Prof. Axon's full lecture notes are available in PDF and PPT formats.
The main ideas for today are
You can find this material described in your textbook:
The goal for this week is to understand the spectra of light emitted or absorbed by molecules. We'll take it in two parts: last time, we examined the bonds between atoms which create stable molecules. Today, we'll investigate how multi-atomic structures can change their internal energy and emit or absorb light.
As you recall from Modern Physics I, Schroedinger's equation becomes for the quadratic potential
which can be solved by a wave function of the form
where the constants are (for the ground state)
Or, in terms of the frequency of oscillations,
For excited states, the energies turn out to be
Flashback Time! Reduced mass
Physicists just LOVE to simplify complicated situations. Usually, making a complex matter into a simple one involves approximations. However, in some specific cases, one can replace a complicated system with a simpler one which acts EXACTLY the same way.
One example involves certain two-body systems:
In each case, there is a similar system which involves the motion of just a single object:
It turns out that we can convert the two-body system EXACTLY to an equivalent one-body system if we replace the original quantities like so:
original reduced
---------------------------------------------------
mass m1 total mass M = (m1 + m2)
m1 * m2
mass m2 reduced mass μ = -------
m1 + m2
position r1 origin = 0
position r2 separation R = (r2 - r1)
---------------------------------------------------
End of flashback.
We can estimate some properties of a diatomic molecule just by looking carefully at the graph of potential energy as a function of nuclear separation.
We can also figure out properties if we know something about the spectrum of emitted radiation.
Calculate mu, then find the spring constant k
Well?
You see that the spectra of molecules can be used to identify atoms with different isotopes .... if the spectral resolution is high enough.
Alas: some books use the convention that ROTATIONAL angular momentum quantum numbers be denoted by the letter J, but others use the same old letter L that can stand for orbital angular momentum.
Warning: "h" in slide above should be h-bar = h/(2*pi)
It turns out that transitions between rotational states can only occur under certain conditions. This gives rise to a set of selection rules.
Warning: "h" in slide above should be h-bar = h/(2*pi)
Warning: "h" in slide above should be h-bar = h/(2*pi)
Warning: "h" in slide above should be h-bar = h/(2*pi)
Which transitions are most likely to be seen in some given sample of molecules?
Note that not all transitions occur in the figure above. Why? Again, there are certain selection rules which describe the most likely transitions. The result is a spectrum with a particular pattern ....
If we look very closely at the spectra, there are some complications.
Please see Prof. Axon's original notes in his lectures for the explanation of this phenomenon.