the bands (in the visible range, anyway):
(lambda) = (G)*[n₁²/(n₁² - 2²)]
where G is some experimental constant.

There are other series of bands as well, each named after the scientist who first observed it (e.g. Lyman for far UV, Ritz-Paschen for near IR, Brackett for mid-IR, and Pfund for far IR). In general, the equation that predicts the wavelength of the bands of the hydrogen spectrum is:

(nu-bar) = R_H(1/n₂² - 1/n₁²), where n₁² > n₂²

Aside: nu-bar is supposed to represent the symbol for "wave number", or "reciprocal centimeter" if one prefers it that way. Also, R_H is a constant, named after Rydberg (who discovered this generalized equation), and is equal to 109737.31568549... cm^-1.

This is all very well, but why is the spectrum this way?

Bohr came up with a radically new model (at the time) of the atom to explain the observation: he claimed that
1. an atom consists of a positively charge nucleus, and a number of negatively charged electrons revolve around this centre in a circular orbit such that charge is conserved.
2. only certain discrete orbits of electrons can occur; that is, angular momentum is quantized:
mvr_orbit = n*(h-bar)
where h-bar is Planck's constant divided by 2pi.
Isolating for v gives v = n(h-bar)/(mr)
Now, since the force of attraction between an electron and the hydrogen atom's nucleus is electrostatic in nature, and the electron stays in a circular orbit, we get:
F_{electrostatic} = F_centripetal
e²/[4(pi)(epsilon)₀r²] = m_electronv²/r
where (epsilon)₀ is the permittivity of free space.
Solving for radius r gives us an expression in terms of n (well, n² actually), which is approximately
r = (52.9 pm)n²

In other words, 52.9 pm is the smallest radius that the electron can have when orbitting around the nucleus. We call this the Bohr radius, a₀ for short.

If we calculate the energy of the hydrogen atom, we get:
E_{H atom} = -m_electrone⁴/[8(epsilon)₀²h²](1/n²)

3. Radiation emitted or absorbed by the hydrogen atom during electronic transitions from one orbit to another follows what is known as the Bohr Resonance Condition:
(Delta)E = h(nu)
which, if we combine with the expression for E_{H atom}, gives us something that looks alot like what Rydberg got.

Unfortunately, the Bohr model works for nothing else but the hydrogen atom, numerically speaking. So let's set this aside for now.

Two other famous physicists we ought to mention:

De Broglie - his idea is basically this: light (electromagnetic radiation) is a wave with particle-like properties. So why can't matter (particles, if you will) have wave-like properties, too? His equation: (lambda) = h/p, where p stands for momentum. The nice thing about this is that the wave-like properties of particles can justify the fixed orbits required by the Bohr model: an orbit is stable only if the circumference of the orbit is an integer multiple of the particle's wavelength, or else destructive interference will occur and the orbit won't be stable.

Heisenberg - famous for his "Uncertainty Principle", which claims that making exact measurements of both the position and momentum of a particle is impossible, since we require light to make the measurements, and light itself interacts with and changes the behaviour of the particle under observation. More quantitatively speaking,
[(Delta)p]*[(Delta)x] >= (h-bar)/(2*pi)

where the "Delta" stands for "uncertainty" and not "difference".

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It took surprisingly long (more than one day) to type this up semi-coherently...please point out any errors! And...I'll introduce Schrodinger next time.