Guilt. With Pictures.

Yet another week has gone by without a review post. So here's an attempt at compensation. Also known as "alot of pictures".

Note: The fourth-from-the-last image is from Engel's Quantum Chemistry and Spectroscopy. The others are from my notes (and Paint. Gotta have Paint). I realize that these are truly pitiful compared with the gorgeous images that my readers create, but I am posting them anyway.








Oh, and a bit of housekeeping today: blogroll is updated now. Let's put it this way: it is now a larger, but probably still proper, subset of the blogs that I do actually read.

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Delay

I regret to say that the introductory spectroscopy review will be delayed due to something called that-stupid-report-I-have-to-write-for-that-stupid-course.

Posts will resume as soon as possible...

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More Quantum Mechanics: Schrödinger's Sans Cat, and Some Other Stuff.

Thanks for the comments from last time! Now let me try out some of these newfangled things so the post can display Greek characters properly...

Recall from the previous post that there were three things of note that was "new" about the nature of atoms:

• Energy quantization (ΔE = hν)
  
• Wave-particle duality (in particular, that matter can behave as waves)
  
• Heisenberg's Uncertainty Principle.
  

Combining the three gives rise to a shiny new theory: quantum mechanics.

In particular, Schrödinger describe the behaviour of an electron (a particle) in an atom using a wave function. It is similar to the Bohr model in that the electron described must be in discrete states (i.e. states where the electron waves would not destructively interfere with itself, kind of like "nλ = 2&pi r" except alot more complicated).

(Notice that what we mean by "the behaviour of an electron" is rather vague at the moment. It'll kind of clear up itself later.)

If we assume that the energy of the electron in a particular scenario is not time-dependent, we can skip alot of math about Hilbert spaces (and other cool things), ending up at the Time Independent Schrödinger Equation (TISE for short):
HΨ = EΨ
where H really needs a hat and is an operator (called a "Hamiltonian"), and E (as usual) stands for the energy (of whatever system, e.g. electron in an atom, we are describing), and is constant.

A tiny, miniscule bit on the math:
In the equation HΨ = EΨ, H is an operator, i.e. it acts on a function (namely Ψ). It is a function whose input and output are functions themselves, if you will. For example, the symbol "d/dx" is an operator that can act on a differentiable function f(x). Yes, this means that the Ψ's do not "cancel".
Ψ is a function that satisfies the equation: i.e. applying a specified operation on it will give the function itself, multiplied by some constant value. A function that satisfies this is called an "eigenfunction", while the constant is called an "eigenvalue".

Now we have a cute (but powerful) equation. And since we started this mess because of studying hydrogen emission spectra, let's apply it to the hydrogen atom.

Since we are interested in the energy of the electron, it makes sense to define the Hamiltonian H as an operation that gives the total energy (kinetic and electrostatic energy) of the electron given location (say in spherical coordinates, (r, &theta, &phi)). For the hydrogen atom,

(Kinetic energy) K = -[(h-bar)/&mu]&Delta
(Electrostatic potential energy) U = -e²/(4&pi&epsilon₀r)

where &Delta is the Laplacian operator and e is the fundamental charge.
(I actually don't remember what &mu is. Guess: reduced mass of the system...)

So altogether, H = K + U.

Now, we seek functions &Psi that satisfies the TISE. There are lots of them, by the way. Each one describes the different allowed quantized states of the electron, and each with its own associated energy E. In general:

&Psi_{n, l, ml}(r, &theta,&phi) = R_n,l(r)Y_{l, ml}(&theta, &phi)

Where we call the part dependent only on the radius r the "radial" part, and the part dependent on angles &theta, &phi the "angular" part.

The 3 little subscripts by the &Psi denote the 3 numbers associated with each &Psi:

n - principle quantum number
(n = 1, 2, 3, ...)
This number exclusively determines the energy of the electron in the hydrogen atom:
E_n = -R_H/n², where R_H is Rydberg's constant in your unit of choice (= 2.179824*10^-18 J)
Orbitals (see below) with the same value of n has the same energy, and are said to be in the same "shell" (remember, we are still at the hydrogen atom).

l - orbital (azimuthal) quantum number
(l = 0, 1, ..., n-1)
This number determines the shape of the orbital. If one recalls the s, p, d, f, (g, h, ...) orbitals, well, l = 0 describes the s orbital, l = 1 describes the p orbital, etc.
Orbitals with the same value of l are said to be in the same "subshell".

m_l - magnetic quantum number
(m_l = -l, ..., l)
This number describes the orientation and energy in a magnetic field of an orbital.

(Yes, there's a fourth quantum number, for describing the electron's spin, but that's not part of the function.)

The thing about &Psi is that it doesn't give an explicit location of where the electron in the hydrogen atom is going to be at or anything specific liek that, because of Heisenberg's uncertainty principle. Instead, &Psi*&Psi (not to be confused with the illustrious blogger of the same name) gives the probability that the electron will be at a given location.

Now, the R (radial) part of &Psi gives the "amplitude" of the function at a given distance r (away from the nucleus, or the origin). Often, it is represented by graphing the x-axis, in units of r/a₀. The y-axis is usually arbitrary. The interesting thing is, the wave has positive and negative parts (peaks and troughs, if you will), and thus points where the wave crosses the x-axis. These points are what we call radial nodes - the electron is never present at those points.

The Y (angular) part gives something that's more important for chemists when we study bonding - the shape of the area where the electron is likely going to be. This part has signs, related to amplitude, too.

Emphasis: the signs of the function has nothing to do with charge!

This is all very well, except there's alot more elements on the periodic table than just hydrogen. Like helium. But - there are two electrons, and a nucleus! It turns out that we can't solve a system like that exactly, because there are too many particles around (what people call the "three-body problem"). So, if we want to describe other atoms, we have to resort to approximations. Lots and lots of approximations.

Qualitatively speaking, we use the solutions &Psi for the hydrogen atom for all the other atoms, just as a model ("hydrogen-like atomic orbitals", if you will). For multielectron systems, however, the energies of the atomic orbitals will depend on l as well, not just on n, due to the repulsion that the electrons in the atom exert on each other (called the "shielding effect"). This gives rise to the differences in properties, as well as the periodic nature, of the elements on the periodic table.

Let me just mention one more thing: electron spin and Pauli's Exclusion Principle.
It turns out that particles (e.g. electrons) possess an instrinsic angular momentum (not associated with the revolution about another particle, but only with the particle itself). It is described by what we call the spin quantum number, m_s. For electrons, this is either +1/2 or -1/2. According to Pauli, each electron in an atom has a unique set of quantum numbers (n, l, m_l, m_s), which means that no matter how "large" or "small" an orbital may seem on a graph, only two electrons can be "in" that orbital.

Next up - molecular orbitals, maybe even with pictures (MOs without pictures make no sense). I really Should mention term symbols (and Hund's Rules), but I don't really understand them at all, so there's no point in blurbing about them. Maybe when I actually take a proper quantum chemistry course...

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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.

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